#  Lattice paths and branched continued fractions
III. Generalizations of the
Laguerre, rook and Lah polynomials

Bishal Deb1,2, Alexander Dyachenko3, Mathias Pétréolle1 and Alan D. Sokal1,4
1Department of Mathematics, University College London, London WC1E 6BT, UK
2Sorbonne Université and Université Paris Cité, CNRS, Laboratoire de
Probabilités,
Statistique et Modélisation, 75005 Paris, France
3Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow
125047, Russia
4Department of Physics, New York University, New York, NY 10003, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(November 28, 2023)

###### Abstract

We introduce a triangular array $\widehat{\sf L}^{(\alpha)}$ of 5-variable
homogeneous polynomials that enumerate Laguerre digraphs (digraphs in which
each vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weights
for peaks, valleys, double ascents, double descents, and loops. These
polynomials generalize the classical Laguerre polynomials as well as the rook
and Lah polynomials. We show that this triangular array is totally positive
and that the sequence of its row-generating polynomials is Hankel-totally
positive, under suitable restrictions on the values given to the
indeterminates. This implies, in particular, the coefficientwise Hankel-total
positivity of the monic unsigned univariate Laguerre polyomials. Our proof
uses the method of production matrices as applied to exponential Riordan
arrays. Our main technical lemma concerns the total positivity of a large
class of quadridiagonal production matrices; it generalizes the tridiagonal
comparison theorem. In some cases these polynomials are given by a branched
continued fraction. Our constructions are motivated in part by recurrences for
the multiple orthogonal polynomials associated to weights based on modified
Bessel functions of the first kind $I_{\alpha}$.

Key Words: Laguerre polynomial, rook polynomial, Lah polynomial, Laguerre
digraph, production matrix, quadridiagonal matrix, exponential Riordan array,
Hankel matrix, total positivity, Hankel-total positivity, continued fraction,
branched continued fraction, multiple orthogonal polynomial, modified Bessel
function.

Mathematics Subject Classification (MSC 2020) codes: 05A15 (Primary); 05A05,
05A19, 05A20, 05C30, 05C38, 15B48, 30B70, 33C10, 33C45, 42C05 (Secondary).

###### Contents

  1. 1 Introduction and statement of main results
    1. 1.1 Univariate Laguerre polynomials
    2. 1.2 Combinatorial interpretation in terms of Laguerre digraphs
    3. 1.3 A first multivariate generalization
    4. 1.4 A second multivariate generalization
    5. 1.5 Total positivity
    6. 1.6 Statement of main results: Univariate case
    7. 1.7 Statement of main results: Multivariate case
    8. 1.8 Plan of this paper
  2. 2 Preliminaries
    1. 2.1 Partially ordered commutative rings and total positivity
    2. 2.2 Production matrices
    3. 2.3 Production matrices and total positivity
    4. 2.4 Binomial row-generating matrices
    5. 2.5 Exponential Riordan arrays
  3. 3 Production matrices: Univariate case
    1. 3.1 Coefficient matrix ${\sf L}^{(\alpha)}$: Proof of Proposition 1.2(a)
    2. 3.2 Binomial row-generating matrix ${\sf L}^{(\alpha)}B_{x}$: Proof of Proposition 1.2(b)
    3. 3.3 Total positivity of the coefficient matrix ${\sf L}^{(-1+\lambda)}$
  4. 4 Multivariate Laguerre polynomials:  Exponential generating functions and production matrices
    1. 4.1 Exponential generating functions
    2. 4.2 Production matrices: Proof of Proposition 1.5(a,b)
  5. 5 Total positivity of the production matrices: Univariate case
    1. 5.1 The tridiagonal matrix $P^{\circ}$: Proof of Proposition 1.3(a)
    2. 5.2 The quadridiagonal matrix $P$: Proof of Proposition 1.3(b)
  6. 6 Total positivity of the production matrices: Multivariate case
    1. 6.1 The tridiagonal matrix $P^{\circ\flat}$: Proof of Proposition 1.6(a)
    2. 6.2 The quadridiagonal matrix $P^{\flat}$: Proof of Proposition 1.6(b)
  7. 7 The connection with multiple orthogonal polynomials
  8. A Generalized and modified $\bm{m}$-Stieltjes–Rogers polynomials
    1. A.1 Classical Stieltjes–Rogers polynomials
    2. A.2 $m$-Stieltjes–Rogers polynomials (branched S-fractions)
    3. A.3 Generalized $m$-Stieltjes–Rogers polynomials of type $j$
    4. A.4 Application to the univariate Laguerre production matrix
      1. A.4.1 Case $j=0$
      2. A.4.2 Case $j=1$
      3. A.4.3 Case $j=2$
  9. B When is $\bm{B_{\xi}^{-1}PB_{\xi}}$ $\bm{(r,1)}$-banded?
  10. C Total positivity for a variant class of quadridiagonal matrices

##  1 Introduction and statement of main results

In a seminal 1980 paper, Flajolet [29] showed that the coefficients in the
Taylor expansion of the generic Stieltjes-type (resp. Jacobi-type) continued
fraction — which he called the Stieltjes–Rogers (resp. Jacobi–Rogers)
polynomials — can be interpreted as the generating polynomials for Dyck (resp.
Motzkin) paths with specified height-dependent weights. More recently it was
independently discovered by several authors [34, 56, 41, 70] that Thron-type
continued fractions also have an interpretation of this kind: namely, their
Taylor coefficients — which we call, by analogy, the Thron–Rogers polynomials
— can be interpreted as the generating polynomials for Schröder paths with
specified height-dependent weights.

In a recent paper [59] we presented an infinite sequence of generalizations of
the Stieltjes–Rogers and Thron–Rogers polynomials, which are parametrized by
an integer $m\geq 1$ and reduce to the classical Stieltjes–Rogers and
Thron–Rogers polynomials when $m=1$; they are the generating polynomials of
$m$-Dyck and $m$-Schröder paths, respectively, with height-dependent weights,
and are also the Taylor coefficients of certain branched continued fractions.
We proved that these generalizations all possess the fundamental property of
coefficientwise Hankel-total positivity [66, 70], jointly in all the
(infinitely many) indeterminates. These facts were known when $m=1$ [66, 70]
but were new when $m>1$. By specializing the indeterminates we were able to
give many examples of Hankel-totally positive sequences whose generating
functions do not possess nice classical continued fractions. (The concept of
Hankel-total positivity [66, 70] will be explained in more detail later in
this Introduction.) Similar considerations apply to the $m$-Jacobi–Rogers
polynomials, which are the generating polynomials of $m$-Łukasiewicz paths;
but here the Hankel-total positivity is a more delicate matter, for which a
sufficient (but not necessary) condition is the total positivity of a lower-
Hessenberg production matrix.

In the present paper we will apply the $m$-Jacobi–Rogers theory with $m=2$ to
prove the coefficientwise Hankel-total positivity for some multivariate
generalizations of the Laguerre, rook and Lah polynomials. Our constructions
are motivated in part by the work of Coussement and Van Assche [19] on the
multiple orthogonal polynomials associated to weights based on modified Bessel
functions of the first kind $I_{\alpha}$; we will explain this unexpected
connection in Section 7. Our method — to use production matrices for a
binomial row-generating matrix, exploiting the theory of exponential Riordan
arrays — was employed previously by ourselves [58, 67, 13] and Zhu [90, 92];
but the present case is more delicate than these previous applications because
the total positivity of the quadridiagonal production matrix (Section 6.2) is
decidedly nontrivial.

The starting point of this work was the fact, due to Gantmakher and Krein
[36], that a Hankel matrix of real numbers is totally positive if and only if
the underlying sequence is a Stieltjes moment sequence (see Section 1.5
below), combined with the classical fact that, for each $\alpha\geq-1$ and
$x\geq 0$, the sequence $({{\mathcal{L}}_{n}^{(\alpha)}}(x))_{n\geq 0}$ of
monic unsigned Laguerre polynomials [defined in (1.5) below] is a Stieltjes
moment sequence [cf. (7.1) below]. This led Sylvie Corteel and one of the
authors (A.D.S.) to conjecture, a few years ago [18], that the sequence
$({{\mathcal{L}}_{n}^{(\alpha)}}(x))_{n\geq 0}$ is coefficientwise Hankel-
totally positive (see Section 1.5 for the definition). The purpose of this
paper is to prove this conjecture, along with a vast multivariate
generalization. The univariate conjecture was also recently proven by Zhu [90,
92].

###  1.1 Univariate Laguerre polynomials

The Laguerre polynomials ${L_{n}^{(\alpha)}}$ are conventionally defined as
[63, 77, 3, 40]

${L_{n}^{(\alpha)}}(x)\;=\;\sum_{k=0}^{n}\binom{n+\alpha}{n-k}\,{(-x)^{k}\over
k!}$ (1.1)

and have the generating function

$\sum_{n=0}^{\infty}{L_{n}^{(\alpha)}}(x)\>t^{n}\;=\;(1-t)^{-(1+\alpha)}\,e^{-xt/(1-t)}\;.$
(1.2)

However, for our purposes it is more convenient to work with the monic
unsigned Laguerre polynomials

$\displaystyle{{\mathcal{L}}_{n}^{(\alpha)}}(x)\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;n!\,{L_{n}^{(\alpha)}}(-x)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}{n!\over
k!}\,\binom{n+\alpha}{n-k}\,x^{k}\;=\;\sum_{k=0}^{n}\binom{n}{k}\,(n+\alpha)^{\underline{n-k}}\,x^{k}$
(1.5) $\displaystyle=$
$\displaystyle(1+\alpha)^{\overline{n}}\;{{\tensor[_{1\\!}]{F}{{}_{1}}\\!}\\!\biggl{(}\\!\\!\begin{array}[]{c}-n\\\\[2.84526pt]
1+\alpha\end{array}\\!\bigg{|}\,-x\\!\biggr{)}}$

[where $\rho^{\underline{n}}\stackrel{{\scriptstyle\rm
def}}{{=}}\rho(\rho-1)\cdots(\rho-n+1)$ and
$\rho^{\overline{n}}\stackrel{{\scriptstyle\rm
def}}{{=}}\rho(\rho+1)\cdots(\rho+n-1)$], or the reversed monic unsigned
Laguerre polynomials

$\displaystyle{{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x)\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;n!\,x^{n}\,{L_{n}^{(\alpha)}}(-1/x)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}\,\binom{n+\alpha}{k}\,k!\,x^{k}\;=\;\sum_{k=0}^{n}\binom{n}{k}\,(n+\alpha)^{\underline{k}}\,x^{k}$
(1.8) $\displaystyle=$
$\displaystyle{{\tensor[_{2\\!}]{F}{{}_{0}}\\!}\biggl{(}\\!\\!\begin{array}[]{c}-n,\,-n-\alpha\\\\[2.84526pt]
\hbox{---}\end{array}\\!\bigg{|}\,x\\!\biggr{)}}\;.$

Note that ${{\mathcal{L}}_{n}^{(\alpha)}}(x)$ and
${{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x)$ are polynomials jointly in $x$
and $\alpha$, with nonnegative integer coefficients. The first few
${{\mathcal{L}}_{n}^{(\alpha)}}(x)$ are thus

${\mathcal{L}}_{0}^{(\alpha)}(x)&=1\\\
{\mathcal{L}}_{1}^{(\alpha)}(x)=(1+\alpha)\,+\,x\\\
{\mathcal{L}}_{2}^{(\alpha)}(x)=(1+\alpha)(2+\alpha)\,+\,2(2+\alpha)x\,+\,x^{2}\\\
{\mathcal{L}}_{3}^{(\alpha)}(x)=(1+\alpha)(2+\alpha)(3+\alpha)\,+\,3(2+\alpha)(3+\alpha)x\,+\,3(3+\alpha)x^{2}\,+\,x^{3}\qquad$
(1.9)

It follows from (1.2) that these polynomials have the exponential generating
functions

$\displaystyle\sum_{n=0}^{\infty}{{\mathcal{L}}_{n}^{(\alpha)}}(x)\>{t^{n}\over
n!}$ $\displaystyle=$ $\displaystyle(1-t)^{-(1+\alpha)}\,e^{xt/(1-t)}$ (1.10)
$\displaystyle\sum_{n=0}^{\infty}{{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x)\>{t^{n}\over
n!}$ $\displaystyle=$ $\displaystyle(1-xt)^{-(1+\alpha)}\,e^{t/(1-xt)}$ (1.11)

A major role will be played in what follows by the coefficient matrix of the
monic unsigned Laguerre polynomials: it is the unit-lower-triangular matrix
${\sf L}^{(\alpha)}$ with entries

$({\sf L}^{(\alpha)})_{n,k}\;=\;{n!\over
k!}\,\binom{n+\alpha}{n-k}\;=\;\binom{n}{k}\,(n+\alpha)^{\underline{n-k}}\;=\;\binom{n}{k}\,(1+\alpha+k)^{\overline{n-k}}\;.$
(1.12)

This matrix is an exponential Riordan array (see Section 2.5)
${\mathcal{R}}[F,G]$ with $F(t)=(1-t)^{-(1+\alpha)}$ and $G(t)=t/(1-t)$:
compare (1.10) with (2.33) below. It is also the matrix of generalized
Stieltjes–Rogers polynomials of the first kind (see Section A.1) for the
Stieltjes-type continued fraction associated to the sequence
${\bigl{(}(1+\alpha)^{\overline{n}}\bigr{)}_{n\geq 0}}$: see Proposition 3.1.

An important role will also be played by the binomial row-generating matrix
${\sf L}^{(\alpha)}B_{x}$, where $B_{x}$ is the weighted binomial matrix

$(B_{x})_{ij}\;=\;\binom{i}{j}\,x^{i-j}$ (1.13)

(note that it too is unit-lower-triangular). The matrix ${\sf
L}^{(\alpha)}B_{x}$ is an exponential Riordan array ${\mathcal{R}}[e^{xG}F,G]$
with $F$ and $G$ as above (see Corollary 2.20). The zeroth column of ${\sf
L}^{(\alpha)}B_{x}$ consists of the monic unsigned Laguerre polynomials
${\mathcal{L}}_{n}^{(\alpha)}(x)$. More generally, it can be shown (see
Proposition 3.2) that

$({\sf L}^{(\alpha)}B_{x})_{n,k}\;=\;{1\over k!}\,{d^{k}\over
dx^{k}}\,{{\mathcal{L}}_{n}^{(\alpha)}}(x)\;=\;\binom{n}{k}\,{\mathcal{L}}_{n-k}^{(\alpha+k)}(x)\;.$
(1.14)

Several specializations of the Laguerre polynomials to integer values of
$\alpha$ correspond to well-known combinatorial objects:

1) The rook polynomial $R_{B}(x)$ of a chessboard $B$ is the generating
polynomial for placements of zero or more nonattacking rooks on $B$, with a
weight $x$ for each rook.111 See [64, chapters 7 and 8] [72, sections 2.3 and
2.4] for further discussion of rook polynomials.  In particular, for an
$m\times n$ rectangular chessboard, the number of ways of placing $k$ non-
attacking rooks is $\binom{m}{k}\binom{n}{k}k!$: we choose $k$ rows, $k$
columns, and a permutation connecting them. It follows that the rook
polynomial of an $n\times n$ square chessboard is the reversed monic unsigned
Laguerre polynomial
${\overline{\mathcal{L}}}_{n}^{(0)}(x)=\sum_{k=0}^{n}\binom{n}{k}^{2}k!\,x^{k}$;
and more generally, for any integer $\alpha\geq-n$, the rook polynomial of an
$n\times(n+\alpha)$ rectangular chessboard is
${\overline{\mathcal{L}}}_{n}^{(\alpha)}(x)=\sum_{k=0}^{n}\binom{n}{k}\binom{n+\alpha}{k}k!\,x^{k}$.222
See [55, A144084] for the triangular array corresponding to the rook
polynomial of the $n\times n$ chessboard, and see [55, A002720] for the row
sums. Note also that a rook configuration of an $m\times n$ chessboard can
equivalently be viewed as a matching of the complete bipartite graph
$K_{m,n}$; so the rook polynomial of the $m\times n$ chessboard is also the
matching polynomial [50, p. 334] of $K_{m,n}$.  The first few rook polynomials
for square chessboards are

${\overline{\mathcal{L}}}_{0}^{(0)}(x)&=1\\\
{\overline{\mathcal{L}}}_{1}^{(0)}(x)=1+x\\\
{\overline{\mathcal{L}}}_{2}^{(0)}(x)=1+4x+2x^{2}\\\
{\overline{\mathcal{L}}}_{3}^{(0)}(x)=1+9x+18x^{2}+6x^{3}\\\
{\overline{\mathcal{L}}}_{4}^{(0)}(x)=1+16x+72x^{2}+96x^{3}+24x^{4}\qquad$
(1.15)

2) The Lah number $\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}$ is the number of
partitions of an $n$-element set into $k$ nonempty linearly ordered blocks
(also called lists); we set
$\genfrac{\lfloor}{\rfloor}{0.0pt}{}{0}{k}=\delta_{k0}$. The Lah polynomial is
then defined as
${\textrm{Lah}}_{n}(x)=\sum_{k=0}^{n}\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}\>x^{k}$.
It is easy to see that the Lah numbers have the explicit expression

$\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}\;=\;{n!\over
k!}\binom{n-1}{n-k}\;=\;\begin{cases}\delta_{k0}&\textrm{if
$n=0$}\\\\[5.69054pt] {n!\over k!}\binom{n-1}{k-1}&\textrm{if $n\geq
1$}\end{cases}$ (1.16)

It follows that ${\textrm{Lah}}_{n}(x)$ is the monic unsigned Laguerre
polynomial ${\mathcal{L}}^{(-1)}_{n}(x)$.333 See [55, A105278/A008297/A066667]
for further information on the Lah numbers and Lah polynomials.  The first few
Lah polynomials are

${\mathcal{L}}_{0}^{(-1)}(x)&=1\\\
{\mathcal{L}}_{1}^{(-1)}(x)=\hphantom{0+}x\\\
{\mathcal{L}}_{2}^{(-1)}(x)=\hphantom{0+}2x+x^{2}\\\
{\mathcal{L}}_{3}^{(-1)}(x)=\hphantom{0+}6x+6x^{2}+x^{3}\\\
{\mathcal{L}}_{4}^{(-1)}(x)=\hphantom{0+}24x+36x^{2}+12x^{3}+x^{4}\qquad$
(1.17)

3) More generally, for any integer $r\geq 0$, the $r$-Lah number
${\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}}_{\\!r}$ is the number of
partitions of an $(n+r)$-element set into $k+r$ nonempty linearly ordered
blocks, with the restriction that $r$ distinguished elements must belong to
distinct blocks [54]. The $\bm{r}$-Lah polynomial is then defined as
${\textrm{Lah}}^{[r]}_{n}(x)=\sum_{k=0}^{n}{\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}}_{\\!r}\>x^{k}$.
Clearly, ${\textrm{Lah}}^{[0]}_{n}(x)={\textrm{Lah}}_{n}(x)$ and
${\textrm{Lah}}^{[1]}_{n}(x)={\textrm{Lah}}_{n+1}(x)/x$. It is not difficult
to show [54, Theorem 3.7] that the $r$-Lah numbers have the explicit
expression

${\genfrac{\lfloor}{\rfloor}{0.0pt}{}{n}{k}}_{\\!r}\;=\;{n!\over
k!}\binom{n+2r-1}{n-k}\;=\;\begin{cases}\delta_{k0}&\textrm{if
$n=0$}\\\\[5.69054pt] {n!\over k!}\binom{n+2r-1}{k+2r-1}&\textrm{if $n\geq
1$}\end{cases}$ (1.18)

It follows that ${\textrm{Lah}}^{[r]}_{n}(x)$ is the monic unsigned Laguerre
polynomial ${\mathcal{L}}^{(2r-1)}_{n}(x)$.444 See [55,
A143497/A143498/A143499] for further information on the $r$-Lah numbers and
$r$-Lah polynomials.

