# Full indefinite Stieltjes moment problem and Padé approximants

Volodymyr Derkach  Department of Mathematics
Vasyl′ Stus Donetsk National University
600-Richchya Str 21, Vinnytsia, 21021, Ukraine<EMAIL_ADDRESS>and  Ivan
Kovalyov  Department of Mathematics
Dragomanov National Pedagogical University
Kiev, Pirogova 9, 01601, Ukraine<EMAIL_ADDRESS>
###### Abstract.

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur
algorithm. Naturally associated with indefinite Stieltjes moment problem are
generalized Stieltjes continued fraction and a system of difference equations,
which, in turn, lead to factorization of resolvent matrices of indefinite
Stieltjes moment problem. A criterion for such a problem to be indeterminate
in terms of continued fraction is found and a complete description of its
solutions is given in the indeterminate case. Explicit formulae for diagonal
and sub-diagonal Padé approximants for formal power series corresponding to
indefinite Stieltjes moment problem and convergence results for Padé
approximants are presented.

###### Key words and phrases:

Indefinite Stieltjes moment problem, Generalized Stieltjes function,
Generalized Stieltjes polynomials, Schur algorithm, Resolvent matrix

###### 1991 Mathematics Subject Classification:

Primary 30E05; Secondary 15B57, 30B70, 46C20

The present research was supported by Ministry of Education and Science of
Ukraine (projects # 0118U003138, 0118U002060), and a grant of the Volkswagen
Foundation. V. D. gratefully acknowledges financial support by the German
Research Foundation (DFG), grant TR 903/22-1

##  1\. Introduction

The classical Stieltjes moment problem consists in the following: given a
sequence of real numbers $s_{j}$ $(j\in\mathbb{Z}_{+}:=\mathbb{N}\cup\\{0\\})$
find a positive measure $\sigma$ with a support on $\mathbb{R}_{+}$, such that

(1.1) $\int_{\mathbb{R}_{+}}t^{j}d\sigma(t)=s_{j},\quad j\in\mathbb{Z}_{+}.$

It follows easily from (1.1) that the inequalities

(1.2) $S_{n}:=\left(s_{i+j}\right)_{i,j=0}^{n-1}\geq 0\quad\mbox{ and }\quad
S_{n}^{+}:=\left(s_{i+j+1}\right)_{i,j=0}^{n-1}\geq 0,\quad
n\in\mathbb{Z}_{+},$

are necessary for solvability of the moment problem (1.1). Moreover, the
inequalities (1.2) are also sufficient for solvability of the moment problem
(1.1), see [1, Appendix, Section A1]. Let

(1.3) $D_{n}:=\mbox{det }S_{n}\quad\mbox{ and }\quad\quad D_{n}^{+}:=\mbox{det
}S_{n}^{+},\quad n\in\mathbb{Z}_{+}.$

For every solution $\sigma$ of the Stieltjes moment problem its Stieltjes
transform

(1.4) $f(z)=\int_{\mathbb{R}_{+}}\frac{d\sigma(t)}{t-z}\qquad
z\in\mathbb{C}\backslash\mathbb{R}_{+}$

belongs to the class $\mathbf{N}$ of functions holomorphic on
$\mathbb{C}\backslash\mathbb{R}$ with nonnegative imaginary part in
$\mathbb{C}_{+}$ and such that $f(\overline{z})=\overline{f(z)}$ for
$z\in\mathbb{C}_{+}$. Moreover, $f$ belongs to the Stieltjes class
$\mathbf{S}$ of functions $f\in\mathbf{N}$, which admit holomorphic and
nonnegative continuation to $\mathbb{R}_{-}$. By M.G. Krein’s criterion [27]

$f\in\mathbf{S}\Longleftrightarrow f\in\mathbf{N}\quad\mbox{and}\quad
zf\in\mathbf{N}.$

Notice, that by the Hamburger–Nevanlinna theorem [1, Theorem 3.2.1] the
Stieltjes moment problem can be reformulated as the following interpolation
problem at $\infty$ for the Stieltjes transform
$f(z)=\int_{\mathbb{R}_{+}}\frac{d\sigma(t)}{t-z}$ of the measure $\sigma$

(1.5)
$f(z)=-\frac{s_{0}}{z}-\frac{s_{1}}{z^{2}}-\cdots-\frac{s_{2n}}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right),\quad\quad
z\widehat{\rightarrow}\infty.$

The notation $z\widehat{\rightarrow}\infty$ means that $z\rightarrow\infty$
nontangentially, that is inside the sector $\varepsilon<\arg
z<\pi-\varepsilon$ for some $\varepsilon>0$.