In summary, the rook polynomials correspond to $\alpha=0$ for square
chessboards and $\alpha=1,2,3,\ldots$ for rectangular chessboards, the Lah
polynomials correspond to $\alpha=-1$, and the $r$-Lah polynomials correspond
to $\alpha=2r-1$.

###  1.2 Combinatorial interpretation in terms of Laguerre digraphs

Four decades ago, Foata and Strehl [31] introduced a beautiful combinatorial
interpretation of the Laguerre polynomials, which will form the starting point
for our work.555 See also the recent paper of Strehl [76] for further
applications of this formalism.  Let us define a Laguerre digraph to be a
digraph in which each vertex has out-degree 0 or 1 and in-degree 0 or 1. It
follows that each weakly connected component of a Laguerre digraph is either a
directed path of some length $\ell\geq 0$ (where a path of length 0 is an
isolated vertex) or else a directed cycle of some length $\ell\geq 1$ (where a
cycle of length 1 is a loop): see Figure 1. For each integer $n\geq 0$, let us
write ${\mathbf{LD}}_{n}$ for the set of Laguerre digraphs on the vertex set
$[n]\stackrel{{\scriptstyle\rm def}}{{=}}\\{1,\ldots,n\\}$; and for $n\geq
k\geq 0$, let us write ${\mathbf{LD}}_{n,k}$ for the set of Laguerre digraphs
on the vertex set $[n]$ with $k$ paths. For $G\in{\mathbf{LD}}_{n}$, we write
${\rm e}(G)$ [resp. ${\rm cyc}(G)$, ${\rm pa}(G)$] for the number of edges
(resp. cycles, paths) in $G$, and observe that ${\rm e}(G)=n-{\rm pa}(G)$.
Foata and Strehl [31] then showed that

${{\mathcal{L}}_{n}^{(\alpha)}}(x)\;=\;\sum_{G\in{\mathbf{LD}}_{n}}x^{{\rm
pa}(G)}\,(1+\alpha)^{{\rm cyc}(G)}$ (1.19)

or equivalently

${{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x)\;=\;\sum_{G\in{\mathbf{LD}}_{n}}x^{{\rm
e}(G)}\,(1+\alpha)^{{\rm cyc}(G)}\;.$ (1.20)

Indeed, the proof of (1.19)/(1.20) is an easy argument using the exponential
formula [73, chapter 5], or equivalently, the theory of species [7]. Let us do
(1.19): The number of directed paths on $n\geq 1$ vertices is $n!$, so with a
weight $x$ per path they have exponential generating function $xt/(1-t)$. The
number of directed cycles on $n\geq 1$ vertices is $(n-1)!$, so with a weight
$1+\alpha$ per cycle they have exponential generating function
$-(1+\alpha)\log(1-t)$. A Laguerre digraph is a disjoint union of paths and
cycles, so by the exponential formula it has exponential generating function

$\exp\Bigl{[}{xt\over
1-t}\,-\,(1+\alpha)\log(1-t)\Bigr{]}\;=\;(1-t)^{-(1+\alpha)}\,e^{xt/(1-t)}\;,$
(1.21)

which coincides with (1.10).666 Foata and Strehl [31] also gave a direct
combinatorial proof that (1.19)/(1.20) is equivalent to (1.5)/(1.8); this
requires a bit more work [31, Lemma 2.1].  The principal virtue of the
combinatorial model (1.19)/(1.20) is that it treats $\alpha$ as an
indeterminate; it need not be an integer.777 The formulae (1.19)/(1.20) also
make clear (as Rota [53, p. 206] realized many years ago) that the natural
parameter for the Laguerre polynomials is $\lambda\stackrel{{\scriptstyle\rm
def}}{{=}}1+\alpha$, not $\alpha$. But it would be too confusing to try to
change the notation at this late date.

1234567891011 Figure 1: A Laguerre digraph on $11$ vertices with $2$ cycles
(one of which is a loop) and $2$ paths (one of which is an isolated vertex).

The identity (1.19) can equivalently be understood as a combinatorial
representation for the coefficient matrix ${\sf L}^{(\alpha)}$:

$({\sf L}^{(\alpha)})_{n,k}\;=\;\sum_{G\in{\mathbf{LD}}_{n,k}}(1+\alpha)^{{\rm
cyc}(G)}\;.$ (1.22)

This formula will be the starting point for our multivariate generalizations,
in which we will enumerate Laguerre digraphs with additional statistics.

Remarks. 1\. There is a one-to-one correspondence between Laguerre digraphs
and rook configurations on an $n\times n$ chessboard: for a rook at position
$(i,j)$ we draw an edge $i\to j$. This explains why $\alpha=0$ corresponds to
the rook polynomial, for which there is no special weighting of cycles.
However, we prefer the digraph point of view, where the cycles (and therefore
the role of $\alpha$) can be seen more clearly.

2\. Note also from (1.19) that the case $\alpha=-1$ leads to Laguerre digraphs
with no cycles — that is, only directed paths — with a weight $x$ per path.
These digraphs are in obvious one-to-one correspondence with partitions of
$[n]$ into linearly ordered blocks; this explains why $\alpha=-1$ corresponds
to the Lah polynomials. $\blacksquare$

###  1.3 A first multivariate generalization

It is now natural to generalize the Laguerre polynomials by introducing
further statistics into the digraph model. Here is one way: Let us write ${\rm
e}_{-}(G),{\rm e}_{0}(G),{\rm e}_{+}(G)$ for the number of edges $i\to j$ in
$G$ where $j<i$, $j=i$ or $j>i$, respectively. We then generalize the
coefficient matrix (1.22) by introducing separate variables
$v_{-},v_{0},v_{+}$ for the three types of edges:

${\sf L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{G\in{\mathbf{LD}}_{n,k}}v_{-}^{{\rm e}_{-}(G)}\,v_{0}^{{\rm
e}_{0}(G)}\,v_{+}^{{\rm e}_{+}(G)}\,(1+\alpha)^{{\rm cyc}(G)}\;.$ (1.23)

This polynomial is homogeneous of degree $n-k$ in $v_{-},v_{0},v_{+}$. We call
the matrix ${\sf L}^{(\alpha)}(v_{-},v_{0},v_{+})$ the (first) multivariate
Laguerre coefficient matrix. In Section 4.1 we will compute the bivariate
exponential generating function for this matrix: see (4.15). By specializing
$v_{-}=v_{0}=v_{+}=v$, we recover a rescaled version of the univariate
Laguerre coefficient matrix

${\sf L}^{(\alpha)}(v,v,v)_{n,k}\;=\;({\sf L}^{(\alpha)})_{n,k}\,v^{n-k}\;.$
(1.24)

Let us then introduce the row-generating polynomials of the multivariate
Laguerre coefficient matrix:

$\displaystyle{{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},v_{0},v_{+})$
$\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle\sum_{k=0}^{n}{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\>x^{k}$ (1.25)
$\displaystyle{{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x;v_{-},v_{0},v_{+})$
$\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle\sum_{k=0}^{n}{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\>x^{n-k}$ (1.26)

Of course, by homogeneity (1.26) can also be understood as

${{\overline{\mathcal{L}}}_{n}^{(\alpha)}}(x;v_{-},v_{0},v_{+})\;=\;\sum_{k=0}^{n}{\sf
L}^{(\alpha)}(xv_{-},xv_{0},xv_{+})_{n,k}\;.$ (1.27)

Remark. In the rook model, the three $v$ variables correspond to giving
different weights for rooks below the diagonal, on the diagonal, or above the
diagonal. To our knowledge this type of weighting has not previously been
considered, even in the rook case $\alpha=0$. However, two special cases
generalize previously known results:

1) If we set $v_{+}=0$, then the only possible connected components of $G$ are
decreasing paths (of length $\ell\geq 0$) and loops. It follows that

${\sf
L}^{(\alpha)}(v_{-},v_{0},0)_{n,k}\;=\;\sum_{i=0}^{n}\binom{n}{i}\,[(1+\alpha)v_{0}]^{i}\>\genfrac{\\{}{\\}}{0.0pt}{}{n-i}{k}\,v_{-}^{n-i-k}\;,$
(1.28)

where $\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}$ denotes the number of partitions of
an $n$-element set into $k$ nonempty blocks: we choose $i$ loops and then
partition the remaining vertices into $k$ nonempty blocks. Further
specializing to $\alpha=0$ and $v_{0}=0$ yields the well-known [72, Corollary
2.4.2] formula for the counting of rook configurations on an $n\times n$
triangular board:

${\sf L}^{(0)}(1,0,0)_{n,k}\;=\;\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\;,$ (1.29)

where a configuration with $k$ paths has $n-k$ edges and hence $n-k$ rooks.
Similarly, for $\alpha=0$ and $v_{0}=1$ we get

${\sf L}^{(0)}(1,1,0)_{n,k}\;=\;\genfrac{\\{}{\\}}{0.0pt}{}{n+1}{k+1}\;,$
(1.30)

which follows from (1.28) by a standard identity [39, eq. (6.15)].

2) At the other extreme, when $k=0$, paths are forbidden, and we obtain the
generating polynomial for permutations of $[n]$ with a weight $v_{+}$ for each
cycle ascent (= excedance), $v_{-}$ for each cycle descent (= anti-excedance),
$v_{0}$ for each fixed point, and $1+\alpha$ for each cycle. For the
corresponding exponential generating function, see (4.15) specialized to
$u=0$. In particular, when $v_{-}=v_{0}=v_{+}=v$ we obtain

${\sf L}^{(\alpha)}(v,v,v)_{n,0}\;=\;(1+\alpha)^{\overline{n}}\,v^{n}\;.$
(1.31)

And when $\alpha=0$ and $v_{0}=v_{-}$, we obtain

${\sf
L}^{(0)}(v_{-},v_{-},v_{+})_{n,0}\;=\;\sum_{j=0}^{n-1}\genfrac{\langle}{\rangle}{0.0pt}{}{n}{j}\,v_{+}^{j}\,v_{-}^{n-j}\;,$
(1.32)

where the Eulerian number $\genfrac{\langle}{\rangle}{0.0pt}{}{n}{j}$ is the
number of permutations of $[n]$ with $j$ excedances (or $j$ descents). More
generally, these two specializations can be combined: we obtain the generating
polynomial for permutations by number of excedances and number of cycles [71,
eq. (2.12)] [65, Proposition 3.3]

${\sf
L}^{(\alpha)}(v_{-},v_{-},v_{+})_{n,0}\;=\;\sum_{j=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{j}\,(v_{+}-v_{-})^{n-j}\,v_{-}^{j}\,(1+\alpha)^{\overline{j}}\;.$
(1.33)

And most generally, we can introduce additional fixed points, to obtain

${\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,0}\;=\;\sum_{i=0}^{n}\binom{n}{i}\,[(1+\alpha)(v_{0}-v_{-})]^{i}\>{\sf
L}^{(\alpha)}(v_{-},v_{-},v_{+})_{n-i,0}\;.$ (1.34)

$\blacksquare$

###  1.4 A second multivariate generalization

But we can go farther: instead of weighting edges according to whether they
are increasing, decreasing or fixed points, we can weight pairs of successive
edges according to whether they are peaks ($+-$), valleys ($-+$), double
ascents ($++$), double descents ($--$) or fixed points ($00$). To define these
concepts for a Laguerre digraph, we first need to make a convention about
boundary conditions at the two ends of a path. We will here use 0–0 boundary
conditions: that is, we extend the Laguerre digraph $G$ on the vertex set
$[n]$ to a digraph $\widehat{G}$ on the vertex set $[n]\cup\\{0\\}$ by
decreeing that any vertex $i\in[n]$ that has in-degree (resp. out-degree) 0 in
$G$ will receive an incoming (resp. outgoing) edge from (resp. to) the vertex
0. In this way each vertex $i\in[n]$ will have a unique predecessor
$p(i)\in[n]\cup\\{0\\}$ and a unique successor $s(i)\in[n]\cup\\{0\\}$. We
then say that a vertex $i\in[n]$ is a

  * •

peak (p) if $p(i)<i>s(i)$;

  * •

valley (v) if $p(i)>i<s(i)$;

  * •

double ascent (da) if $p(i)<i<s(i)$;

  * •

double descent (dd) if $p(i)>i>s(i)$;

  * •

fixed point (fp) if $p(i)=i=s(i)$.

(Note that “fixed point” is a synonym of “loop”.) When these concepts are
applied to the cycles of a Laguerre digraph, we obtain the usual cycle
classification of indices in a permutation as cycle peaks, cycle valleys,
cycle double rises, cycle double falls and fixed points [84, 71]. When applied
to the paths of a Laguerre digraph, we obtain the usual linear classification
of indices in a permutation (written in word form) as peaks, valleys, double
ascents or double descents [72, p. 45]. Note that, because of the 0–0 boundary
conditions, an isolated vertex is always a peak, the initial vertex of a path
is always a peak or double ascent, and the final vertex of a path is always a
peak or double descent; moreover, each path contains at least one peak.

We write ${\rm p}(G),{\rm v}(G),{\rm da}(G),{\rm dd}(G),{\rm fp}(G)$ for the
number of vertices $i\in[n]$ that are, respectively, peaks, valleys, double
ascents, double descents or fixed points. We then introduce the second
multivariate Laguerre coefficient matrix

${\widehat{{\sf L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm
dd}},{y_{\rm fp}})_{n,k}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{G\in{\mathbf{LD}}_{n,k}}y_{\rm p}^{{\rm p}(G)}y_{\rm
v}^{{\rm v}(G)}y_{\rm da}^{{\rm da}(G)}y_{\rm dd}^{{\rm dd}(G)}y_{\rm
fp}^{{\rm fp}(G)}\,(1+\alpha)^{{\rm cyc}(G)}\>.$ (1.35)

This polynomial is homogeneous of degree $n$ in ${y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}}$. In Section 4.1 we will compute
the bivariate exponential generating function for this matrix (and indeed for
a matrix that further generalizes it).

Because each path contains at least one peak, we can, if we wish, remove a
factor $y_{\rm p}^{k}$ and define a unit-lower-triangular matrix by

${\widehat{{\sf L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})_{n,k}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;{\widehat{{\sf L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})_{n,k}/y_{\rm p}^{k}\;.$ (1.36)

This polynomial is homogeneous of degree $n-k$ in ${y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}}$.

We can recover the first multivariate Laguerre coefficient matrix by looking
only at the second step of each pair: specializing ${y_{\rm p}}={y_{\rm
dd}}=v_{-}$, ${y_{\rm v}}={y_{\rm da}}=v_{+}$ and ${y_{\rm fp}}=v_{0}$, we
obtain the previous counting with an extra weight $v_{-}$ associated to the
final vertex of each path:

${\widehat{{\sf L}}}^{(\alpha)}(v_{-},v_{+},v_{+},v_{-},v_{0})_{n,k}\;=\;{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\>v_{-}^{k}$ (1.37)

or equivalently

${\widehat{{\sf
L}}}^{(\alpha)\flat}(v_{-},v_{+},v_{+},v_{-},v_{0})_{n,k}\;=\;{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\;.$ (1.38)

Alternatively, we can look only at the first step of each pair: specializing
${y_{\rm v}}={y_{\rm dd}}=v_{-}$, ${y_{\rm p}}={y_{\rm da}}=v_{+}$ and
${y_{\rm fp}}=v_{0}$, we obtain the previous counting with an extra weight
$v_{+}$ associated to the initial vertex of each path:

${\widehat{{\sf L}}}^{(\alpha)}(v_{+},v_{-},v_{+},v_{-},v_{0})_{n,k}\;=\;{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\>v_{+}^{k}$ (1.39)

or equivalently

${\widehat{{\sf
L}}}^{(\alpha)\flat}(v_{+},v_{-},v_{+},v_{-},v_{0})_{n,k}\;=\;{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\;.$ (1.40)

Once again we can introduce the row-generating polynomials:

$\displaystyle{\widehat{{\mathcal{L}}}_{n}^{(\alpha)}}(x;{y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}})$
$\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle\sum_{k=0}^{n}{\sf L}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})_{n,k}\>x^{k}$ (1.41)
$\displaystyle{\overline{\widehat{{\mathcal{L}}}}_{n}^{(\alpha)}}(x;{y_{\rm
p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}})$
$\displaystyle\stackrel{{\scriptstyle\rm def}}{{=}}$
$\displaystyle\sum_{k=0}^{n}{\sf L}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})_{n,k}\>x^{n-k}$ (1.42)

###  1.5 Total positivity

The main results of this paper concern the total positivity of various
matrices associated to the univariate and multivariate Laguerre polynomials.
Recall first that a finite or infinite matrix of real numbers is called
totally positive (TP) if all its minors are nonnegative, and totally positive
of order $r$ (TPr) if all its minors of size $\leq r$ are nonnegative.
Background information on totally positive matrices can be found in [42, 35,
60, 28]; they have application to many fields of pure and applied
mathematics.888 See [67, footnote 4] for many references.  In particular, it
is known [36, Théorème 9] [60, section 4.6] that an infinite Hankel matrix
$(a_{i+j})_{i,j\geq 0}$ of real numbers is totally positive if and only if the
underlying sequence $(a_{n})_{n\geq 0}$ is a Stieltjes moment sequence, i.e.
the moments of a positive measure on $[0,\infty)$.

But this is only the beginning of the story, because we are here principally
concerned, not with sequences and matrices of real numbers, but with sequences
and matrices of polynomials (with integer or real coefficients) in one or more
indeterminates ${\mathbf{x}}$: they will typically be generating polynomials
that enumerate some combinatorial objects with respect to one or more
statistics. We equip the polynomial ring ${\mathbb{R}}[{\mathbf{x}}]$ with the
coefficientwise partial order: that is, we say that $P$ is nonnegative (and
write $P\succeq 0$) in case $P$ is a polynomial with nonnegative coefficients.
We then say that a matrix with entries in ${\mathbb{R}}[{\mathbf{x}}]$ is
coefficientwise totally positive if all its minors are polynomials with
nonnegative coefficients; and analogously for coefficientwise total positivity
of order $r$. We say that a sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ with entries
in ${\mathbb{R}}[{\mathbf{x}}]$ is coefficientwise Hankel-totally positive if
its associated infinite Hankel matrix is coefficientwise totally positive; and
likewise for the version of order $r$. Similar definitions apply to the
formal-power-series ring ${\mathbb{R}}[[{\mathbf{x}}]]$. Most generally, we
can consider sequences and matrices with values in an arbitrary partially
ordered commutative ring (a precise definition will be given in Section 2.1);
total positivity and Hankel-total positivity are then defined in the obvious
way. Coefficientwise Hankel-total positivity of a sequence of polynomials
$(P_{n}({\mathbf{x}}))_{n\geq 0}$ implies the pointwise Hankel-total
positivity (i.e. the Stieltjes moment property) for all ${\mathbf{x}}\geq 0$,
but it is vastly stronger.

For instance, it is known (see Section 7) that, for each $\alpha\geq-1$ and
$x\geq 0$, the sequence $({{\mathcal{L}}_{n}^{(\alpha)}}(x))_{n\geq 0}$ of
monic unsigned Laguerre polynomials is a Stieltjes moment sequence. So every
minor of the Hankel matrix $({\mathcal{L}}_{i+j}^{(\alpha)}(x))_{i,j\geq 0}$
is a polynomial in $x$ and $\alpha$ that is nonnegative whenever
$\alpha\geq-1$ and $x\geq 0$. But much more turns out to be true: every minor
of the Hankel matrix $({\mathcal{L}}_{i+j}^{(\alpha)}(x))_{i,j\geq 0}$ is in
fact a polynomial in $x$ and $\alpha$ that is coefficientwise nonnegative in
$x$ and $\lambda\stackrel{{\scriptstyle\rm def}}{{=}}1+\alpha$. That is, the
sequence $({\mathcal{L}}_{n}^{(-1+\lambda)}(x))_{n\geq 0}$ is coefficientwise
Hankel-totally positive in $x$ and $\lambda$: this will be our first main
result (Theorem 1.1(c)). We will then extend this result to the multivariate
polynomials introduced in Sections 1.3 and 1.4.

Our proofs of total positivity and Hankel-total positivity will be based on
the method of production matrices [22, 23]. We will review this theory in
Sections 2.2 and 2.3, so now we state only the bare-bones definitions. Let
$P=(p_{ij})_{i,j\geq 0}$ be an infinite matrix with entries in a commutative
ring $R$; we assume that $P$ is either row-finite (i.e. has only finitely many
nonzero entries in each row) or column-finite. Now define an infinite matrix
$A=(a_{nk})_{n,k\geq 0}$ by

$a_{nk}\;=\;(P^{n})_{0k}\;.$ (1.43)

We call $P$ the production matrix and $A$ the output matrix, and we write
$A={\mathcal{O}}(P)$. The two key facts here are the following [70]: if $R$ is
a partially ordered commutative ring, $P$ is totally positive of order $r$,
and $a$ is a nonnegative element of $R$, then ${\mathcal{O}}(P+aI)$ is totally
positive of order $r$ and the zeroth column of ${\mathcal{O}}(P+aI)$ is
Hankel-totally positive of order $r$. See Section 2.3 for precise statements
and proofs.

###  1.6 Statement of main results: Univariate case

Our first main result is the following:

######  Theorem 1.1 (Total positivity of the univariate Laguerre polynomials).

  * (a)

The Laguerre coefficient matrix ${\sf L}^{(-1+\lambda)}$ is totally positive
in the ring ${\mathbb{Z}}[\lambda]$ equipped with the coefficientwise order.

  * (b)

The binomial row-generating matrix ${\sf L}^{(-1+\lambda)}\,B_{x}$ is totally
positive in the ring ${\mathbb{Z}}[x,\lambda]$ equipped with the
coefficientwise order.

  * (c)

The sequence of monic unsigned Laguerre polynomials
$({\mathcal{L}}_{n}^{(-1+\lambda)}(x))_{n\geq 0}$ is Hankel-totally positive
in the ring ${\mathbb{Z}}[x,\lambda]$ equipped with the coefficientwise order.

  * (d)

The sequence of reversed monic unsigned Laguerre polynomials
$({\overline{\mathcal{L}}}_{n}^{(-1+\lambda)}(x))_{n\geq 0}$ is Hankel-totally
positive in the ring ${\mathbb{Z}}[x,\lambda]$ equipped with the
coefficientwise order.

Theorem 1.1(c,d) proves a conjecture made a few years ago by Sylvie Corteel
and one of us [18]. Theorem 1.1 was also proven recently by Zhu [90,
Proposition 4.14] [92, Proposition 5.11]; his methods are very similar to
ours.999 Zhu [90, Proposition 4.14] [92, Proposition 5.11] states a slightly
weaker form of Theorem 1.1 in which $\lambda$ is a nonnegative real number
rather than an indeterminate. But his methods actually prove the stronger
result claimed here.

The result in part (a) is fairly easy, and we will give two proofs using
different methods: one direct proof (Section 3.3), and one proof using
production matrices. Part (b) is then an immediate consequence of part (a)
together with the coefficientwise total positivity of the binomial matrix
$B_{x}$ (Lemma 2.2). Also, part (d) is trivially equivalent to part (c) by
virtue of the relation
${\overline{\mathcal{L}}}_{n}^{(-1+\lambda)}(x)=x^{n}\,{\mathcal{L}}_{n}^{(-1+\lambda)}(1/x)$.
So our main effort will be directed to proving part (c): we will do this by
constructing the production matrix for the binomial row-generating matrix
${\sf L}^{(-1+\lambda)}\,B_{x}$ and then proving its total positivity. More
precisely, we will prove the following:

######  Proposition 1.2 (Production matrices for the univariate Laguerre
polynomials).

  * (a)

The production matrix of the Laguerre coefficient array ${\sf L}^{(\alpha)}$
is the tridiagonal unit-lower-Hessenberg matrix
$P^{\circ}=(p^{\circ}_{ij})_{i,j\geq 0}$ defined by

$p^{\circ}_{n,n+1}&=1\\\ p^{\circ}_{n,n}=2n+1+\alpha\\\
p^{\circ}_{n,n-1}=n(n+\alpha)\\\ p^{\circ}_{n,k}=0\qquad\textrm{if $k<n-1$ or
$k>n+1$}$ (1.44)

  * (b)

The production matrix of the binomial row-generating matrix ${\sf
L}^{(\alpha)}B_{x}$ is the quadridiagonal unit-lower-Hessenberg matrix
$P=(p_{ij})_{i,j\geq 0}$ defined by

$p_{n,n+1}&=1\\\ p_{n,n}=(2n+1+\alpha)\,+\,x\\\
p_{n,n-1}=n(n+\alpha)\,+\,2nx\\\ p_{n,n-2}=n(n-1)x\\\
p_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$ (1.45)

Remarks. 1\. When $\alpha\in\\{-1,0,1\\}$, the production matrix (1.45) arises
from a 2-branched S-fraction [59] — more precisely, as the production matrix
for the generalized $m$-Stieltjes–Rogers polynomials of type $j$ with $m=2$
and $j\in\\{0,1,2\\}$. See Appendix A.3 for the theory of these generalized
$m$-Stieltjes–Rogers polynomials, and Appendix A.4 for the application to
(1.45).

2\. It will follow easily from our general theory (Lemma 2.7) that the
production matrices of Proposition 1.2(a,b) are related by
$P=B_{x}^{-1}P^{\circ}B_{x}$. What is far from obvious is why $P$ is
quadridiagonal, since it is not in general true that $B_{x}^{-1}TB_{x}$ is
quadridiagonal whenever $T$ is tridiagonal. In Appendix B we explain why $P$
is quadridiagonal in the present case, by answering the more general question:
Which lower-Hessenberg matrices $P$ have the property that
$B_{\xi}^{-1}PB_{\xi}$ is $(r,1)$-banded? $\blacksquare$

######  Proposition 1.3 (Total positivity of the univariate production
matrices).

  * (a)

The matrix $P^{\circ}=(p^{\circ}_{ij})_{i,j\geq 0}$ defined by (1.44), with
the change of variable $\alpha=-1+\lambda$, is totally positive in the ring
${\mathbb{Z}}[\lambda]$ equipped with the coefficientwise order.

  * (b)

The matrix $P=(p_{ij})_{i,j\geq 0}$ defined by (1.45), with the change of
variable $\alpha=-1+\lambda$, is totally positive in the ring
${\mathbb{Z}}[x,\lambda]$ equipped with the coefficientwise order.

Combining Propositions 1.2 and 1.3 with the general theory of production
matrices proves Theorem 1.1.

We will prove Proposition 1.2 in Section 3, and Proposition 1.3 in Section 5.

###  1.7 Statement of main results: Multivariate case

We now state the corresponding results for the first and second multivariate
Laguerre polynomials that were introduced in Sections 1.3 and 1.4,
respectively. We begin by considering the second multivariate Laguerre
coefficient matrix, for which it is convenient to use the form ${\widehat{{\sf
L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}})$ that was defined in (1.36). Our main result, generalizing Theorem 1.1,
is the following:

######  Theorem 1.4 (Total positivity of the second multivariate Laguerre
polynomials).

Let $\lambda,{y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}}$
be elements of a partially ordered commutative ring $R$ that satisfy
$\lambda\geq 0$, $\lambda{y_{\rm fp}}\geq\lambda{y_{\rm p}}$, ${y_{\rm p}}\geq
0$, ${y_{\rm v}}\geq 0$ and ${y_{\rm da}}+{y_{\rm dd}}\geq{y_{\rm p}}+{y_{\rm
v}}$; and let $x$ be an indeterminate. Then:

  * (a)

The second multivariate Laguerre coefficient array ${\widehat{{\sf
L}}}^{(-1+\lambda)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm
dd}},{y_{\rm fp}})$ is totally positive in the ring $R$.

  * (b)

The binomial row-generating matrix ${\widehat{{\sf
L}}}^{(-1+\lambda)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm
dd}},{y_{\rm fp}})\,B_{x}$ is totally positive in the ring $R[x]$ equipped
with the coefficientwise order.

  * (c)

The sequence of row-generating polynomials
$\widehat{{\mathcal{L}}}_{n}^{(-1+\lambda)}(x;{y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})$ defined by (1.41) is Hankel-totally positive
in the ring $R[x]$ equipped with the coefficientwise order.

  * (d)

The sequence of reversed row-generating polynomials
$\overline{\widehat{{\mathcal{L}}}}_{n}^{(-1+\lambda)}(x;{y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}})$ defined by (1.42) is Hankel-
totally positive in the ring $R[x]$ equipped with the coefficientwise order.

These statements are also true when the roles of ${y_{\rm p}}$ and ${y_{\rm
v}}$ are interchanged.

As in the univariate case, part (b) is an immediate consequence of part (a),
and part (d) is trivially equivalent to part (c). We therefore concentrate on
proving (a) and (c).

As before, we will prove Theorem 1.4 by exhibiting the production matrices and
then proving their total positivity.

######  Proposition 1.5 (Production matrices for the multivariate Laguerre
polynomials).

  * (a)

The production matrix of the multivariate Laguerre coefficient array
${\widehat{{\sf L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})$ is the tridiagonal unit-lower-Hessenberg
matrix $P^{\circ\flat}=(p^{\circ\flat}_{ij})_{i,j\geq 0}$ defined by

$p^{\circ\flat}_{n,n+1}&=1\\\ p^{\circ\flat}_{n,n}=(1+\alpha){y_{\rm
fp}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})\\\
p^{\circ\flat}_{n,n-1}=n(n+\alpha){y_{\rm p}}{y_{\rm v}}\\\
p^{\circ\flat}_{n,k}=0\qquad\textrm{if $k<n-1$ or $k>n+1$}$ (1.46)

  * (b)

The production matrix of the binomial row-generating matrix ${\widehat{{\sf
L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}})B_{x}$ is the quadridiagonal unit-lower-Hessenberg matrix
$P^{\flat}=(p^{\flat}_{ij})_{i,j\geq 0}$ defined by

$p^{\flat}_{n,n+1}&=1\\\ p^{\flat}_{n,n}=(1+\alpha){y_{\rm fp}}\,+\,n({y_{\rm
da}}+{y_{\rm dd}})\,+\,x\\\ p^{\flat}_{n,n-1}=n(n+\alpha){y_{\rm p}}{y_{\rm
v}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})x\\\ p^{\flat}_{n,n-2}=n(n-1){y_{\rm
p}}{y_{\rm v}}x\\\ p^{\flat}_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$
(1.47)

######  Proposition 1.6 (Total positivity of the multivariate production
matrices).

  * (a)

The matrix $P^{\circ\flat}=(p^{\circ\flat}_{ij})_{i,j\geq 0}$ defined by
(1.46), with the variables substituted to elements of a partially ordered
commutative ring $R$ that satisfy $\alpha\geq-1$, ${(1+\alpha){y_{\rm
fp}}\geq(1+\alpha){y_{\rm p}}}$, ${y_{\rm p}}\geq 0$, ${y_{\rm v}}\geq 0$ and
${y_{\rm da}}+{y_{\rm dd}}\geq{y_{\rm p}}+{y_{\rm v}}$, is totally positive in
the ring $R$.

  * (b)

The matrix $P^{\flat}=(p^{\flat}_{ij})_{i,j\geq 0}$ defined by (1.47), with
the variables substituted to elements of a partially ordered commutative ring
$R$ that satisfy $\alpha\geq-1$, ${(1+\alpha){y_{\rm fp}}=(1+\alpha){y_{\rm
p}}}$, ${y_{\rm p}}\geq 0$, ${y_{\rm v}}\geq 0$, ${y_{\rm da}}+{y_{\rm
dd}}\geq{y_{\rm p}}+{y_{\rm v}}$ and $x\geq 0$, is totally positive in the
ring $R$.

These statements are also true when the roles of ${y_{\rm p}}$ and ${y_{\rm
v}}$ are interchanged.

We will prove Proposition 1.5 in Section 4.2, and Proposition 1.6 in Section
6. Here the most difficult part is Proposition 1.6(b): namely, proving the
total positivity of the quadridiagonal matrix (1.47) under the specified
conditions. We will do this by proving the total positivity of a much more
general quadridiagonal matrix (Theorem 6.1); and see also Appendix C (Theorem
C.1) for a variant of this result.

Remarks. 1\. The zeroth column of the multivariate Laguerre coefficient array
${\widehat{{\sf L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}})$ is the sequence of generating polynomials for
permutations weighted according to the cycle classification, and the
tridiagonal production matrix (1.46) is the production matrix for the
corresponding J-fraction, namely, [71, Theorem 2.4] specialized to
$x_{1}=u_{1}={y_{\rm p}}$, $x_{2}=u_{2}={y_{\rm dd}}$, $y_{1}=v_{1}={y_{\rm
v}}$, $y_{2}=v_{2}={y_{\rm da}}$, $w_{n}={y_{\rm fp}}$, $\lambda=1+\alpha$.

2\. Note that in part (a) we require only the inequality ${(1+\alpha){y_{\rm
fp}}\geq(1+\alpha){y_{\rm p}}}$, while in part (b) we require the equality
${(1+\alpha){y_{\rm fp}}=(1+\alpha){y_{\rm p}}}$. However, this will not
preclude us from proving Theorem 1.4 assuming only the inequality, as our
general theory (Corollaries 2.12 and 2.16) entitles us to use the production
matrix $P^{\flat}+aI$ with $a=\lambda({y_{\rm fp}}-{y_{\rm p}})\geq 0$.
$\blacksquare$

We can now recover the first multivariate Laguerre coefficient matrix ${\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})$ defined in (1.23), and its corresponding
row-generating polynomials
${{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},v_{0},v_{+})$ defined in (1.25), by
specializing either

${y_{\rm p}}={y_{\rm dd}}=v_{-}$, $\;{y_{\rm v}}={y_{\rm da}}=v_{+}$,
$\;{y_{\rm fp}}=v_{0}\;$ as in (1.37)/(1.38) (1.48)

or

$\hbox{${y_{\rm v}}={y_{\rm dd}}=v_{-}$, $\;{y_{\rm p}}={y_{\rm da}}=v_{+}$,
$\;{y_{\rm fp}}=v_{0}\;$ as in (\ref{eq.second-to-first.2})/(\ref{eq.second-
to-first.2a})}\;.$ (1.49)

By applying these specializations to Theorem 1.4, we obtain:

######  Corollary 1.7 (Total positivity of the first multivariate Laguerre
polynomials).

  * (a)

The first multivariate Laguerre coefficient matrix ${\sf
L}^{(-1+\lambda)}(v_{-},v_{0},v_{+})$, defined in (1.23), is totally positive
when the variables are substituted to elements of a partially ordered
commutative ring $R$ that satisfy $\lambda\geq 0$, $v_{-}\geq 0$, $v_{+}\geq
0$, and either $v_{0}\geq v_{-}$ or $v_{0}\geq v_{+}$.

Equivalently, the matrices ${\sf L}^{(-1+\lambda)}(v_{-},v_{-}+w,v_{+})$ and
${\sf L}^{(-1+\lambda)}(v_{-},v_{+}+w,v_{+})$ are totally positive in the ring
${\mathbb{Z}}[v_{-},v_{+},w,\lambda]$ equipped with the coefficientwise order.

  * (b)

The sequences
$\bigl{(}{{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},v_{-},v_{+})\bigr{)}_{n\geq
0}$ and
$\bigl{(}{{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},v_{+},v_{+})\bigr{)}_{n\geq
0}$ of row-generating polynomials, defined in (1.25), are Hankel-totally
positive, in the ring $R[x]$ equipped with the coefficientwise order, when the
variables are substituted to elements of a partially ordered commutative ring
$R$ that satisfy $\lambda\geq 0$, $v_{-}\geq 0$, $v_{+}\geq 0$, and either
$v_{0}\geq v_{-}$ or $v_{0}\geq v_{+}$.

Equivalently, the sequences
$\bigl{(}{{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},{v_{-}+w},v_{+})\bigr{)}_{n\geq
0}$ and
$\bigl{(}{{\mathcal{L}}_{n}^{(\alpha)}}(x;v_{-},v_{+}+w,v_{+})\bigr{)}_{n\geq
0}$ are Hankel-totally positive in the ring
${\mathbb{Z}}[x,v_{-},v_{+},w,\lambda]$ equipped with the coefficientwise
order.

The multivariate case (Theorem 1.4 and Propositions 1.5 and 1.6) obviously
subsumes the univariate case (Theorem 1.1 and Propositions 1.2 and 1.3), so it
is in principle redundant to consider the latter. But since the proofs in the
univariate case are quite a bit simpler, and since this case exhibits some
special features that do not carry over to the fully multivariate case, we
think it useful to present the univariate case first.

###  1.8 Plan of this paper

Although the present paper is a follow-up to our papers [59, 58], we have
endeavored, for the convenience of the reader, to make it as self-contained as
possible. We have therefore begun, in Section 2, with a brief review of the
key definitions and results from [59, 58] (plus a few other things) that will
be needed in the sequel. We then proceed as follows: In Section 3 we determine
the production matrices for the univariate Laguerre coefficient matrix and
binomial row-generating matrix. In Section 4 we compute the exponential
generating function for the multivariate Laguerre coefficient matrix (1.35)
and employ this to determine the corresponding production matrices, using the
theory of exponential Riordan arrays. In Sections 5 and 6 we prove the total
positivity of the univariate and multivariate production matrices,
respectively. In particular, in Section 6.2 (Theorem 6.1) we prove the
coefficientwise total positivity of a rather general class of quadridiagonal
matrices; we think that this result, and its method of proof, may be of some
independent interest. In Section 7 we explain the unexpected connection with
multiple orthogonal polynomials; in particular, we explain how we were led to
guess the production matrices (1.45) and (1.47).

In Appendix A we review the theory of classical and branched S-fractions, and
then introduce the generalized $m$-Stieltjes–Rogers polynomials and modified
$m$-Stieltjes–Rogers polynomials of type $j\geq 0$; this expands and
supersedes some of the discussion in [59, sections 5, 7 and 9]. We also apply
this theory to the univariate Laguerre production matrix (1.45). In Appendix B
we answer the question: Which lower-Hessenberg matrices $P$ have the property
that $B_{\xi}^{-1}PB_{\xi}$ is $(r,1)$-banded? In Appendix C we prove the
total positivity of a class of quadridiagonal matrices, giving a variant of
what was done in Section 6.2.