In the pioneering paper [40] by T. Stieltjes it was shown that if $D_{n},$
$D_{n}^{+}$ are positive for all $n\in{\mathbb{Z}}_{+}$ and

(1.6) $l_{n}:=\frac{D_{n}^{2}}{D_{n}^{+}D_{n-1}^{+}},\quad
m_{n}:=\frac{(D_{n-1}^{+})^{2}}{D_{n}D_{n-1}},\quad n\in\mathbb{N}.$

then the Laurent series $-\sum_{j=0}^{\infty}\frac{s_{j}}{z^{j+1}}$ can be
uniquely developed in the continued fraction

(1.7) $\displaystyle\frac{\displaystyle 1}{\displaystyle-
zm_{1}+\displaystyle\frac{1}{\displaystyle l_{1}+\dots\frac{1}{\displaystyle-
zm_{n}+\frac{1}{\displaystyle l_{n}+\dots}}}}.$

The moment problem (1.1) is called determinate, if it has unique solution, and
indeterminate otherwise. A solution $\sigma$ of (1.1) is called canonical, if
the set of polynomials is dense in $L^{2}(\sigma)$. As was shown in [40] the
moment problem (1.1) is indeterminate, if and only if

(1.8) $M:=\sum_{i=1}^{\infty}m_{i}<\infty\quad\mbox{and}\quad
L:=\sum_{i=1}^{\infty}l_{i}<\infty.$

As was shown in [27, Appendix, $13.2^{\circ}$], in the indeterminate case
there exists an entire matrix valued function $W(z)=(w_{i,j}(z))_{i,j=1}^{2}$
such that the set of all solutions of the problem (1.1) can be parametrized by
the formula

(1.9)
$\int_{\mathbb{R}_{+}}\frac{d\sigma(t)}{t-z}=T_{W(z)}[\tau(z)]:=\frac{w_{11}(z)\tau(z)+w_{12}(z)}{w_{21}(z)\tau(z)+w_{22}(z)},$

where $\tau$ ranges over the Stieltjes class $\mathbf{S}$. A subset of its
canonical solutions was described already in [40].

Now let us remind the indefinite versions of the classes $\mathbf{N}$ and
$\mathbf{S}$, see [30].

######  Definition 1.1.

([30]) A function $f$ meromorphic on $\mathbb{C}\backslash\mathbb{R}$ with the
set of holomorphy ${\mathfrak{h}}_{f}$ is said to be in the generalized
Nevanlinna class $\mathbf{N}_{\kappa}$ $(\kappa\in\mathbb{N})$, if for every
set $z_{j}\in\mathbb{C}_{+}\cap{\mathfrak{h}}_{f}$
($z_{i}\neq\overline{z}_{j}$, $i,j=1,\ldots,n$) the form

$\sum_{i,j=1}^{n}\frac{f(z_{i})-\overline{f(z_{j})}}{z_{i}-\overline{z}_{j}}\xi_{i}\overline{\xi}_{j}$

has at most $\kappa$ and for some choice of $z_{j}$ ($j=1,\ldots,n$) exactly
$\kappa$ negative squares. A functions $f\in\mathbf{N}_{\kappa}$ is said to
belong to the generalized Stieltjes class $\mathbf{N}_{\kappa}^{+}$, if
$zf\in\mathbf{N}$.

Similarly, in [9, 10] the class $\mathbf{N}_{\kappa}^{k}$
($\kappa,k\in\mathbb{N}$) was introduced as the set of functions
$f\in\mathbf{N}_{\kappa}$, such that $zf(z)$ belongs to the class
$\mathbf{N}_{k}$, see also [18, 20], where the class $\mathbf{N}_{0}^{k}$ was
studied.