The first proof of the production matrices in this paper was found in 2019 by
one of us (M.P.) using a bijection from Laguerre digraphs to labeled
2-Łukasiewicz paths. We hope to revisit this powerful method in a subsequent
paper.

##  2 Preliminaries

In this section we review some definitions and results from [59, 58, 67, 13,
70] that will be needed in what follows. After a brief introduction to total
positivity in a partially ordered commutative ring (Section 2.1), we provide a
précis of the theory of production matrices (Section 2.2) and its application
to total positivity (Section 2.3); these latter results form the theoretical
foundation for our work. Then we introduce the concept of binomial row-
generating matrices (Section 2.4). Finally, we review the theory of
exponential Riordan arrays (Section 2.5); this theory will be our main
technical tool.

For the benefit of readers who are familiar with the papers [59, 58, 67, 13],
let us mention two things that are new here (and crucial for the present
paper): the tridiagonal comparison theorem (Proposition 2.6), and adding a
multiple of the identity to the production matrix (Corollaries 2.12 and 2.16).

###  2.1 Partially ordered commutative rings and total positivity

In this paper all rings will be assumed to have an identity element 1 and to
be nontrivial ($1\neq 0$).

A partially ordered commutative ring is a pair $(R,{\mathcal{P}})$ where $R$
is a commutative ring and ${\mathcal{P}}$ is a subset of $R$ satisfying

  * (a)

$0,1\in{\mathcal{P}}$.

  * (b)

If $a,b\in{\mathcal{P}}$, then $a+b\in{\mathcal{P}}$ and $ab\in{\mathcal{P}}$.

  * (c)

${\mathcal{P}}\cap(-{\mathcal{P}})=\\{0\\}$.

We call ${\mathcal{P}}$ the nonnegative elements of $R$, and we define a
partial order on $R$ (compatible with the ring structure) by writing $a\leq b$
as a synonym for $b-a\in{\mathcal{P}}$. Please note that, unlike the practice
in real algebraic geometry [9, 46, 61, 51], we do not assume here that squares
are nonnegative; indeed, this property fails completely for our prototypical
example, the ring of polynomials with the coefficientwise order, since
$(1-x)^{2}=1-2x+x^{2}$ is not coefficientwise nonnegative.

Now let $(R,{\mathcal{P}})$ be a partially ordered commutative ring and let
${\mathbf{x}}=\\{x_{i}\\}_{i\in I}$ be a collection of indeterminates. In the
polynomial ring $R[{\mathbf{x}}]$ and the formal-power-series ring
$R[[{\mathbf{x}}]]$, let ${\mathcal{P}}[{\mathbf{x}}]$ and
${\mathcal{P}}[[{\mathbf{x}}]]$ be the subsets consisting of polynomials
(resp. series) with nonnegative coefficients. Then
$(R[{\mathbf{x}}],{\mathcal{P}}[{\mathbf{x}}])$ and
$(R[[{\mathbf{x}}]],{\mathcal{P}}[[{\mathbf{x}}]])$ are partially ordered
commutative rings; we refer to this as the coefficientwise order on
$R[{\mathbf{x}}]$ and $R[[{\mathbf{x}}]]$.

A finite or infinite matrix with entries in a partially ordered commutative
ring is called totally positive (TP) if all its minors are nonnegative; it is
called totally positive of order $\bm{r}$ (TPr) if all its minors of size
$\leq r$ are nonnegative. It follows immediately from the Cauchy–Binet formula
that the product of two TP (resp. TPr) matrices is TP (resp. TPr).101010 For
infinite matrices, we need some condition to ensure that the product is well-
defined. For instance, the product $AB$ is well-defined whenever $A$ is row-
finite (i.e. has only finitely many nonzero entries in each row) or $B$ is
column-finite.  This fact is so fundamental to the theory of total positivity
that we shall henceforth use it without comment.

We say that a sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ with entries in a
partially ordered commutative ring is Hankel-totally positive (resp. Hankel-
totally positive of order $\bm{r}$) if its associated infinite Hankel matrix
$H_{\infty}({\bm{a}})=(a_{i+j})_{i,j\geq 0}$ is TP (resp. TPr). We say that
${\bm{a}}$ is Toeplitz-totally positive (resp. Toeplitz-totally positive of
order $\bm{r}$) if its associated infinite Toeplitz matrix
$T_{\infty}({\bm{a}})=(a_{i-j})_{i,j\geq 0}$ (where
$a_{n}\stackrel{{\scriptstyle\rm def}}{{=}}0$ for $n<0$) is TP (resp.
TPr).111111 When $R={\mathbb{R}}$, Toeplitz-totally positive sequences are
traditionally called Pólya frequency sequences (PF), and Toeplitz-totally
positive sequences of order $r$ are called Pólya frequency sequences of order
$r$ (PFr). See [42, chapter 8] for a detailed treatment.

We will need a few easy facts about the total positivity of special matrices:

######  Lemma 2.1 (Bidiagonal matrices).

Let $A$ be a matrix with entries in a partially ordered commutative ring, with
the property that all its nonzero entries belong to two consecutive diagonals.
Then $A$ is totally positive if and only if all its entries are nonnegative.

Proof. The nonnegativity of the entries (i.e. TP1) is obviously a necessary
condition for TP. Conversely, for a matrix of this type it is easy to see that
every nonzero minor is simply a product of some entries. $\square$

######  Lemma 2.2 (Binomial matrix).

In the ring ${\mathbb{Z}}$, the binomial matrix
$B=\bigl{(}{\textstyle\binom{n}{k}}\bigr{)}_{n,k\geq 0}$ is totally positive.
More generally, the weighted binomial matrix

$B_{x,y}\;=\;\bigl{(}x^{n-k}y^{k}{\textstyle\binom{n}{k}}\bigr{)}_{\\!n,k\geq
0}$ (2.1)

is totally positive in the ring ${\mathbb{Z}}[x,y]$ equipped with the
coefficientwise order.

We also write $B_{x}$ as a shorthand for $B_{x,1}$.

Proof of Lemma 2.2.  It is well known that the binomial matrix $B$ is totally
positive, and this can be proven by a variety of methods: e.g. using
production matrices [42, pp. 136–137, Example 6.1] [60, pp. 108–109], by
diagonal similarity to a totally positive Toeplitz matrix [60, p. 109], by
exponentiation of a nonnegative lower-subdiagonal matrix [28, p. 63], or by an
application of the Lindström–Gessel–Viennot lemma [33, p. 24].

Then $B_{x,y}=DBD^{\prime}$ where $D=\mathop{\rm
diag}\nolimits\bigl{(}(x^{n})_{n\geq 0}\bigr{)}$ and $D^{\prime}=\mathop{\rm
diag}\nolimits\bigl{(}(x^{-k}y^{k})_{k\geq 0}\bigr{)}$. By Cauchy–Binet,
$B_{x,y}$ is totally positive in the ring ${\mathbb{Z}}[x,x^{-1},y]$ equipped
with the coefficientwise order. But because $B$ is lower-triangular, the
elements of $B_{x,y}$ actually lie in the subring ${\mathbb{Z}}[x,y]$.
$\square$

See also Example 2.10 below for an ab initio proof of Lemma 2.2 using
production matrices.

######  Lemma 2.3 (Introducing indeterminates).

Let $A=(a_{ij})_{i,j\geq 0}$ be a lower-triangular matrix with entries in a
partially ordered commutative ring $R$, and let ${\bm{x}}=(x_{i})_{i\geq 1}$.
Define the lower-triangular matrix $B=(b_{ij})_{i,j\geq 0}$ by

$b_{ij}\;=\;x_{j+1}x_{j+2}\cdots x_{i}\,a_{ij}\;.$ (2.2)

Then:

  * (a)

If $A$ is TPr and ${\bm{x}}$ are indeterminates, then $B$ is TPr in the ring
$R[{\bm{x}}]$ equipped with the coefficientwise order.

  * (b)

If $A$ is TPr and ${\bm{x}}$ are nonnegative elements of $R$, then $B$ is TPr
in the ring $R$.

Proof. (a) Let ${\bm{x}}=(x_{i})_{i\geq 1}$ be commuting indeterminates, and
let us work in the ring $R[{\bm{x}},{\bm{x}}^{-1}]$ equipped with the
coefficientwise order. Let $D=\mathop{\rm
diag}\nolimits(1,\,x_{1},\,x_{1}x_{2},\,\ldots)$. Then $D$ is invertible, and
$D^{-1}=\mathop{\rm
diag}\nolimits(1,\,x_{1}^{-1},\,x_{1}^{-1}x_{2}^{-1},\,\ldots)$ has
nonnegative elements. It follows that $B=DAD^{-1}$ is TPr in the ring
$R[{\bm{x}},{\bm{x}}^{-1}]$ equipped with the coefficientwise order. But the
matrix elements $b_{ij}$ actually belong to the subring $R[{\bm{x}}]\subseteq
R[{\bm{x}},{\bm{x}}^{-1}]$. So $B$ is TPr in the ring $R[{\bm{x}}]$ equipped
with the coefficientwise order.

(b) follows from (a) by specializing indeterminates. $\square$

Finally, we need some special facts about the total positivity of tridiagonal
matrices. We recall that a contiguous principal minor of a matrix $A$ is a
minor $\det A_{II}$ for $I=\\{r,{r+1},{r+2},\ldots,s\\}$ for some $r\leq s$.

######  Lemma 2.4.

Every nonzero $k\times k$ minor of a tridiagonal matrix is a product of off-
diagonal elements and contiguous principal minors, with sizes adding up to
$k$.

The standard proof for real matrices [60, p. 98] is valid in any commutative
ring.

######  Corollary 2.5 (Total positivity of tridiagonal matrices).

Let $A$ be a tridiagonal matrix with entries in a partially ordered
commutative ring. Then $A$ is totally positive of order $r$ if and only if all
its off-diagonal elements and all its contiguous principal minors of size
$\leq r$ are nonnegative.

In other words, the well-known [60, Theorem 4.3] criterion for total
positivity of tridiagonal matrices with real entries extends without change to
matrices with entries in an arbitrary partially ordered commutative ring.

This has the following important consequence:

######  Proposition 2.6 (Tridiagonal comparison theorem, weak form).

Let $A$ and $D$ be matrices with entries in a partially ordered commutative
ring, with $A$ being tridiagonal and $D$ being diagonal. If $A$ is totally
positive of order $r$ and $D$ is nonnegative, then $A+D$ is totally positive
of order $r$.

Proof. We apply Corollary 2.5. The off-diagonal elements of $A+D$ are the same
as those of $A$. The principal minors of $A+D$ are

$\det\,(A+D)_{II}\;=\;\sum_{J\subseteq I}(\det A_{JJ})\prod_{i\in I\setminus
J}d_{i}\;,$ (2.3)

where $d_{i}$ are the diagonal elements of $D$. $\square$

Remarks. 1\. There is a stronger form of the tridiagonal comparison theorem
[70] [91, Proposition 3.1] in which we can increase the diagonal elements and
decrease the off-diagonal elements (while keeping them nonnegative). But we
will not need this stronger result here.

2\. In the special case where $A=LU$ with $L$ nonnegative lower-bidiagonal and
$U$ nonnegative upper-bidiagonal, Proposition 2.6 can be proven by a
Lindström–Gessel–Viennot argument [2, Chapter 32] using the digraph shown in
Figure 2. $\blacksquare$

$u_{0}$$v_{0}$$a_{0}$$d_{0}$$e_{0}$$u_{1}$$v_{1}$$a_{1}$$d_{1}$$e_{1}$$u_{2}$$v_{2}$$a_{2}$$d_{2}$$e_{2}$$u_{3}$$v_{3}$$a_{3}$$d_{3}$$e_{3}$$\vdots$$\vdots$$b_{1}$$c_{1}$$b_{2}$$c_{2}$$b_{3}$$c_{3}$
Figure 2: Digraph representation of the matrix $LU+D$ where $L$ is the lower-
bidiagonal matrix with $a_{0},a_{1},\ldots$ on the diagonal and
$b_{1},b_{2},\ldots$ on the subdiagonal, $U$ is the upper-bidiagonal matrix
with $c_{1},c_{2},\ldots$ on the superdiagonal and $d_{0},d_{1},\ldots$ on the
diagonal, and $D$ is a diagonal matrix with entries $e_{0},e_{1},\ldots\;$.

###  2.2 Production matrices

The method of production matrices [22, 23] has become in recent years an
important tool in enumerative combinatorics. In the special case of a
tridiagonal production matrix, this construction goes back to Stieltjes’ [74,
75] work on continued fractions: the production matrix of a classical
S-fraction or J-fraction is tridiagonal. In the present paper, by contrast, we
shall need production matrices that are lower-Hessenberg (i.e. vanish above
the first superdiagonal) but are not in general tridiagonal. We therefore
begin by reviewing briefly the basic theory of production matrices. The
important connection of production matrices with total positivity will be
treated in the next subsection.

Let $P=(p_{ij})_{i,j\geq 0}$ be an infinite matrix with entries in a
commutative ring $R$. In order that powers of $P$ be well-defined, we shall
assume that $P$ is either row-finite (i.e. has only finitely many nonzero
entries in each row) or column-finite.

Let us now define an infinite matrix $A=(a_{nk})_{n,k\geq 0}$ by

$a_{nk}\;=\;(P^{n})_{0k}$ (2.4)

(in particular, $a_{0k}=\delta_{0k}$). Writing out the matrix multiplications
explicitly, we have

$a_{nk}\;=\;\sum_{i_{1},\ldots,i_{n-1}}p_{0i_{1}}\,p_{i_{1}i_{2}}\,p_{i_{2}i_{3}}\,\cdots\,p_{i_{n-2}i_{n-1}}\,p_{i_{n-1}k}\;,$
(2.5)

so that $a_{nk}$ is the total weight for all $n$-step walks in ${\mathbb{N}}$
from $i_{0}=0$ to $i_{n}=k$, in which the weight of a walk is the product of
the weights of its steps, and a step from $i$ to $j$ gets a weight $p_{ij}$.
Yet another equivalent formulation is to define the entries $a_{nk}$ by the
recurrence

$a_{nk}\;=\;\sum_{i=0}^{\infty}a_{n-1,i}\,p_{ik}\qquad\hbox{for $n\geq 1$}$
(2.6)

with the initial condition $a_{0k}=\delta_{0k}$.

We call $P$ the production matrix and $A$ the output matrix, and we write
$A={\mathcal{O}}(P)$. Note that if $P$ is row-finite, then so is
${\mathcal{O}}(P)$; if $P$ is lower-Hessenberg, then ${\mathcal{O}}(P)$ is
lower-triangular; if $P$ is lower-Hessenberg with invertible superdiagonal
entries, then ${\mathcal{O}}(P)$ is lower-triangular with invertible diagonal
entries; and if $P$ is unit-lower-Hessenberg (i.e. lower-Hessenberg with
entries 1 on the superdiagonal), then ${\mathcal{O}}(P)$ is unit-lower-
triangular. In all the applications in this paper, $P$ will be lower-
Hessenberg, most often unit-lower-Hessenberg.

The matrix $P$ can also be interpreted as the adjacency matrix for a weighted
directed graph on the vertex set ${\mathbb{N}}$ (where the edge $i\to j$ is
omitted whenever $p_{ij}=0$). Then $P$ is row-finite (resp. column-finite) if
and only if every vertex has finite out-degree (resp. finite in-degree); and
$P$ is lower-Hessenberg if and only if all edges $i\to j$ satisfy $j\leq i+1$.

This iteration process can be given a compact matrix formulation. Let us
define the augmented production matrix

$\widetilde{P}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\left[\begin{array}[]{c}1\;\;0\;\;0\;\;0\;\;\cdots\;\vphantom{\Sigma}\\\
\hline\cr P\end{array}\right]\;,$ (2.7)

or in other words

$(\widetilde{P})_{n,k}\;=\;\begin{cases}\delta_{0k}&\textrm{if $n=0$}\\\
p_{n-1,k}&\textrm{if $n\geq 1$}\end{cases}$ (2.8)

Then the recurrence (2.6) together with the initial condition
$a_{0k}=\delta_{0k}$ can be written as

$A\;=\;\left[\begin{array}[]{c}1\;\;0\;\;0\;\;0\;\;\cdots\;\vphantom{\Sigma}\\\
\hline\cr AP\end{array}\right]\;=\;\left[\begin{array}[]{c|c}1&{\bm{0}}\\\
\hline\cr{\bm{0}}&A\end{array}\right]\left[\begin{array}[]{c}1\;\;0\;\;0\;\;0\;\;\cdots\;\vphantom{\Sigma}\\\
\hline\cr P\end{array}\right]\;=\;\left[\begin{array}[]{c|c}1&{\bm{0}}\\\
\hline\cr{\bm{0}}&A\end{array}\right]\widetilde{P}\;.$ (2.9)

This identity can be iterated to give the factorization

$A\;=\;\cdots\,\left[\begin{array}[]{c|c}I_{3}&{\bm{0}}\\\
\hline\cr&\\\\[-11.38109pt]
{\bm{0}}&\widetilde{P}\end{array}\right]\left[\begin{array}[]{c|c}I_{2}&{\bm{0}}\\\
\hline\cr&\\\\[-11.38109pt]
{\bm{0}}&\widetilde{P}\end{array}\right]\left[\begin{array}[]{c|c}I_{1}&{\bm{0}}\\\
\hline\cr&\\\\[-11.38109pt]
{\bm{0}}&\widetilde{P}\end{array}\right]\widetilde{P}$ (2.10)

where $I_{k}$ is the $k\times k$ identity matrix; and conversely, (2.10)
implies (2.9).

Now let $\Delta=(\delta_{i+1,j})_{i,j\geq 0}$ be the matrix with 1 on the
superdiagonal and 0 elsewhere. Then for any matrix $M$ with rows indexed by
${\mathbb{N}}$, the product $\Delta M$ is simply $M$ with its zeroth row
removed and all other rows shifted upwards. (Some authors use the notation
$\overline{M}\stackrel{{\scriptstyle\rm def}}{{=}}\Delta M$.) The recurrence
(2.6) can then be written as

$\Delta\,{\mathcal{O}}(P)\;=\;{\mathcal{O}}(P)\,P\;.$ (2.11)

It follows that if $A$ is a row-finite matrix that has a row-finite inverse
$A^{-1}$ and has first row $a_{0k}=\delta_{0k}$, then $P=A^{-1}\Delta A$ is
the unique matrix such that $A={\mathcal{O}}(P)$. This holds, in particular,
if $A$ is lower-triangular with invertible diagonal entries and $a_{00}=1$;
then $A^{-1}$ is lower-triangular and $P=A^{-1}\Delta A$ is lower-Hessenberg.
And if $A$ is unit-lower-triangular, then $P=A^{-1}\Delta A$ is unit-lower-
Hessenberg.

Later we shall need the following easy but fundamental fact, which shows how
the production matrix transforms when the output matrix $A$ is right-
multiplied by another matrix $B$:

######  Lemma 2.7 (Right-multiplication lemma).

Let $P=(p_{ij})_{i,j\geq 0}$ be a row-finite matrix (with entries in a
commutative ring $R$), with output matrix $A={\mathcal{O}}(P)$; and let
$B=(b_{ij})_{i,j\geq 0}$ be a lower-triangular matrix with invertible (in $R$)
diagonal entries. Then

$AB\;=\;b_{00}\,{\mathcal{O}}(B^{-1}PB)\;.$ (2.12)

That is, up to a factor $b_{00}$, the matrix $AB$ has production matrix
$B^{-1}PB$.

Proof. Since $P$ is row-finite, so is $A={\mathcal{O}}(P)$; then the matrix
products $AB$ and $B^{-1}PB$ arising in the lemma are well-defined. Now

$a_{nk}\;=\;\sum_{i_{1},\ldots,i_{n-1}}p_{0i_{1}}\,p_{i_{1}i_{2}}\,p_{i_{2}i_{3}}\,\cdots\,p_{i_{n-2}i_{n-1}}\,p_{i_{n-1}k}\;,$
(2.13)

while

${\mathcal{O}}(B^{-1}PB)_{nk}\;=\;\sum_{j,i_{1},\ldots,i_{n-1},i_{n}}(B^{-1})_{0j}\,p_{ji_{1}}\,p_{i_{1}i_{2}}\,p_{i_{2}i_{3}}\,\cdots\,p_{i_{n-2}i_{n-1}}\,p_{i_{n-1}i_{n}}\,b_{i_{n}k}\;.$
(2.14)

But $B$ is lower-triangular with invertible diagonal entries, so $B$ is
invertible and $B^{-1}$ is lower-triangular, with
$(B^{-1})_{0j}=b_{00}^{-1}\delta_{j0}$. It follows that
$AB=b_{00}\,{\mathcal{O}}(B^{-1}PB)$. $\square$

Remark. If $b_{00}=1$, then $(AB)_{00}=1$ and hence $AB$ can be the output
matrix of some production matrix; and in this case (2.12) is an immediate
consequence of (2.11). Indeed, right-multiplying (2.11) by $B$ yields

$\Delta\,({\mathcal{O}}(P)\,B)\;=\;{\mathcal{O}}(P)\,PB\;=\;({\mathcal{O}}(P)\,B)\,B^{-1}PB\;.$
(2.15)

$\blacksquare$

We will frequently apply this lemma with $B$ taken to be the binomial matrix
$B_{x}$. Unfortunately we know very little about what happens to the
production matrix when the output matrix is left-multiplied by another matrix
$B$. But we do know one special case: see Lemma 2.11 below.

###  2.3 Production matrices and total positivity

Let $P=(p_{ij})_{i,j\geq 0}$ be a matrix with entries in a partially ordered
commutative ring $R$. We will use $P$ as a production matrix; let
$A={\mathcal{O}}(P)$ be the corresponding output matrix. As before, we assume
that $P$ is either row-finite or column-finite.

When $P$ is totally positive, it turns out [70] that the output matrix
${\mathcal{O}}(P)$ has two total-positivity properties: firstly, it is totally
positive; and secondly, its zeroth column is Hankel-totally positive. More
generally, the same properties hold whenever $P=P_{0}+aI$, where $P_{0}$ is
totally positive and $a$ is nonnegative. Since [70] is not yet publicly
available, we shall present briefly here (with proof) the main results that
will be needed in the sequel.

The fundamental fact that drives the whole theory is the following:

######  Proposition 2.8 (Minors of the output matrix).

Every $k\times k$ minor of the output matrix $A={\mathcal{O}}(P)$ can be
written as a sum of products of minors of size $\leq k$ of the production
matrix $P$.

In this proposition the matrix elements ${\mathbf{p}}=\\{p_{ij}\\}_{i,j\geq
0}$ should be interpreted in the first instance as indeterminates: for
instance, we can fix a row-finite or column-finite set
$S\subseteq{\mathbb{N}}\times{\mathbb{N}}$ and define the matrix
$P^{S}=(p^{S}_{ij})_{i,j\in{\mathbb{N}}}$ with entries

$p^{S}_{ij}\;=\;\begin{cases}p_{ij}&\textrm{if $(i,j)\in S$}\\\\[2.84526pt]
0&\textrm{if $(i,j)\notin S$}\end{cases}$ (2.16)

Then the entries (and hence also the minors) of both $P$ and $A$ belong to the
polynomial ring ${\mathbb{Z}}[{\mathbf{p}}]$, and the assertion of Proposition
2.8 makes sense. Of course, we can subsequently specialize the indeterminates
${\mathbf{p}}$ to values in any commutative ring $R$.

Proof of Proposition 2.8.  For any infinite matrix $X=(x_{ij})_{i,j\geq 0}$,
let us write $X_{N}=(x_{ij})_{0\leq i\leq N-1,\,j\geq 0}$ for the submatrix
consisting of the first $N$ rows (and all the columns) of $X$. Every $k\times
k$ minor of $A$ is of course a $k\times k$ minor of $A_{N}$ for some $N$, so
it suffices to prove that the claim about minors holds for all the $A_{N}$.
But this is easy: the fundamental identity (2.9) implies

$A_{N}\;=\;\left[\begin{array}[]{c|c}1&{\bm{0}}\\\
\hline\cr{\bm{0}}&A_{N-1}\end{array}\right]\,\left[\begin{array}[]{c}1\;\;0\;\;0\;\;0\;\;\cdots\;\vphantom{\Sigma}\\\
\hline\cr P\end{array}\right]\;.$ (2.17)

So the result follows by induction on $N$, using the Cauchy–Binet formula.
$\square$

If we now specialize the indeterminates ${\mathbf{p}}$ to values in some
partially ordered commutative ring $R$, we can immediately conclude:

######  Theorem 2.9 (Total positivity of the output matrix).

Let $P$ be an infinite matrix that is either row-finite or column-finite, with
entries in a partially ordered commutative ring $R$. If $P$ is totally
positive of order $r$, then so is $A={\mathcal{O}}(P)$.

Remarks. 1\. In the case $R={\mathbb{R}}$, Theorem 2.9 is due to Karlin [42,
pp. 132–134]; see also [60, Theorem 1.11]. Karlin’s proof is different from
ours.

2\. Our quick inductive proof of Proposition 2.8 follows an idea of Zhu [86,
proof of Theorem 2.1], which was in turn inspired in part by Aigner [1, pp.
45–46]. The same idea recurs in recent work of several authors [87, Theorem
2.1] [11, Theorem 2.1(i)] [12, Theorem 2.3(i)] [47, Theorem 2.1] [14, Theorems
2.1 and 2.3] [37]. However, all of these results concerned only special cases:
[1, 86, 12, 47] treated the case in which the production matrix $P$ is
tridiagonal; [87] treated a (special) case in which $P$ is upper-bidiagonal;
[11] treated the case in which $P$ is the production matrix of a Riordan
array; [14, 37] treated (implicitly) the case in which $P$ is upper-triangular
and Toeplitz. But the argument is in fact completely general, as we have just
seen; there is no need to assume any special form for the matrix $P$.

3\. A slightly different version of this proof was presented in [59, 58]. The
simplified reformulation given here, using the augmented production matrix, is
due to Mu and Wang [52]. $\blacksquare$

######  Example 2.10 (Binomial matrix).

Let $P$ be the upper-bidiagonal Toeplitz matrix $xI+y\Delta$, where $x$ and
$y$ are indeterminates. By Lemma 2.1, $P$ is TP in the ring
${\mathbb{Z}}[x,y]$ equipped with the coefficientwise order. An easy
computation shows that ${\mathcal{O}}(xI+y\Delta)=B_{x,y}$, the weighted
binomial matrix with entries $(B_{x,y})_{nk}=x^{n-k}y^{k}\binom{n}{k}$. So
Theorem 2.9 implies that $B_{x,y}$ is TP in the ring ${\mathbb{Z}}[x,y]$
equipped with the coefficientwise order. This gives an ab initio proof of
Lemma 2.2. $\blacksquare$

More generally, we have:

######  Lemma 2.11.

${\mathcal{O}}(aI+bP)\>=\>B_{a,b}\,{\mathcal{O}}(P)$.

Note that Example 2.10 is the special case $P=\Delta$.

Proof of Lemma 2.11.  Since

$(aI+bP)^{n}\;=\;\sum_{j=0}^{n}a^{n-j}\,b^{j}\,\binom{n}{j}\,P^{j}\;,$ (2.18)

we have

${\mathcal{O}}(aI+bP)_{nk}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;[(aI+bP)^{n}]_{0k}&=\sum_{j=0}^{n}a^{n-j}\,b^{j}\,\binom{n}{j}\,(P^{j})_{0k}\qquad\\\
=\sum_{j=0}^{n}(B_{a,b})_{nj}\,{\mathcal{O}}(P)_{jk}{}\\\
=[B_{a,b}\,{\mathcal{O}}(P)]_{nk}\;.\\\ $ (2.19)

$\square$

Since the binomial matrix $B_{a,b}$ is totally positive by Lemma 2.2, we
conclude:

######  Corollary 2.12 (Total positivity of output matrix, improved).

Let $P$ be an infinite matrix that is either row-finite or column-finite, with
entries in a partially ordered commutative ring $R$. Then $P$ is TPr
$\implies$ ${\mathcal{O}}(P)$ is TPr $\iff$ ${\mathcal{O}}(aI+bP)$ is TPr for
all $a,b\geq 0$.

Here $a$ and $b$ can be nonnegative elements of $R$; or $a$ and $b$ can be
indeterminates and we work in the ring $R[a,b]$ equipped with the
coefficientwise order. Corollary 2.12 allows us to prove the total positivity
of the output matrix ${{\mathcal{O}}(aI+bP)}$ whenever $P$ is TP, even if
$aI+bP$ is not TP. This will play a crucial role in our proof of Theorem 1.4,
by allowing us to take $\lambda{y_{\rm fp}}\geq\lambda{y_{\rm p}}$ rather than
just $\lambda{y_{\rm fp}}=\lambda{y_{\rm p}}$.

Remark. Lemma 2.11 and Corollary 2.12 are a special case of an idea of Zhu
[88]; see [70] for a general version. $\blacksquare$

Now define
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)$
to be the zeroth-column sequence of ${\mathcal{O}}(P)$, i.e.

$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)_{n}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;{\mathcal{O}}(P)_{n0}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;(P^{n})_{00}\;.$ (2.20)

Then the Hankel matrix of
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)$
has matrix elements

$\displaystyle\\!\\!\\!\\!\\!\\!\\!H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))_{nn^{\prime}}\;=\;\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)_{n+n^{\prime}}\;=\;(P^{n+n^{\prime}})_{00}\;=\;\sum_{k=0}^{\infty}(P^{n})_{0k}\,(P^{n^{\prime}})_{k0}\;=\;$
$\displaystyle\sum_{k=0}^{\infty}(P^{n})_{0k}\,((P^{\rm
T})^{n^{\prime}})_{0k}\;=\;\sum_{k=0}^{\infty}{\mathcal{O}}(P)_{nk}\,{\mathcal{O}}(P^{\rm
T})_{n^{\prime}k}\;=\;\big{[}{\mathcal{O}}(P)\,{{\mathcal{O}}(P^{\rm T})}^{\rm
T}\big{]}_{nn^{\prime}}\;.\qquad$ (2.21)

(Note that the sum over $k$ has only finitely many nonzero terms: if $P$ is
row-finite, then there are finitely many nonzero $(P^{n})_{0k}$, while if $P$
is column-finite, there are finitely many nonzero
$(P^{n^{\prime}})_{k0}$.121212 Or to put it another way: If $P$ is row-finite,
then ${\mathcal{O}}(P)$ is row-finite; ${\mathcal{O}}(P^{\rm T})$ need not be
row- or column-finite, but the product
${\mathcal{O}}(P)\,{{\mathcal{O}}(P^{\rm T})}^{\rm T}$ is anyway well-defined.
Likewise, if $P$ is column-finite, then ${{\mathcal{O}}(P^{\rm T})}^{\rm T}$
is column-finite; ${\mathcal{O}}(P)$ need not be row- or column-finite, but
the product ${\mathcal{O}}(P)\,{{\mathcal{O}}(P^{\rm T})}^{\rm T}$ is anyway
well-defined. ) We have therefore proven:

######  Lemma 2.13 (Identity for Hankel matrix of the zeroth column).

Let $P$ be a row-finite or column-finite matrix with entries in a commutative
ring $R$. Then

$H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))\;=\;{\mathcal{O}}(P)\,{{\mathcal{O}}(P^{\rm
T})}^{\rm T}\;.$ (2.22)

Combining Proposition 2.8 with Lemma 2.13 and the Cauchy–Binet formula, we
obtain:

######  Corollary 2.14 (Hankel minors of the zeroth column).

Every $k\times k$ minor of the infinite Hankel matrix
$H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))=((P^{n+n^{\prime}})_{00})_{n,n^{\prime}\geq
0}$ can be written as a sum of products of the minors of size $\leq k$ of the
production matrix $P$.

And specializing the indeterminates ${\mathbf{p}}$ to nonnegative elements in
a partially ordered commutative ring, in such a way that $P$ is row-finite or
column-finite, we deduce:

######  Theorem 2.15 (Hankel-total positivity of the zeroth column).

Let $P=(p_{ij})_{i,j\geq 0}$ be an infinite row-finite or column-finite matrix
with entries in a partially ordered commutative ring $R$, and define the
infinite Hankel matrix
$H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))=((P^{n+n^{\prime}})_{00})_{n,n^{\prime}\geq
0}$. If $P$ is totally positive of order $r$, then so is
$H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))$.

Once again we can we can improve these results to replace $P$ by $aI+bP$. It
suffices to note that

$H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(aI+bP))&={\mathcal{O}}(aI+bP)\,{{\mathcal{O}}(aI+bP^{\rm
T})}^{\rm T}\\\ =B_{a,b}\,{\mathcal{O}}(P)\,{{\mathcal{O}}(P^{\rm T})}^{\rm
T}\,(B_{a,b})^{\rm T}\\\
=B_{a,b}\,H_{\infty}(\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P))\,(B_{a,b})^{\rm
T}$ (2.23)

by Lemmas 2.11 and 2.13. Since the binomial matrix $B_{a,b}$ is totally
positive by Lemma 2.2, we conclude:

######  Corollary 2.16 (Hankel-total positivity of zeroth column, improved).

Let $P$ be an infinite matrix that is either row-finite or column-finite, with
entries in a partially ordered commutative ring $R$. Then $P$ is TPr
$\implies$ ${\mathcal{O}}(P)$ and ${\mathcal{O}}(P^{\rm T})$ are TPr
$\implies$
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)$
is Hankel-TPr $\iff$
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(aI+bP)$
is Hankel-TPr for all $a,b\geq 0$.

Once again, $a$ and $b$ can be nonnegative elements of $R$, or $a$ and $b$ can
be indeterminates and we work in the ring $R[a,b]$ equipped with the
coefficientwise order. Corollary 2.16 allows us to prove the Hankel-total
positivity of the output sequence
${\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(aI+bP)}$
whenever $P$ is TP, even if $aI+bP$ is not TP. It will play an important role
in our proof of Theorem 1.4.

Remark. We see from (LABEL:eq.proof.lemma.aI+bP.b) specialized to $k=0$ that
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(aI+bP)$
is a binomial transform of
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)$:

$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(aI+bP)_{n}\;=\;\sum_{j=0}^{n}a^{n-j}\,b^{j}\,\binom{n}{j}\,\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.6}{$\scriptscriptstyle\mathcal{O}$}}_{0}(P)_{j}\;.$
(2.24)

Now, it is known [89, 70] that the binomial transform preserves Hankel-TPr of
arbitrary sequences in a partially ordered commutative ring. So Corollary 2.16
is just a special case of this general result combined with Theorem 2.15.
$\blacksquare$

###  2.4 Binomial row-generating matrices

Let $A=(a_{nk})_{n,k\geq 0}$ be a row-finite matrix with entries in a
commutative ring $R$. (In most applications, including all those in the
present paper, the matrix $A$ will be lower-triangular.) We define its row-
generating polynomials in the usual way:

$A_{n}(x)\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{k=0}^{\infty}a_{nk}\,x^{k}\;,$ (2.25)

where the sum is actually finite because $A$ is row-finite. More generally,
let us define its binomial partial row-generating polynomials

$A_{n,k}(x)&\stackrel{{\scriptstyle\rm
def}}{{=}}\sum_{\ell=k}^{\infty}a_{n\ell}\,\binom{\ell}{k}\,x^{\ell-k}\\\
={1\over k!}\,{d^{k}\over dx^{k}}\,A_{n}(x)\;.$ (2.26)

(Note that the operator $(1/k!)\,d^{k}\\!/\\!dx^{k}$ has a well-defined action
on the polynomial ring $R[x]$ even if $R$ does not contain the rationals,
since $(1/k!)(d^{k}\\!/\\!dx^{k})x^{n}=\binom{n}{k}x^{n-k}$.) The polynomials
$A_{n,k}(x)$ are the matrix elements of the binomial row-generating matrix
$AB_{x}$:

$(AB_{x})_{nk}\;=\;A_{n,k}(x)\;,$ (2.27)

where $B_{x}=B_{x,1}$ is the weighted binomial matrix defined in (1.13). The
zeroth column of the matrix $AB_{x}$ consists of the row-generating
polynomials $A_{n}(x)=A_{n,0}(x)$.

In this paper the matrix $A$ will be either the Laguerre coefficient matrix
${\sf L}^{(\alpha)}$ or one of its multivariate generalizations.

We can now explain the method that we will use to prove Theorems 1.1 and 1.4:

######  Proposition 2.17.

Let $P$ be a row-finite matrix with entries in a partially ordered commutative
ring $R$, and let $A={\mathcal{O}}(P)$.

  * (a)

If $P$ is totally positive of order $r$, then so is $A$.

  * (b)

If the matrix $B_{x}^{-1}PB_{x}$ is totally positive of order $r$ in the ring
$R[x]$ equipped with the coefficientwise order, then the sequence
$(A_{n}(x))_{n\geq 0}$ of row-generating polynomials is Hankel-totally
positive of order $r$ in the ring $R[x]$ equipped with the coefficientwise
order.

Indeed, (a) is just a restatement of Theorem 2.9; and (b) is an immediate
consequence of Lemma 2.7 and Theorem 2.15 together with the fact that the
zeroth column of the matrix $AB_{x}$ consists of the row-generating
polynomials $A_{n}(x)$.