In the present paper we consider the following problems.

Full indefinite moment problem $MP_{\kappa}(\textbf{s})$. Given
$\kappa\in\mathbb{Z}_{+}$, and an infinite sequence
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}$ of real numbers, describe
the set $\mathcal{M}_{\kappa}({\mathbf{s}})$ of functions
$f\in\mathbf{N}_{\kappa}$, which satisfy (1.10) for all $\ell\in{\mathbb{N}}$.

Full indefinite moment problem $MP_{\kappa}^{k}(\textbf{s})$. Given
$\kappa,k\in\mathbb{Z}_{+}$, and an infinite sequence
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}$of real numbers, describe
the set
$\mathcal{M}_{\kappa}^{k}({\mathbf{s}}):=\mathcal{M}_{\kappa}({\mathbf{s}})\cap\mathbf{N}_{\kappa}^{k}$.

Indefinite moment problems $MP_{\kappa}(\textbf{s})$ and
$MP_{\kappa}^{+}(\textbf{s})$ were studied in [31], [32] by the methods of
extension theory of Pontryagin space symmetric operators developed in [30],
[31]. In particular, it was shown in [31] that the moment problem
$MP^{+}_{\kappa}(\textbf{s})$ is solvable if and only if the number
$\nu_{-}(S_{n})$ of negative eigenvalues of $S_{n}$ does not exceed $\kappa$
and $S_{n}^{+}>0$ for all $n\in\mathbb{N}$. Further applications of the
operator approach to the moment problem $MP_{\kappa}^{k}(\textbf{s})$ were
given in [10]. A reproducing kernel approach to the moment problems
$MP_{\kappa}(\textbf{s})$ was presented in [21]. A step-by-step algorithm of
solving the moment problems $MP_{\kappa}(\textbf{s})$ was elaborated in [6],
[7] and [2]. Applications of the Schur algorithm to degenerate moment problem
in the class $\mathbf{N}_{\kappa}$ were given in [14].

The step-by-step algorithm reduces the moment problems
$MP_{\kappa}(\textbf{s})$ and $MP_{\kappa}^{k}(\textbf{s})$ to the following
truncated moment problems:

Truncated indefinite moment problem $MP_{\kappa}(\textbf{s},\ell)$. Given
$\ell,\kappa\in\mathbb{Z}_{+}$, and a finite sequence
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\ell}$ of real numbers, describe the
set $\mathcal{M}_{\kappa}({\mathbf{s}},\ell)$ of functions
$f\in\mathbf{N}_{\kappa}$, which satisfy the asymptotic expansion

(1.10)
$f(z)=-\frac{s_{0}}{z}-\cdots-\frac{s_{\ell}}{z^{\ell+1}}+o\left(\frac{1}{z^{\ell+1}}\right),\quad(z=iy,\,\,y{\uparrow}\infty).$

Truncated indefinite moment problem $MP_{\kappa}^{k}(\textbf{s},\ell)$. Given
$\ell,\kappa,k\in\mathbb{Z}_{+}$, and a sequence
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\ell}$ of real numbers, describe the
set $\mathcal{M}_{\kappa}^{k}({\mathbf{s}},\ell)$ of functions
$f\in\mathbf{N}_{\kappa}^{k}$, which satisfy (1.10). A truncated moment
problem is called even or odd regarding to the oddness of the number $\ell+1$
of given moments.

Let ${\mathcal{H}}$ be the set of all infinite real sequences ${\bf
s}=\\{s_{j}\\}_{j=0}^{\infty}$ and let ${\mathcal{H}}_{\kappa}$ be the set of
sequences ${\bf s}\in{\mathcal{H}}$, such that

(1.11) $\nu_{-}(S_{n})=\kappa\quad\mbox{for all $n$ big enough}$

Denote by ${\mathcal{H}}_{\kappa}^{k}$ the set of real sequences ${\bf
s}\in{\mathcal{H}}_{\kappa}$, such that
$\\{s_{j+1}\\}_{j=0}^{\infty}\in{\mathcal{H}}_{k}$, i.e.

(1.12) $\nu_{-}(S_{n}^{+})=k\quad\mbox{for all $n$ big enough}.$

A number $n_{j}\in\mathbb{N}$ is said to be a normal index of the sequence
${\bf s}\in{\mathcal{H}}$, if det$\,S_{n_{j}}\neq 0$. The ordered set of
normal indices

$n_{1}<n_{2}<\dots<n_{N}$

of the sequence ${\bf s}$ is denoted by $\mathcal{N}({\bf s})$.

As was shown in [7] for every ${\bf s}\in{\mathcal{H}}_{\kappa}$ there exists
a sequence of real numbers $b_{i}\in{\mathbb{R}}\backslash\\{0\\}$,
$i\in\mathbb{N}$ and real monic polynomials

(1.13)
$a_{i}(z)=z^{\ell_{i}}+a_{\ell_{i}-1}^{(i)}z^{\ell_{i}-1}+\ldots+a_{1}^{(i)}z+a_{0}^{(i)}$

of degree $\ell_{i}=n_{i+1}-n_{i}$, $i\in\mathbb{N}$, such that the
convergents of the continued fraction

(1.14)
$\frac{-b_{0}}{\displaystyle{a}_{0}(z)-\frac{b_{1}}{\displaystyle{a}_{1}(z)-{\dots-\frac{b_{n}}{a_{n}(z)-\dots}}}}$

for sufficiently large $n$ have the asymptotic expansion (1.10) for every
$\ell\in{\mathbb{N}}$. This fact was known already to L. Kronecker [33] and
then it was reinvented in [6]. The pairs $(a_{i},b_{i})$ are called atoms, see
[24] and the continued fraction (1.14) is called the $P$–fraction, [36].

Consider the three-term recurrence relation

(1.15) $b_{j}y_{{j-1}}(z)-a_{j}(z)y_{{j}}(z)+y_{{j+1}}(z)=0,$

associated with the sequence of atoms $\\{a_{i},b_{i}\\}$, $i\in\mathbb{N}$,
and define polynomials $P_{{j}}(z)$ and $Q_{{j}}(z)$ as solutions of the
system (1.15) subject to the initial conditions

(1.16) $P_{-1}(z)\equiv 0,\mbox{ }P_{0}(z)\equiv 1,\mbox{
}Q_{-1}(z)\equiv-1,\mbox{ }Q_{0}(z)\equiv 0.$

The polynomials $P_{{j}}$ and $Q_{{j}}$ are called Lanzcos polynomials of the
first and second kind. Moreover, the $n$-th convergent of the continued
fraction (1.14) takes the form (see [24, Section 8.3.7]).

$f^{[n]}(z)=-\frac{Q_{{n}}(z)}{P_{{n}}(z)},\quad n\in{\mathbb{N}}.$

As was shown in [7] the set $\mathcal{M}_{\kappa}({\mathbf{s}},2n_{j}-2)$ can
be described in terms of the Lanzcos polynomials of the first and second kind.

A sequence ${\bf s}\in{\mathcal{H}}_{\kappa}^{k}$ is called regular (see
[15]), and is designated as $\textbf{s}\in{\mathcal{H}}_{\kappa}^{k,reg}$, if

(1.17) $D_{n_{j}}^{+}=\det S_{n_{j}}^{+}\neq 0\qquad\mbox{for all}\quad
j\in{\mathbb{N}}.$

In [16] it was shown that an even indefinite Stieltjes moment problems
$MP_{\kappa}^{k}(\textbf{s},2n_{N}-1)$ for regular sequence ${\bf s}$ is
solvable if and only if

(1.18) $\kappa_{N}:=n_{-}(S_{n_{N}})\leq\kappa\quad\mbox{and}\quad
k_{N}^{+}:=n_{-}(S_{n_{N}}^{+})\leq k.$

For this problem one step of the Schur algorithm was split in [16] into two
substeps and this leads to the expansion of
$f\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s},2n_{N}-2)$ into a generalized
Stieltjes continued fraction