Remark. The binomial row-generating matrix $AB_{x}$ can also be considered as
a modified Wronskian matrix. To see this, let
$N\in{\mathbb{N}}\cup\\{\infty\\}$, and fix formal power series
$f_{1}(x),\ldots,f_{N}(x)\in R[[x]]$. We define the $N\times\infty$ Wronskian
matrix

$W(f_{1},\ldots,f_{N})(x)\;=\;\big{(}f_{n}^{(k)}(x)\bigr{)}_{1\leq n\leq
N,\,k\geq 0}$ (2.28)

(where (k) denotes the $k$th derivative) and the $N\times\infty$ modified
Wronskian matrix

$\widetilde{W}(f_{1},\ldots,f_{N})(x)\;=\;\big{(}f_{n}^{(k)}(x)/k!\bigr{)}_{1\leq
n\leq N,\,k\geq 0}\;;$ (2.29)

these are matrices with entries in $R[[x]]$. Of course, we have

$W(f_{1},\ldots,f_{N})(x)\;=\;\widetilde{W}(f_{1},\ldots,f_{N})(x)\>D$ (2.30)

where $D=\mathop{\rm diag}\nolimits\big{(}(k!)_{k\geq 0}\big{)}$. The key fact
is that

$\widetilde{W}(f_{1},\ldots,f_{N})(x)\;=\;\widetilde{W}(f_{1},\ldots,f_{N})(0)\>B_{x}\;.$
(2.31)

In particular, if the ring $R$ carries a partial order, the total positivity
of order $r$ of the matrix $\widetilde{W}(f_{1},\ldots,f_{N})(0)$ in the ring
$R$ implies the total positivity of order $r$ of the matrix
$\widetilde{W}(f_{1},\ldots,f_{N})(x)$ in the ring $R[[x]]$ equipped with the
coefficientwise order. $\blacksquare$

###  2.5 Exponential Riordan arrays

Let $R$ be a commutative ring containing the rationals, and let
$F(t)=\sum_{n=0}^{\infty}f_{n}t^{n}/n!$ and
$G(t)=\sum_{n=1}^{\infty}g_{n}t^{n}/n!$ be formal power series with
coefficients in $R$; we set $g_{0}=0$. Then the exponential Riordan array [24,
23, 5] associated to the pair $(F,G)$ — or equivalently to the pair of
sequences ${\bm{f}}=(f_{n})_{n\geq 0}$ and ${\bm{g}}=(g_{n})_{n\geq 1}$ — is
the infinite lower-triangular matrix
${\mathcal{R}}[F,G]=({\mathcal{R}}[F,G]_{nk})_{n,k\geq 0}$ defined by

${\mathcal{R}}[F,G]_{nk}\;=\;{n!\over k!}\>[t^{n}]\,F(t)G(t)^{k}\;.$ (2.32)

That is, the $k$th column of ${\mathcal{R}}[F,G]$ has exponential generating
function $F(t)G(t)^{k}/k!$. It follows that the bivariate exponential
generating function of ${\mathcal{R}}[F,G]$ is

$\sum_{n,k=0}^{\infty}{\mathcal{R}}[F,G]_{nk}\>{t^{n}\over
n!}\,u^{k}\;=\;F(t)\,e^{uG(t)}\;.$ (2.33)

Please note that the diagonal elements of ${\mathcal{R}}[F,G]$ are
${\mathcal{R}}[F,G]_{nn}=f_{0}g_{1}^{n}$, so the matrix ${\mathcal{R}}[F,G]$
is invertible in the ring $R^{{\mathbb{N}}\times{\mathbb{N}}}_{\rm lt}$ of
lower-triangular matrices if and only if $f_{0}$ and $g_{1}$ are invertible in
$R$.

We shall use an easy but important result that is sometimes called the
_fundamental theorem of exponential Riordan arrays_ (FTERA):

######  Lemma 2.18 (Fundamental theorem of exponential Riordan arrays).

Let ${\bm{b}}=(b_{n})_{n\geq 0}$ be a sequence with exponential generating
function $B(t)=\sum_{n=0}^{\infty}b_{n}t^{n}/n!$. Considering ${\bm{b}}$ as a
column vector and letting ${\mathcal{R}}[F,G]$ act on it by matrix
multiplication, we obtain a sequence ${\mathcal{R}}[F,G]{\bm{b}}$ whose
exponential generating function is $F(t)\,B(G(t))$.

Proof. We compute

$\sum_{k=0}^{n}{\mathcal{R}}[F,G]_{nk}\,b_{k}&=\sum_{k=0}^{\infty}{n!\over
k!}\,[t^{n}]\,F(t)G(t)^{k}\,b_{k}\\\
=n!\>[t^{n}]\>F(t)\sum_{k=0}^{\infty}b_{k}\,{G(t)^{k}\over k!}\\\
=n!\>[t^{n}]\>F(t)\,B(G(t))\;.$ (2.34)

$\square$

Let us now consider the product of two exponential Riordan arrays
${\mathcal{R}}[F_{1},G_{1}]$ and ${\mathcal{R}}[F_{2},G_{2}]$. Applying the
FTERA to the $k$th column of ${\mathcal{R}}[F_{2},G_{2}]$, whose exponential
generating function is $F_{2}(t)G_{2}(t)^{k}/k!$, we readily obtain:

######  Lemma 2.19 (Product of two exponential Riordan arrays).

We have

${\mathcal{R}}[F_{1},G_{1}]\,{\mathcal{R}}[F_{2},G_{2}]\;=\;{\mathcal{R}}[(F_{2}\circ
G_{1})F_{1},\,G_{2}\circ G_{1}]$ (2.35)

where $\circ$ denotes composition of formal power series.

In particular, if we let ${\mathcal{R}}[F_{2},G_{2}]$ be the weighted binomial
matrix $B_{\xi}={\mathcal{R}}[e^{\xi t},t]$, we obtain:

######  Corollary 2.20 (Binomial row-generating matrix of an exponential
Riordan array).

We have

${\mathcal{R}}[F,G]\,B_{\xi}\;=\;{\mathcal{R}}[{\rm e}^{\xi G}F,G]\;.$ (2.36)

We can now determine the production matrix of an exponential Riordan array
${\mathcal{R}}[F,G]$. Let ${\bm{a}}=(a_{n})_{n\geq 0}$ and
${\bm{z}}=(z_{n})_{n\geq 0}$ be sequences in a commutative ring $R$, with
ordinary generating functions $A(s)=\sum_{n=0}^{\infty}a_{n}s^{n}$ and
$Z(s)=\sum_{n=0}^{\infty}z_{n}s^{n}$. We then define the exponential AZ matrix
associated to the sequences ${\bm{a}}$ and ${\bm{z}}$ to be the lower-
Hessenberg matrix with entries

$\mathop{\rm EAZ}\nolimits({\bm{a}},{\bm{z}})_{nk}\;=\;{n!\over
k!}\>(z_{n-k}\,+\,k\,a_{n-k+1})$ (2.37)

(where $z_{-1}\stackrel{{\scriptstyle\rm def}}{{=}}0$), or equivalently (if
$R$ contains the rationals)

$\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{z}})\;=\;D\,T_{\infty}({\bm{z}})\,D^{-1}\>+\>D\,T_{\infty}({\bm{a}})\,D^{-1}\,\Delta$
(2.38)

where $D=\mathop{\rm diag}\nolimits\bigl{(}(n!)_{n\geq 0}\bigr{)}$. We also
write $\mathop{\rm EAZ}\nolimits(A,Z)$ as a synonym for $\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{z}})$.

######  Theorem 2.21 (Production matrices of exponential Riordan arrays).

Let $L$ be a lower-triangular matrix (with entries in a commutative ring $R$
containing the rationals) with invertible diagonal entries and $L_{00}=1$, and
let $P=L^{-1}\Delta L$ be its production matrix. Then $L$ is an exponential
Riordan array if and only if $P$ is an exponential AZ matrix.

More precisely, $L={\mathcal{R}}[F,G]$ if and only if $P=\mathop{\rm
EAZ}\nolimits(A,Z)$, where the generating functions $\big{(}F(t),G(t)\big{)}$
and $\big{(}A(s),Z(s)\big{)}$ are connected by

$G^{\prime}(t)\;=\;A(G(t))\;,\qquad{F^{\prime}(t)\over F(t)}\;=\;Z(G(t))$
(2.39)

or equivalently

$A(s)\;=\;G^{\prime}(\bar{G}(s))\;,\qquad
Z(s)\;=\;{F^{\prime}(\bar{G}(s))\over F(\bar{G}(s))}$ (2.40)

where $\bar{G}(s)$ is the compositional inverse of $G(t)$.

Proof (mostly contained in [5, pp. 217–218]). Suppose that
$L={\mathcal{R}}[F,G]$. The hypotheses on $L$ imply that $f_{0}=1$ and that
$g_{1}$ is invertible in $R$; so $G(t)$ has a compositional inverse. Now let
$P=(p_{nk})_{n,k\geq 0}$ be a matrix; its column exponential generating
functions are, by definition, $P_{k}(t)=\sum_{n=0}^{\infty}p_{nk}\,t^{n}/n!$.
Applying the FTERA to each column of $P$, we see that ${\mathcal{R}}[F,G]P$ is
a matrix whose column exponential generating functions are
$\big{(}F(t)\,P_{k}(G(t))\big{)}_{k\geq 0}$. On the other hand,
$\Delta\,{\mathcal{R}}[F,G]$ is the matrix ${\mathcal{R}}[F,G]$ with its
zeroth row removed and all other rows shifted upwards, so it has column
exponential generating functions

${d\over dt}\,\big{(}F(t)\,G(t)^{k}/k!\big{)}\;=\;{1\over
k!}\>\Big{[}F^{\prime}(t)\,G(t)^{k}\>+\>k\,F(t)\,G(t)^{k-1}\,G^{\prime}(t)\Big{]}\;.$
(2.41)

Comparing these two results, we see that
$\Delta\,{\mathcal{R}}[F,G]={\mathcal{R}}[F,G]\,P$ if and only if

$P_{k}(G(t))\;=\;{1\over
k!}\>{F^{\prime}(t)\,G(t)^{k}\>+\>k\,F(t)\,G(t)^{k-1}\,G^{\prime}(t)\over
F(t)}\;,$ (2.42)

or in other words

$P_{k}(t)\;=\;{1\over k!}\>\biggl{[}{F^{\prime}(\bar{G}(t))\over
F(\bar{G}(t))}\,t^{k}\>+\>k\,t^{k-1}\,G^{\prime}(\bar{G}(t))\biggr{]}\;.$
(2.43)

Therefore

$p_{nk}&={n!\over k!}\>[t^{n}]\,\biggl{[}{F^{\prime}(\bar{G}(t))\over
F(\bar{G}(t))}\,t^{k}\>+\>k\,t^{k-1}\,G^{\prime}(\bar{G}(t))\biggr{]}\\\
={n!\over k!}\>\biggl{[}[t^{n-k}]\>{F^{\prime}(\bar{G}(t))\over
F(\bar{G}(t))}\>+\>k\,[t^{n-k+1}]\>G^{\prime}(\bar{G}(t))\biggr{]}\\\
={n!\over k!}\>(z_{n-k}\,+\,k\,a_{n-k+1})$ (2.44)

where ${\bm{a}}=(a_{n})_{n\geq 0}$ and ${\bm{z}}=(z_{n})_{n\geq 0}$ are given
by (2.40).

Conversely, suppose that $P=\mathop{\rm EAZ}\nolimits(A,Z)$. Define $F(t)$ and
$G(t)$ as the unique solutions (in the formal-power-series ring $R[[t]]$) of
the differential equations (2.39) with initial conditions $F(0)=1$ and
$G(0)=0$. Then running the foregoing computation backwards shows that
$\Delta\,{\mathcal{R}}[F,G]={\mathcal{R}}[F,G]\,P$. $\square$

A central role will be played later in this paper by a simple but remarkable
identity for $B_{x}^{-1}\mathop{\rm EAZ}\nolimits({\bm{a}},{\bm{z}})\,B_{x}$,
where $B_{x}$ is the $x$-binomial matrix defined in (1.13):

######  Lemma 2.22 (Identity for $B_{x}^{-1}\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{z}})\,B_{x}$).

Let ${\bm{a}}=(a_{n})_{n\geq 0}$, ${\bm{z}}=(z_{n})_{n\geq 0}$ and $x$ be
indeterminates. Then

$B_{x}^{-1}\mathop{\rm EAZ}\nolimits({\bm{a}},{\bm{z}})\,B_{x}\;=\;\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{z}}+x{\bm{a}})$ (2.45)

as an identity in ${\mathbb{Z}}[{\bm{a}},{\bm{z}}]$.

The special case ${\bm{z}}={\bm{0}}$ of this lemma was proven in [58, Lemma
3.6]; a simpler proof was given in [67, Lemma 2.16]. Here we give the easy
generalization to include ${\bm{z}}$, taken from [13]:

Proof. We work temporarily in the ring ${\mathbb{Q}}[{\bm{a}},{\bm{z}}]$ and
use the matrix definition (2.38):

$\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{z}})\;=\;D\,T_{\infty}({\bm{z}})\,D^{-1}\>+\>D\,T_{\infty}({\bm{a}})\,D^{-1}\,\Delta$
(2.46)

where $D=\mathop{\rm diag}\nolimits\bigl{(}(n!)_{n\geq 0}\bigr{)}$. Since
$\mathop{\rm EAZ}\nolimits({\bm{a}},{\bm{z}})=\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{0}})+\mathop{\rm
EAZ}\nolimits({\bm{0}},{\bm{z}})$, it suffices to consider separately the two
contributions.

The key observation is that $B_{x}=D\,T_{\infty}\big{(}(x^{n}/n!)_{n\geq
0}\big{)}\,D^{-1}$. Now two Toeplitz matrices always commute:
$T_{\infty}({\bm{a}})\,T_{\infty}({\bm{b}})=T_{\infty}({\bm{a}}\star{\bm{b}})=T_{\infty}({\bm{b}})\,T_{\infty}({\bm{a}})$.
It follows that $DT_{\infty}({\bm{z}})D^{-1}$ and
$DT_{\infty}({\bm{a}})D^{-1}$ commute with $B_{x}$. Therefore

$B_{x}^{-1}\mathop{\rm EAZ}\nolimits({\bm{0}},{\bm{z}})\,B_{x}\;=\;\mathop{\rm
EAZ}\nolimits({\bm{0}},{\bm{z}})\;.$ (2.47)

On the other hand, the classic recurrence for binomial coefficients implies

$\Delta B_{x}\;=\;B_{x}\,(xI+\Delta)$ (2.48)

(cf. Example 2.10). Therefore

$B_{x}^{-1}\mathop{\rm
EAZ}\nolimits({\bm{a}},{\bm{0}})B_{x}&=B_{x}^{-1}\,DT_{\infty}({\bm{a}})D^{-1}\,\Delta\,B_{x}\\\
=B_{x}^{-1}\,DT_{\infty}({\bm{a}})D^{-1}\,B_{x}\,(xI+\Delta)\\\
=DT_{\infty}({\bm{a}})D^{-1}\,(xI+\Delta)\\\ =\mathop{\rm
EAZ}\nolimits({\bm{a}},x{\bm{a}})\;.$ (2.49)

Adding (2.47) and (2.49) yields (2.45). $\square$

The identity (2.45) can alternatively be proven by combining Lemma 2.7 and
Corollary 2.20 with Theorem 2.21: see [13] for details.

##  3 Production matrices: Univariate case

In this section we will prove Proposition 1.2, which gives the production
matrices for the Laguerre coefficient matrix ${\sf L}^{(\alpha)}$ [defined in
(1.12)] and for the binomial row-generating matrix ${\sf L}^{(\alpha)}B_{x}$.
The proofs are in fact quite easy. We also give a simple direct proof of the
coefficientwise total positivity of the Laguerre coefficient matrix ${\sf
L}^{(-1+\lambda)}$,

###  3.1 Coefficient matrix ${\sf L}^{\bm{(\alpha)}}$: Proof of Proposition
1.2(a)

We begin by proving Proposition 1.2(a), which asserts that the production
matrix for the Laguerre coefficient matrix ${\sf L}^{(\alpha)}$ is the
tridiagonal unit-lower-Hessenberg matrix $P^{\circ}=(p^{\circ}_{ij})_{i,j\geq
0}$ defined by

$p^{\circ}_{n,n+1}&=1\\\ p^{\circ}_{n,n}=2n+1+\alpha\\\
p^{\circ}_{n,n-1}=n(n+\alpha)\\\ p^{\circ}_{n,k}=0\qquad\textrm{if $k<n-1$ or
$k>n+1$}$ (3.1)

We give two proofs: one by direct computation, and one using the theory of
exponential Riordan arrays.

First Proof. Recall that the Laguerre coefficient matrix ${\sf L}^{(\alpha)}$
has entries

$\ell^{(\alpha)}_{n,k}\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\binom{n}{k}\,(\alpha+1+k)^{\overline{n-k}}\;.$ (3.2)

We need to verify that
$\sum_{j}\ell^{(\alpha)}_{n,j}\,p^{\circ}_{j,k}=\ell^{(\alpha)}_{n+1,k}$, or
in other words that

$\ell^{(\alpha)}_{n,k-1}\>+\>(2k+1+\alpha)\ell^{(\alpha)}_{n,k}\>+\>(k+1)(k+1+\alpha)\ell^{(\alpha)}_{n,k+1}\;=\;\ell^{(\alpha)}_{n+1,k}\;.$
(3.3)

This is a straightforward computation. $\square$

Second Proof. It follows from (1.10) and (2.33) that the matrix ${\sf
L}^{(\alpha)}$ is an exponential Riordan array ${\mathcal{R}}[F,G]$ with
$F(t)=(1-t)^{-(1+\alpha)}$ and $G(t)=t/(1-t)$. Since

$G^{\prime}(t)\;=\;[1+G(t)]^{2}\;,$ (3.4)

from (2.39) we have

$A(s)\;=\;1+2s+s^{2}\;.$ (3.5)

Then

${F^{\prime}(t)\over F(t)}\;=\;{1+\alpha\over
1-t}\;=\;(1+\alpha)\,[1+G(t)]\;,$ (3.6)

so from (2.39) we have

$Z(s)\;=\;(1+\alpha)\,(1+s)\;.$ (3.7)

By Theorem 2.21, the production matrix for ${\sf L}^{(\alpha)}$ is the
exponential AZ matrix $\mathop{\rm EAZ}\nolimits(A,Z)$. Inserting (3.5)/(3.7)
into (2.37) yields (3.1). $\square$

We now wish to make an observation about the connection of the Laguerre
coefficient matrix ${\sf L}^{(\alpha)}$ with continued fractions. The zeroth
column of ${\sf L}^{(\alpha)}$ is the sequence of rising powers (= Stirling
cycle polynomials) $(\lambda^{\overline{n}})_{n\geq 0}$, where
$\lambda\stackrel{{\scriptstyle\rm def}}{{=}}\alpha+1$. And this sequence has
a well-known classical S-fraction for its ordinary generating function, which
was found more than two-and-a-half centuries ago by Euler [26, section 26]
[27]131313 The paper [26], which is E247 in Eneström’s [25] catalogue, was
probably written circa 1746; it was presented to the St. Petersburg Academy in
1753, and published in 1760. The paper [27], which is E616 in Eneström’s [25]
catalogue, was apparently presented to the St. Petersburg Academy in 1776, and
published posthumously in 1788. :

$\sum_{n=0}^{\infty}\lambda^{\overline{n}}\>t^{n}\;=\;\cfrac{1}{1-\cfrac{\lambda
t}{1-\cfrac{t}{1-\cfrac{(\lambda+1)t}{1-\cfrac{2t}{1-\cdots}}}}}$ (3.8)

with coefficients $\alpha_{2k-1}=\lambda+k-1$ and $\alpha_{2k}=k$. Now, any
S-fraction has an associated production matrix

$P\;=\;\begin{bmatrix}\alpha_{1}&1&&&\\\
\alpha_{1}\alpha_{2}&\alpha_{2}+\alpha_{3}&1&&\\\
&\alpha_{3}\alpha_{4}&\alpha_{4}+\alpha_{5}&1&\\\
&&\ddots&\ddots&\ddots\end{bmatrix}$ (3.9)

(see eq. (A.7) in Appendix A.1), which has as its output matrix the triangular
array of generalized Stieltjes–Rogers polynomials of the first kind. And the
production matrix corresponding to the coefficients
$\alpha_{2k-1}=\lambda+k-1$ and $\alpha_{2k}=k$ is precisely (3.1). In view of
Proposition 1.2(a), this shows:

######  Proposition 3.1.

The Laguerre coefficient matrix ${\sf L}^{(\alpha)}$ is the matrix ${\sf
S}=(S_{n,\ell}({\bm{\alpha}}))_{n,\ell\geq 0}$ of generalized Stieltjes–Rogers
polynomials of the first kind corresponding to the coefficients
$\alpha_{2k-1}=k+\alpha$ and $\alpha_{2k}=k$.

Remark. Proposition 3.1 looks at first sight a bit bizarre, because the
coefficient matrix for the monic orthogonal polynomials associated to a
Stieltjes moment sequence ${\bm{a}}=(a_{n})_{n\geq 0}$ is in general the
inverse matrix of the corresponding matrix ${\sf S}$ of generalized
Stieltjes–Rogers polynomials141414 This is a well-known fact: see, for
instance [79, p. III-2, Théorème 1]. And for a generalization to unit-lower-
Hessenberg production matrices that are not necessarily tridiagonal, see [68,
Proposition 3.2 and Remark 1 following it]. ; therefore, the coefficient
matrix for the monic unsigned Laguerre polynomials should be the unsigned
inverse matrix of ${\sf S}$, i.e.

${\sf S}^{\sharp}\;\stackrel{{\scriptstyle\rm def}}{{=}}\;Q\,{\sf S}^{-1}\,Q$
(3.10)

where $Q=((-1)^{i}\delta_{ij})_{i,j\geq 0}$ is the diagonal matrix of
alternating 1’s and $-1$’s. This apparent paradox is explained by the curious
fact that the matrix ${\sf L}^{(\alpha)}$ equals its own unsigned inverse:
that is,

${\sf L}^{(\alpha)}\;=\;Q\,({\sf L}^{(\alpha)})^{-1}\,Q\;.$ (3.11)

Proof. From (1.12) we see that ${\sf L}^{(\alpha)}=D\,B^{(\alpha)}\,D^{-1}$
where $B^{(\alpha)}$ is the generalized binomial matrix

$(B^{(\alpha)})_{nk}\;=\;\binom{n+\alpha}{n-k}$ (3.12)

and $D=\mathop{\rm diag}\nolimits\big{(}(n!)_{n\geq 0}\big{)}$. So it suffices
to prove that $B^{(\alpha)}$ equals its own unsigned inverse:

$\sum_{j}\binom{n+\alpha}{n-j}\,\binom{j+\alpha}{j-k}\,(-1)^{j-k}&=\binom{n+\alpha}{n-k}\,\sum_{j}\binom{n-k}{j-k}\,(-1)^{j-k}\\\
=\delta_{nk}$ (3.13)

(cf. [39, eq. (5.21)]). $\blacksquare$

###  3.2 Binomial row-generating matrix ${\sf L}^{\bm{(\alpha)}}\bm{B_{x}}$:
Proof of Proposition 1.2(b)

We now prove Proposition 1.2(b), which asserts that the production matrix for
the binomial row-generating matrix ${\sf L}^{(\alpha)}B_{x}$ is the
quadridiagonal unit-lower-Hessenberg matrix $P=(p_{ij})_{i,j\geq 0}$ defined
by

$p_{n,n+1}&=1\\\ p_{n,n}=(2n+1+\alpha)\,+\,x\\\
p_{n,n-1}=n(n+\alpha)\,+\,2nx\\\ p_{n,n-2}=n(n-1)x\\\
p_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$ (3.14)

This is an easy consequence of the theory of exponential Riordan arrays. In
the preceding subsection we saw that the production matrix $P^{\circ}$ for the
Laguerre coefficient matrix ${\sf L}^{(\alpha)}$ is the exponential AZ matrix
$\mathop{\rm EAZ}\nolimits(A,Z)$ where $A$ and $Z$ are given by (3.5)/(3.7).
By Lemma 2.7 the production matrix for ${\sf L}^{(\alpha)}B_{x}$ is
$P=B_{x}^{-1}P^{\circ}B_{x}$; and by Lemma 2.22 we have
$B_{x}^{-1}P^{\circ}B_{x}=\mathop{\rm EAZ}\nolimits(A,Z+xA)$. Using this
together with (3.5)/(3.7) and (2.37) yields (3.14).

In Appendix B we will explain why $P=B_{x}^{-1}P^{\circ}B_{x}$ is
quadridiagonal, by answering the more general question: Which lower-Hessenberg
matrices $P$ have the property that $B_{\xi}^{-1}PB_{\xi}$ is $(r,1)$-banded?

We also have an explicit formula for the binomial row-generating matrix:

######  Proposition 3.2 (Binomial row-generating matrix of the Laguerre
coefficient array).

The matrix elements of the binomial row-generating matrix ${\sf
L}^{(\alpha)}B_{x}$ are

$({\sf L}^{(\alpha)}B_{x})_{n,k}\;=\;{1\over k!}\,{d^{k}\over
dx^{k}}\,{{\mathcal{L}}_{n}^{(\alpha)}}(x)\;=\;\binom{n}{k}\,{\mathcal{L}}_{n-k}^{(\alpha+k)}(x)\;.$
(3.15)

Proof. The first equality is a special case of the general property (2.26) of
binomial row-generating polynomials. The second equality follows by repeated
use of the differentiation formula

${d\over
dx}{\mathcal{L}}_{n}^{(\alpha)}(x)\;=\;n{\mathcal{L}}_{n-1}^{(\alpha+1)}(x)\;.$
(3.16)

$\square$

It can also be shown, by direct computation using classical identities for the
Laguerre polynomials, that the matrix (3.15) is indeed the output matrix for
the production matrix (3.14). But since this proof is rather lengthy, we
refrain from showing it here.

###  3.3 Total positivity of the coefficient matrix $\bm{{\sf
L}^{(-1+\lambda)}}$

Finally, let us give a simple direct proof of the coefficientwise total
positivity of the Laguerre coefficient matrix ${\sf L}^{(-1+\lambda)}$. Start
from the binomial matrix $a_{nk}=\binom{n}{k}$, which is totally positive by
Lemma 2.2. Now use Lemma 2.3(b) with $x_{i}=\lambda+i-1$ for $i\geq 1$, which
are nonnegative elements of ${\mathbb{Z}}[\lambda]$. Then

$b_{nk}&=x_{k+1}x_{k+2}\,\cdots\,x_{n}\,a_{nk}\\\
=(\lambda+k)^{\overline{n-k}}\,\binom{n}{k}\;,$ (3.17)

which is (1.12) with $\alpha=-1+\lambda$.

##  4 Multivariate Laguerre polynomials:
Exponential generating functions and production matrices

In this section we compute the column and bivariate exponential generating
functions for the multivariate Laguerre coefficient matrix (1.35); we then use
these exponential generating functions, together with the theory of
exponential Riordan arrays, to deduce the corresponding production matrices.

###  4.1 Exponential generating functions

In this subsection we shall compute the column and bivariate exponential
generating functions for the coefficient matrix ${\widehat{{\sf
L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}})$ that was defined in (1.35), which enumerates Laguerre digraphs
according to the status of vertices as peaks, valleys, double ascents, double
descents or fixed points. Indeed, we shall go farther, by assigning different
weights for the vertices belonging to a cycle or to a path. For a Laguerre
digraph $G$, let us write ${\rm pcyc}(G)$, ${\rm vcyc}(G)$, ${\rm dacyc}(G)$,
${\rm ddcyc}(G)$, ${\rm fp}(G)$ for the number of peaks, valleys, double
ascents, double descents and fixed points that belong to a cycle of $G$, and
${\rm ppa}(G)$, ${\rm vpa}(G)$, ${\rm dapa}(G)$, ${\rm ddpa}(G)$ for the
number of peaks, valleys, double ascents and double descents that belong to a
path of $G$ (of course fixed points can only belong to a cycle). We then
assign weights ${y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}}$ to the vertices belonging to a cycle, and weights ${z_{\rm p}},{z_{\rm
v}},{z_{\rm da}},{z_{\rm dd}}$ to the vertices belonging to a path. We
therefore define the generalized second multivariate Laguerre coefficient
matrix

$\displaystyle\\!\\!\\!\\!{\widetilde{{\sf L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm
da}},{z_{\rm dd}})_{n,k}$ $\displaystyle\quad\stackrel{{\scriptstyle\rm
def}}{{=}}\sum_{G\in{\mathbf{LD}}_{n,k}}y_{\rm p}^{{\rm pcyc}(G)}y_{\rm
v}^{{\rm vcyc}(G)}y_{\rm da}^{{\rm dacyc}(G)}y_{\rm dd}^{{\rm ddcyc}(G)}y_{\rm
fp}^{{\rm fp}(G)}\,z_{\rm p}^{{\rm ppa}(G)}z_{\rm v}^{{\rm vpa}(G)}z_{\rm
da}^{{\rm dapa}(G)}z_{\rm dd}^{{\rm ddpa}(G)}\>\times$
$\displaystyle\qquad\qquad\quad\;\,\,(1+\alpha)^{{\rm cyc}(G)}\>.\qquad$ (4.1)

This polynomial is homogeneous of degree $n$ in ${y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm
da}},{z_{\rm dd}}$.

Because of the 0–0 boundary conditions, each path contains at least one peak;
so we can, if we wish, remove a factor $z_{\rm p}^{k}$ and define a unit-
lower-triangular matrix by

${\widetilde{{\sf L}}}^{(\alpha)\flat}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm
dd}})_{n,k}\;\stackrel{{\scriptstyle\rm def}}{{=}}\;{\widetilde{{\sf
L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}})_{n,k}/z_{\rm
p}^{k}\;.$ (4.2)

This polynomial is homogeneous of degree $n-k$ in ${y_{\rm p}},{y_{\rm
v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm
da}},{z_{\rm dd}}$.

We now proceed to compute the exponential generating functions for the
matrices (4.1) and (4.2). We do this by combining the known exponential
generating functions for permutations with cyclic statistics [84, Théorème 1]
and permutations with linear statistics [84, Proposition 4]. These formulae
are as follows:

1) Permutations with cyclic statistics are enumerated by the polynomials

$P_{n}^{\rm cyc}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}},\lambda)\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{\sigma\in{\mathfrak{S}}_{n}}y_{\rm p}^{{\rm
pcyc}(\sigma)}y_{\rm v}^{{\rm vcyc}(\sigma)}y_{\rm da}^{{\rm
dacyc}(\sigma)}y_{\rm dd}^{{\rm ddcyc}(\sigma)}y_{\rm fp}^{{\rm
fp}(\sigma)}\,\lambda^{{\rm cyc}(\sigma)}$ (4.3)

where ${\rm pcyc}(\sigma)$, ${\rm vcyc}(\sigma)$, ${\rm dacyc}(\sigma)$, ${\rm
ddcyc}(\sigma)$, ${\rm fp}(\sigma)$ denote the number of cycle peaks, cycle
valleys, cycle double rises, cycle double falls and fixed points in $\sigma$,
and ${\rm cyc}(\sigma)$ denotes the number of cycles in $\sigma$. By
convention we set $P_{0}^{\rm cyc}=1$. The polynomial $P_{n}^{\rm cyc}$ is
homogeneous of degree $n$ in ${y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm
dd}},{y_{\rm fp}}$. We write

$F(t;{y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}},\lambda)\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{n=0}^{\infty}P_{n}^{\rm cyc}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}},\lambda)\>{t^{n}\over n!}$ (4.4)

for the corresponding exponential generating function.

######  Lemma 4.1.

[84, Théorème 1] We have

$F(t;{y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}},\lambda)&=e^{\lambda{y_{\rm fp}}t}\,\biggl{(}{r_{1}\,-\,r_{2}\over
r_{1}e^{r_{2}t}\,-\,r_{2}e^{r_{1}t}}\biggr{)}^{\\!\lambda}\\\ =F(t;{y_{\rm
p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}},1)^{\lambda}$ (4.5)

where $r_{1}r_{2}={y_{\rm p}}{y_{\rm v}}$ and $r_{1}+r_{2}={y_{\rm
da}}+{y_{\rm dd}}$. Otherwise put, $r_{1}$ and $r_{2}$ are the roots (in
either order) of the quadratic equation $\rho^{2}-({y_{\rm da}}+{y_{\rm
dd}})\rho+{y_{\rm p}}{y_{\rm v}}=0$. Concretely,

$r_{1,2}\;=\;{{y_{\rm da}}+{y_{\rm dd}}\pm\sqrt{({y_{\rm da}}+{y_{\rm
dd}})^{2}-4{y_{\rm p}}{y_{\rm v}}}\over 2}\;.$ (4.6)

2) Permutations with linear statistics and 0–0 boundary conditions are
enumerated by the polynomials

$P_{n}^{\rm lin(00)}({z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm
dd}})\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;\sum_{\sigma\in{\mathfrak{S}}_{n}}z_{\rm p}^{{\rm
ppa}(\sigma)}z_{\rm v}^{{\rm vpa}(\sigma)}z_{\rm da}^{{\rm
dapa}(\sigma)}z_{\rm dd}^{{\rm ddpa}(\sigma)}$ (4.7)

where ${\rm ppa}(\sigma)$, ${\rm vpa}(\sigma)$, ${\rm dapa}(\sigma)$, ${\rm
ddpa}(\sigma)$ denote the number of peaks, valleys, double ascents and double
descents in the permutation $\sigma$ written as a word
$\sigma_{1}\cdots\sigma_{n}$, where we impose the boundary conditions
$\sigma_{0}=\sigma_{n+1}=0$. By convention we restrict attention to $n\geq 1$.
The polynomial $P_{n}^{\rm lin(00)}$ is homogeneous of degree $n$ in ${z_{\rm
p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}}$. We write

$G(t;{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm
dd}})\;\stackrel{{\scriptstyle\rm def}}{{=}}\;\sum_{n=1}^{\infty}P_{n}^{\rm
lin(00)}({z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}})\>{t^{n}\over n!}$
(4.8)

for the corresponding exponential generating function (note that the sum
starts at $n=1$).

######  Lemma 4.2.

[84, Proposition 4] We have

$G(t;{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}})\;=\;{z_{\rm
p}}\>\biggl{(}{e^{r_{1}t}\,-\,e^{r_{2}t}\over
r_{1}e^{r_{2}t}\,-\,r_{2}e^{r_{1}t}}\biggr{)}$ (4.9)

where $r_{1}r_{2}={z_{\rm p}}{z_{\rm v}}$ and $r_{1}+r_{2}={z_{\rm
da}}+{z_{\rm dd}}$ analogously to Lemma 4.1. This function satisfies the
differential equation

$G^{\prime}(t)\;=\;{z_{\rm p}}\,+\,({z_{\rm da}}+{z_{\rm dd}})G(t)\,+\,{z_{\rm
v}}G(t)^{2}\;.$ (4.10)

We can now put these ingredients together to determine the exponential
generating functions for the matrix (4.1). A Laguerre digraph
$G\in{\mathbf{LD}}_{n,k}$ consists of a permutation (that is, a collection of
disjoint cycles) on some subset $S\subseteq[n]$ together with $k$ disjoint
paths on $[n]\setminus S$. Each of these paths can be considered as a
permutation written in word form. By the exponential formula, the exponential
generating function for the $k$th column of the matrix (4.1) is then

$\displaystyle\sum_{n=0}^{\infty}{\widetilde{{\sf L}}}^{(\alpha)}({y_{\rm
p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm
v}},{z_{\rm da}},{z_{\rm dd}})_{n,k}\>{t^{n}\over n!}$
$\displaystyle\qquad=\;F(t;{y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm
dd}},{y_{\rm fp}},1)^{1+\alpha}\;{G(t;{z_{\rm p}},{z_{\rm v}},{z_{\rm
da}},{z_{\rm dd}})^{k}\over k!}\;,\qquad\qquad$ (4.11)

where the $1/k!$ comes because the paths are indistinguishable. The bivariate
exponential generating function is therefore

$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}{\widetilde{{\sf
L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}})_{n,k}\>{t^{n}\over
n!}\,u^{k}$ $\displaystyle\qquad=\;F(t;{y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}},1)^{1+\alpha}\;\exp\bigl{[}u\,G(t;{z_{\rm
p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm dd}})\bigr{]}\;.\qquad\qquad$ (4.12)

Comparing this with (2.32)/(2.33) shows:

######  Proposition 4.3 (Generalized second multivariate Laguerre coefficient
matrix as exponential Riordan array).

The matrix ${\widetilde{{\sf L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm
da}},{y_{\rm dd}},{y_{\rm fp}},{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm
dd}})$, which was defined in (4.1), is an exponential Riordan array
${\mathcal{R}}[F,G]$ where $F$ is given by (4.5) with $\lambda=1+\alpha$ and
$G$ is given by (4.9).

For the matrix ${\widetilde{{\sf L}}}^{(\alpha)\flat}$ defined in (4.2), the
formulae are the same except that $G$ is replaced by

$G^{\flat}(t;{z_{\rm p}},{z_{\rm v}},{z_{\rm da}},{z_{\rm
dd}})\;\stackrel{{\scriptstyle\rm def}}{{=}}\;G(t;{z_{\rm p}},{z_{\rm
v}},{z_{\rm da}},{z_{\rm dd}})/{z_{\rm p}}\;=\;{e^{r_{1}t}\,-\,e^{r_{2}t}\over
r_{1}e^{r_{2}t}\,-\,r_{2}e^{r_{1}t}}\;,$ (4.13)

which satisfies the differential equation

$(G^{\flat})^{\prime}(t)\;=\;1\,+\,({z_{\rm da}}+{z_{\rm
dd}})G^{\flat}(t)\,+\,{z_{\rm p}}{z_{\rm v}}G^{\flat}(t)^{2}\;.$ (4.14)

Remark. For the first multivariate coefficient matrix (1.23), the formulae
simplify: taking ${y_{\rm p}}={y_{\rm dd}}={z_{\rm p}}={z_{\rm dd}}=v_{-}$,
${y_{\rm v}}={y_{\rm da}}={z_{\rm v}}={z_{\rm da}}=v_{-}$ and ${y_{\rm
fp}}=v_{0}$, we get $r_{1}=v_{+}$, $r_{2}=v_{-}$ and hence, using (1.37),

$\displaystyle\sum_{n=0}^{\infty}\sum_{k=0}^{n}{\sf
L}^{(\alpha)}(v_{-},v_{0},v_{+})_{n,k}\>{t^{n}\over n!}\,u^{k}$
$\displaystyle\qquad=\;e^{(1+\alpha)v_{0}t}\>\biggl{(}{v_{+}\,-\,v_{-}\over
v_{+}e^{v_{-}t}\,-\,v_{-}e^{v_{+}t}}\biggr{)}^{\\!1+\alpha}\>\exp\Biggl{[}u\,\biggl{(}{e^{v_{+}t}\,-\,e^{v_{-}t}\over
v_{+}e^{v_{-}t}\,-\,v_{-}e^{v_{+}t}}\biggr{)}\Biggr{]}\;.\qquad\qquad$ (4.15)

###  4.2 Production matrices: Proof of Proposition 1.5(a,b)

We now restrict attention to the case in which the vertices belonging to paths
and cycles are given the same weights, i.e. $({z_{\rm p}},{z_{\rm v}},{z_{\rm
da}},{z_{\rm dd}})=({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}})$. In
this situation it is easy to find the series $A(s)$ and $Z(s)$ that satisfy
the relations (2.39) leading to the production matrix of an exponential
Riordan array. Namely, from (4.10) we have immediately

$A(s)\;=\;{y_{\rm p}}\,+\,({y_{\rm da}}+{y_{\rm dd}})s\,+\,{y_{\rm
v}}s^{2}\;.$ (4.16)

And a straightforward computation shows that the logarithmic derivative of $F$
equals $\lambda{y_{\rm fp}}+\lambda{y_{\rm v}}G$, so we have

$Z(s)\;=\;\lambda{y_{\rm fp}}+\lambda{y_{\rm v}}s\;.$ (4.17)