(1.19) $f(z)=\frac{1}{-zm_{1}(z)+\frac{\displaystyle 1}{\displaystyle
l_{1}+\cdots+\frac{\displaystyle 1}{\displaystyle-
zm_{N}(z)+\frac{\displaystyle 1}{\displaystyle l_{N}+f_{N}(z)}}}},$

where $m_{j}$ are polynomials, $l_{j}\in\mathbb{R}\backslash\\{0\\}$ and
$f_{N}$ is a function from the generalized Stieltjes class
${\mathbf{N}}_{\kappa-\kappa_{N}}^{k-k_{N}}$, such that $f_{N}(z)=o(1)$ as
$z\widehat{\rightarrow}\infty$. If
$f\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s})$ then $f_{N}$ has an induced
asymptotic expansion

(1.20)
$f_{N}(z)=-\frac{s^{(N)}_{0}}{z}-\frac{s^{(N)}_{1}}{z^{2}}-\cdots-\frac{s^{(N)}_{2n}}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right),\quad\quad
z\widehat{\rightarrow}\infty,$

i.e. $f_{N}$ is a solution of an induced moment problem
${MP}_{\kappa-\kappa_{N}}^{k-k_{N}}({\mathbf{s}^{(N)}})$ generated by the
sequence $\textbf{s}^{(N)}=(s^{(N)}_{i})_{i=0}^{\infty}$. Then
$f_{N}\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s}^{(N)})$.

Generalized Stieltjes continued fractions were studied in [15]. Associated to
the continued fraction (1.19) there is a system of difference equations (see
[42, Section 1])

(1.21) $\left\\{\begin{array}[]{l}y_{2j-1}-y_{2j-3}=-zm_{j}(z)y_{2j-2}{,}\\\
y_{2j}-y_{2j-2}=l_{j}y_{2j-1},\\\ \end{array}\right.\quad j\in{\mathbb{N}}.$

Define the generalized Stieltjes polynomials $P_{{j}}^{+}$ and $Q_{{j}}^{+}$
of the first and second kind as solutions of the system (1.21) subject to the
initial conditions

(1.22) $P_{-1}^{+}(z)\equiv 0,\mbox{ }P_{0}^{+}(z)\equiv 1,\mbox{
}Q_{-1}^{+}(z)\equiv 1,\mbox{ }Q_{0}^{+}(z)\equiv 0.$

The formula (1.19) for the set of solutions of the truncated Stieltjes moment
problem ${MP}_{\kappa}^{k}({\mathbf{s}},2n_{N}-1)$ can be rewritten in terms
of the generalized Stieltjes polynomials $P_{{j}}^{+}$ and $Q_{{j}}^{+}$,
$j=2N-1,2N$.

######  Theorem 1.2.

([17]) Let a sequence
${\mathbf{s}}=\\{s_{i}\\}_{i=0}^{\infty}\in\mathcal{H}_{\kappa}^{k}$ be
regular, $\mathcal{N}({\mathbf{s}})=\\{n_{j}\\}_{j=1}^{\infty}$, and let
$P_{{j}}^{+}$ and $Q_{{j}}^{+}$ be generalized Stieltjes polynomials of the
first and second kind. Then:

  1. (i)

A nondegenerate even moment problem $MP_{\kappa}^{k}(\textbf{s},2n_{N}-1)$ is
solvable, if and only if (1.18) holds.

  2. (ii)

$f\in\mathcal{M}_{\kappa}^{k}(\textbf{s},2n_{N}-1)$ if and only if $f$ admits
the representation

(1.23)
$f(z)=\frac{Q^{+}_{2N-1}(z){f_{N}}(z)+Q^{+}_{2N}(z)}{P^{+}_{2N-1}(z){f_{N}}(z)+P^{+}_{2N}(z)},$

where ${f_{N}}(z)$ satisfies the conditions

(1.24)
${f_{N}}\in{\mathbf{N}}_{\kappa-\kappa_{N}}^{k-k_{N}^{+}}\quad\mbox{and}\quad{f_{N}}(z)=o(1),\quad
z\widehat{\rightarrow}\infty.$

In what follows for every $2\times 2$ matrix $W=(w_{ij})_{i,j=1}^{2}$ we
associate the linear-fractional transformation

(1.25) $T_{W}[\tau]:=\frac{w_{11}\tau+w_{12}}{w_{21}\tau+w_{22}}.$

Denote by ${W_{2N}(z)}$ the coefficient matrix of the linear fractional
transform (1.23):