Theorem 2.21 then yields:

######  Proposition 4.4 (Production matrix of the second multivariate Laguerre
coefficient matrix).

The second multivariate Laguerre coefficient matrix ${\widehat{{\sf
L}}}^{(\alpha)}({y_{\rm p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm
fp}})$, which was defined in (1.35), has production matrix
$P^{\circ}=\mathop{\rm EAZ}\nolimits(A,Z)$ where $A$ and $Z$ are given by
(4.16)/(4.17) with $\lambda=1+\alpha$. In detail, $P^{\circ}$ is the
tridiagonal matrix

$p^{\circ}_{n,n+1}&={y_{\rm p}}\\\ p^{\circ}_{n,n}=(1+\alpha){y_{\rm
fp}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})\\\ p^{\circ}_{n,n-1}=n(n+\alpha){y_{\rm
v}}\\\ p^{\circ}_{n,k}=0\qquad\textrm{if $k<n-1$ or $k>n+1$}$ (4.18)

From this it is straightforward to deduce the production matrix for the
binomial row-generating matrix ${\widehat{{\sf L}}}^{(\alpha)}({y_{\rm
p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}})\,B_{x}$. Namely, by
Lemma 2.7 this production matrix is $P=B_{x}^{-1}P^{\circ}B_{x}$; and by Lemma
2.22 we have $B_{x}^{-1}P^{\circ}B_{x}=\mathop{\rm EAZ}\nolimits(A,Z+xA)$. We
have therefore proven:

######  Proposition 4.5 (Production matrix of the second multivariate Laguerre
binomial row-generating matrix).

The binomial row-generating matrix ${\widehat{{\sf L}}}^{(\alpha)}({y_{\rm
p}},{y_{\rm v}},{y_{\rm da}},{y_{\rm dd}},{y_{\rm fp}})\,B_{x}$ has production
matrix $P=\mathop{\rm EAZ}\nolimits(A,Z+xA)$ where $A$ and $Z$ are given by
(4.16)/(4.17) with $\lambda=1+\alpha$. In detail, $P$ is the quadridiagonal
lower-Hessenberg matrix

$p_{n,n+1}&={y_{\rm p}}\\\ p_{n,n}=(1+\alpha){y_{\rm fp}}\,+\,n({y_{\rm
da}}+{y_{\rm dd}})\,+\,{y_{\rm p}}x\\\ p_{n,n-1}=n(n+\alpha){y_{\rm
v}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})x\\\ p_{n,n-2}=n(n-1){y_{\rm v}}x\\\
p_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$ (4.19)

For the matrix ${\widetilde{{\sf L}}}^{(\alpha)\flat}$, the formulae are the
same except that $A$ and $Z$ are replaced by

$\displaystyle A^{\flat}(s)$ $\displaystyle=$ $\displaystyle 1\,+\,({y_{\rm
da}}+{y_{\rm dd}})s\,+\,{y_{\rm p}}{y_{\rm v}}s^{2}$ (4.20) $\displaystyle
Z^{\flat}(s)$ $\displaystyle=$ $\displaystyle\lambda{y_{\rm
fp}}+\lambda{y_{\rm p}}{y_{\rm v}}s$ (4.21)

This leads to the production matrices

$p^{\circ\flat}_{n,n+1}&=1\\\ p^{\circ\flat}_{n,n}=(1+\alpha){y_{\rm
fp}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})\\\
p^{\circ\flat}_{n,n-1}=n(n+\alpha){y_{\rm p}}{y_{\rm v}}\\\
p^{\circ\flat}_{n,k}=0\qquad\textrm{if $k<n-1$ or $k>n+1$}$ (4.22)

and

$p^{\flat}_{n,n+1}&=1\\\ p^{\flat}_{n,n}=(1+\alpha){y_{\rm fp}}\,+\,n({y_{\rm
da}}+{y_{\rm dd}})\,+\,x\\\ p^{\flat}_{n,n-1}=n(n+\alpha){y_{\rm p}}{y_{\rm
v}}\,+\,n({y_{\rm da}}+{y_{\rm dd}})x\\\ p^{\flat}_{n,n-2}=n(n-1){y_{\rm
p}}{y_{\rm v}}x\\\ p^{\flat}_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$
(4.23)

This completes the proof of Proposition 1.5.

##  5 Total positivity of the production matrices:
Univariate case

In this section we will prove the total positivity of the production matrices
for the univariate Laguerre polynomials: namely, the tridiagonal production
matrix $P^{\circ}$ [defined in (1.44)] for the Laguerre coefficient matrix
${\sf L}^{(\alpha)}$, and the quadridiagonal production matrix $P$ [defined in
(1.45)] for the binomial row-generating matrix ${\sf L}^{(\alpha)}\,B_{x}$.
Throughout, we use the parameter $\lambda=1+\alpha$.

The proofs are easy factorizations; in the quadridiagonal case we will also
need to invoke the tridiagonal comparison theorem (Proposition 2.6).

###  5.1 The tridiagonal matrix $\bm{P^{\circ}}$: Proof of Proposition 1.3(a)

The tridiagonal matrix $P^{\circ}$ is given by

$p^{\circ}_{n,n+1}&=1\\\ p^{\circ}_{n,n}=2n+\lambda\\\
p^{\circ}_{n,n-1}=n(n-1+\lambda)\\\ p^{\circ}_{n,k}=0\qquad\textrm{if $k<n-1$
or $k>n+1$}$ (5.1)

As observed preceding Proposition 3.1, this is the production matrix (A.7)
corresponding to the classical S-fraction with coefficients
$\alpha_{2k-1}=k-1+\lambda$ and $\alpha_{2k}=k$ (see Appendix A.1). And as
observed in (A.8), the production matrix (A.7) can be factorized in the form
$LU$ where $L$ is the lower-bidiagonal matrix with 1 on the diagonal and
$\alpha_{2},\alpha_{4},\ldots$ on the subdiagonal, and $U$ is the upper-
bidiagonal matrix with 1 on the superdiagonal and
$\alpha_{1},\alpha_{3},\ldots$ on the diagonal. And finally, as also observed
there, this factorization shows, by Lemma 2.1, that the matrix (A.7) is
coefficientwise totally positive in the indeterminates ${\bm{\alpha}}$.
Putting this all together, we conclude:

######  Proposition 5.1 (Factorization and total positivity of $P^{\circ}$).

  * (a)

The tridiagonal matrix $P^{\circ}$ defined by (5.1) has the factorization
$P^{\circ}=LU$ where

    * $\bullet$

$L$ is the lower-bidiagonal matrix with 1 on the diagonal and $1,2,3,\ldots$
on the subdiagonal, and

    * $\bullet$

$U$ is the upper-bidiagonal matrix with 1 on the superdiagonal and
$\lambda,{\lambda+1},$ ${\lambda+2},\ldots$ on the diagonal.

  * (b)

$P^{\circ}$ is totally positive in the ring ${\mathbb{Z}}[\lambda]$ equipped
with the coefficientwise order.

In particular, this proves Proposition 1.3(a).

Note that we can also write $U=L\Delta+\lambda I$, since $L\Delta$ is the
upper-bidiagonal matrix with 1 on the superdiagonal and $0,1,2,\ldots$ on the
diagonal.

###  5.2 The quadridiagonal matrix $\bm{P}$: Proof of Proposition 1.3(b)

The quadridiagonal matrix $P$ is given by

$p_{n,n+1}&=1\\\ p_{n,n}=(2n+\lambda)\,+\,x\\\
p_{n,n-1}=n(n-1+\lambda)\,+\,2nx\\\ p_{n,n-2}=n(n-1)x\\\
p_{n,k}=0\qquad\textrm{if $k<n-2$ or $k>n+1$}$ (5.2)

######  Proposition 5.2 (Factorization and total positivity of $P$).

  * (a)

The quadridiagonal matrix $P$ defined by (5.2) has the factorization
$P={L(LU_{x}+\lambda I)}$ where

    * $\bullet$

$L$ is the lower-bidiagonal matrix with 1 on the diagonal and $1,2,3,\ldots$
on the subdiagonal, and

    * $\bullet$

$U_{x}=\Delta+xI$ is the upper-bidiagonal matrix with 1 on the superdiagonal
and $x$ on the diagonal.

  * (b)

$P$ is totally positive in the ring ${\mathbb{Z}}[x,\lambda]$ equipped with
the coefficientwise order.

Proof. (a) It is easy to see that $P=P^{\circ}+xLL$. Since
$P^{\circ}=L(L\Delta+\lambda I)$ by Proposition 5.1(a), we have
$P=L[L(\Delta+xI)+\lambda I]$, which proves part (a).

(b) By Lemma 2.1, the bidiagonal matrices $L$ and $U_{x}$ are totally positive
in the ring ${\mathbb{Z}}[x]$ equipped with the coefficientwise order. It
follows that the tridiagonal matrix $LU_{x}$ is also totally positive in the
ring ${\mathbb{Z}}[x]$ equipped with the coefficientwise order. Then the
tridiagonal comparison theorem (Proposition 2.6) shows that the tridiagonal
matrix $LU_{x}+\lambda I$ is totally positive in the ring
${\mathbb{Z}}[x,\lambda]$ equipped with the coefficientwise order. Therefore
the same holds for $P=L(LU_{x}+\lambda I)$. $\square$

This completes the proof of Proposition 1.3(b).

Remarks. 1\. For $\lambda=0$ ($\alpha=-1$), we have $P=LLU_{x}$, which is the
production matrix for the known 2-S-fraction for the Lah polynomials [58,
Theorem 1.5 with $r=2$], which has coefficients

$\alpha_{3n-1}&=x\\\ \alpha_{3n}=n\\\ \alpha_{3n+1}=n$ (5.3)

(For the production matrix associated to a 2-S-fraction, see [59, Propositions
7.2 and 8.2 and eqn. (7.8)] or the case $j=0$ of Proposition A.3 below.)
However, it turns out that this same production matrix has a one-parameter
family of distinct factorizations $P=L_{1}L_{2}U_{x}$, arising from the
2-S-fractions with

$\alpha_{3n-1}&=x\\\ \alpha_{3n}=c_{n}\,n\\\ \alpha_{3n+1}=(2-c_{n})\,n$ (5.4)

where

$c_{n}\;\stackrel{{\scriptstyle\rm def}}{{=}}\;{(n-1)\,-\,(n-2)\kappa\over
n\,-\,(n-1)\kappa}\quad\hbox{for }n\geq 1$ (5.5)

and $\kappa\in[0,1]$: see Proposition A.8 below. The case $\kappa=1$ is the
known 2-S-fraction; all the others appear to be new.

2\. It is easy to see that $L\Delta+I=\Delta L$ (this is a special case of
(A.23) below) and hence $LU_{x}+I=U_{x}L$. This implies that for $\lambda=1$
($\alpha=0$), we have $P=L(LU_{x}+I)=LU_{x}L$. This is the production matrix
for the generalized 2-Stieltjes–Rogers polynomials of type $j=1$ with the
coefficients (5.3): see Proposition A.3 for the general theory, and
Proposition A.12 for this application. $\blacksquare$

##  6 Total positivity of the production matrices:
Multivariate case

In this section we will prove Proposition 1.6 on the total positivity of the
production matrices for the multivariate Laguerre polynomials. The proof of
Proposition 1.6(a), dealing with the tridiagonal matrix $P^{\circ\flat}$, is
an easy factorization combined with an appeal to the tridiagonal comparison
theorem. By contrast, the proof of Proposition 1.6(b), dealing with the
quadridiagonal matrix $P^{\flat}$, is decidedly nontrivial.

###  6.1 The tridiagonal matrix $\bm{P^{\circ\flat}}$: Proof of Proposition
1.6(a)

Proof of Proposition 1.6(a).  Let $Q$ the matrix (A.8) with
$\alpha_{2k-1}=(k+\alpha){y_{\rm p}}$ and $\alpha_{2k}=k{y_{\rm v}}$ for
$k\geq 1$; note that these coefficients are nonnegative in the ring $R$
because of the hypotheses $\alpha\geq-1$, ${y_{\rm p}}\geq 0$ and ${y_{\rm
v}}\geq 0$. It then follows from Lemma 2.1 that $Q$ is totally positive in the
ring $R$. The matrix $Q$ has elements

$q_{n,n+1}&=1\\\ q_{n,n}=(1+\alpha){y_{\rm p}}\,+\,n({y_{\rm p}}+{y_{\rm
v}})\\\ q_{n,n-1}=n(n+\alpha){y_{\rm p}}{y_{\rm v}}\\\
q_{n,k}=0\qquad\textrm{if $k<n-1$ or $k>n+1$}$ (6.1)

Then $P^{\circ\flat}=Q+D$, where $D$ is the diagonal matrix with entries
$d_{n}={(1+\alpha)({y_{\rm fp}}-{y_{\rm p}})}+{n({y_{\rm da}}+{y_{\rm
dd}}-{y_{\rm p}}-{y_{\rm v}})}$. By hypothesis these entries are nonnegative
in the ring $R$, so the tridiagonal comparison theorem (Proposition 2.6)
implies that $P^{\circ\flat}$ is totally positive in the ring $R$. $\square$

###  6.2 The quadridiagonal matrix $\bm{P^{\flat}}$: Proof of Proposition
1.6(b)

We will now prove Proposition 1.6(b), which asserts the total positivity of
the production matrix (1.47) when the variables are substituted to elements of
a partially ordered commutative ring $R$ that satisfy $\lambda\geq 0$,
$\lambda{y_{\rm fp}}=\lambda{y_{\rm p}}$, ${y_{\rm p}}\geq 0$, ${y_{\rm
v}}\geq 0$, ${y_{\rm da}}+{y_{\rm dd}}\geq{y_{\rm p}}+{y_{\rm v}}$ and $x\geq
0$. We will do this by proving the coefficientwise total positivity of a much
more general quadridiagonal lower-Hessenberg matrix $P$, defined by

$P&\stackrel{{\scriptstyle\rm
def}}{{=}}L_{1}UL_{2}\,+\,L_{1}D_{1}\,+\,D_{2}L_{2}{}\\\
=L_{1}(UL_{2}\,+\,D_{1})\,+\,D_{2}L_{2}{}\\\
=(L_{1}U\,+\,D_{2})L_{2}\,+\,L_{1}D_{1}{}$ (6.2)

where

  * •

$L_{1}$ is the lower-bidiagonal matrix with the sequence $a_{0},a_{1},\ldots$
on the diagonal, the sequence $b_{1},b_{2},\ldots$ on the subdiagonal, and
zeroes elsewhere;

  * •

$U$ is the upper-bidiagonal matrix with the sequence $c_{1},c_{2},\ldots$ on
the superdiagonal, the sequence $d_{0},d_{1},\ldots$ on the diagonal, and
zeroes elsewhere;

  * •

$L_{2}$ is the lower-bidiagonal matrix with the sequence $e_{0},e_{1},\ldots$
on the diagonal, the sequence $f_{1},f_{2},\ldots$ on the subdiagonal, and
zeroes elsewhere;

  * •

$D_{1}$ is the diagonal matrix with entries $g_{0},g_{1},\ldots\,$;

  * •

$D_{2}$ is the diagonal matrix with entries $h_{0},h_{1},\ldots\,$;

and ${\mathbf{a}}=(a_{n})_{n\geq 0}$, ${\mathbf{b}}=(b_{n})_{n\geq 1}$,
${\mathbf{c}}=(c_{n})_{n\geq 1}$, ${\mathbf{d}}=(d_{n})_{n\geq 0}$,
${\mathbf{e}}=(e_{n})_{n\geq 0}$, ${\mathbf{f}}=(f_{n})_{n\geq 1}$,
${\mathbf{g}}=(g_{n})_{n\geq 0}$, and ${\mathbf{h}}=(h_{n})_{n\geq 0}$ are all
indeterminates. The entries in the $n$th row of $P=(p_{n,k})_{k\geq 0}$ are
given by

$p_{n,n+1}&=a_{n}c_{n+1}e_{n+1}\\\
p_{n,n}=a_{n}d_{n}e_{n}+b_{n}c_{n}e_{n}+a_{n}c_{n+1}f_{n+1}+a_{n}g_{n}+h_{n}e_{n}\\\
p_{n,n-1}=a_{n}d_{n}f_{n}+b_{n}c_{n}f_{n}+b_{n}d_{n-1}e_{n-1}+b_{n}g_{n-1}+h_{n}f_{n}\\\
p_{n,n-2}=b_{n}d_{n-1}f_{n-1}\\\ p_{n,k}=0\qquad\textrm{if $k<n-2$ or
$k>n+1$}$ (6.3)

where by definition $b_{0}=c_{0}=f_{0}=0$ and
$a_{n}=b_{n}=c_{n}=d_{n}=e_{n}=f_{n}=g_{n}=h_{n}=0$ whenever $n<0$. Our main
result is:

######  Theorem 6.1 (Total positivity of the generalized production matrix).

The matrix $P$ defined by (6.2)/(6.3) is totally positive, coefficientwise in
the indeterminates
${\mathbf{a}},{\mathbf{b}},{\mathbf{c}},{\mathbf{d}},{\mathbf{e}},{\mathbf{f}},{\mathbf{g}},{\mathbf{h}}$.

Proof of Proposition 1.6(b) assuming Theorem 6.1.  Specialize the matrix (6.3)
by setting

$&a_{n}\;=\;c_{n}\;=\;e_{n}\;=\;1\\\ b_{n}\;=\;n{y_{\rm v}}\\\
d_{n}\;=\;n{y_{\rm p}}\\\ f_{n}\;=\;x\\\ g_{n}\;=\;\lambda{y_{\rm
p}}\>(=\lambda{y_{\rm fp}})\\\ h_{n}\;=\;n({y_{\rm da}}+{y_{\rm dd}}-{y_{\rm
p}}-{y_{\rm v}})$ (6.4)

$\square$

Theorem 6.1 will be proven as follows. Define the matrix $Q=(q_{n,k})_{n,k\geq
0}$ by

$Q\;\stackrel{{\scriptstyle\rm
def}}{{=}}\;P\big{|}_{{\mathbf{h}}={\bm{0}}}\;=\;L_{1}(UL_{2}\,+\,D_{1})\;,$
(6.5)

or in other words

$q_{n,n+1}&=a_{n}c_{n+1}e_{n+1}\\\
q_{n,n}=a_{n}d_{n}e_{n}+b_{n}c_{n}e_{n}+a_{n}c_{n+1}f_{n+1}+a_{n}g_{n}\\\
q_{n,n-1}=a_{n}d_{n}f_{n}+b_{n}c_{n}f_{n}+b_{n}d_{n-1}e_{n-1}+b_{n}g_{n-1}\\\
q_{n,n-2}=b_{n}d_{n-1}f_{n-1}\\\ q_{n,k}=0\qquad\textrm{if $k<n-2$ or
$k>n+1$}$ (6.6)

Then

$P\;=\;Q\,+\,D_{2}L_{2}.$ (6.7)

We will begin by proving (Lemma 6.6) that $Q$ is coefficientwise totally
positive; this proof uses the factorization $Q=L_{1}(UL_{2}+D_{1})$ together
with the tridiagonal comparison theorem, in close analogy with Propositions
1.6(a) and 5.2. It follows that for every integer $m\geq 0$, the matrix
consisting of first $m$ rows of $Q$ is also coefficientwise totally positive.

The rest of the proof shows how to restore the terms in $P$ involving
${\mathbf{h}}$. In terms of the row vectors $({\bm{p}}_{n})_{n\geq 0}$,