(1.26) $W_{2N}(z)=\left(\begin{array}[]{cc}Q^{+}_{2N-1}(z)&Q^{+}_{2N}(z)\\\
P^{+}_{2N-1}(z)&P^{+}_{2N}(z)\\\ \end{array}\right).$

Then the formula (1.23) can be rewritten as

(1.27) $f(z)=T_{W_{2N}(z)}[f_{N}(z)].$

The structure of the continued fraction (1.19) leads to the following
factorization of the matrix valued function ${W_{2N}(z)}$:

$W_{2N}(z)=M_{1}(z)L_{1}\ldots M_{N}(z)L_{N},$

where the matrices $M_{j}(z)$ and $L_{j}$ are defined by

(1.28) $M_{j}(z)=\begin{pmatrix}1&0\\\ -zm_{j}(z)&1\\\
\end{pmatrix}{,}\quad\mbox{and}\quad L_{j}=\begin{pmatrix}1&l_{j}\\\ 0&1\\\
\end{pmatrix}\quad j\in{\mathbb{N}}.$

Continued fractions of the form (1.19) with positive and negative masses
$m_{j}$ were studied by Beals, Sattinger and Szmigielski [4] in connection
with the theory of multi-peakon solutions of the Camassa-Holm equation. In
[22] Eckhardt and Kostenko showed that inverse spectral problem for multi-
peakon solutions of the Camassa-Holm equation is solvable in the class of
continued fractions of the form (1.19) with polynomials
$m_{j}(z)=d_{j}z+m_{j}$ of formal degree 1. In [23] the spectral theory of
continued fractions (1.19) was treated within a classical Hamburger moment
problem associated with this sequence.

The full indefinite Stieltjes moment problem $MP_{\kappa}^{k}(\textbf{s})$ for
regular sequences $\textbf{s}\in{\mathcal{H}}_{\kappa}^{k,reg}$ was considered
in [17] via the operator approach. The moment problem
$MP_{\kappa}^{k}(\textbf{s})$ was treated in [17] as a problem of extension
theory for a symmetric operator generated by this generalized Jacobi matrix.
In that paper we used quite advanced tools: the theory of boundary triples
developed in [25, 18, 19], and the M.G. Kreĭn theory of resolvent matrices
extended in [9, 12] to the case of indefinite inner spaces.

In the present paper we are going to use elementary tools in order to make the
presentation available for a wider audience. The main idea is to use the
factorization formula for the coefficients matrix $W_{2j}(z)$

(1.29) $W_{2j}(z)=W_{2N}(z)W^{(N)}_{2(j-N)}(z),\quad
j>N,\,\,j,N\in{\mathbb{N}},$

which allows to reduce the indefinite Stieltjes moment problem
$MP_{\kappa}^{k}(\textbf{s},2j)$ to some classical Stieltjes moment problem
$MP_{0}^{0}(\textbf{s}^{(N)},2(j-N))$ with the resolvent matrix
$W^{(N)}_{2(j-N)}(z)$. Then all the known results for the classical Stieltjes
moment problem can be translated to the indefinite Stieltjes moment problem.
In particular, in Theorem 4.2 it is shown that the problem
$MP_{\kappa}^{k}(\textbf{s})$ is indeterminate if and only if

(1.30) $M:=\sum_{j=1}^{\infty}m_{j}(0)<\infty\quad\mbox{and}\quad
L:=\sum_{j=1}^{\infty}l_{j}<\infty.$

In the classical case, polynomials $m_{j}(z)$ are constants and the criterion
(1.30) coincides with the well known Stieltjes criterion, see [1, Theorem
0.4].

If (1.30) is in force, then the matrix valued functions ${W}_{2j}(z)$ converge
to an entire matrix valued function ${W}_{\infty}^{+}(z)$ of order $1/2$ and
the linear fractional transformation (1.27) generated by the matrix valued
function ${W}_{\infty}(z)$ provides a description of the set
$\mathcal{M}_{\kappa}^{k}(\textbf{s})$.

In Section 5 Padé approximants for formal power series corresponding to an
indefinite Stieltjes moment problem are calculated. As was shown in [8] the
diagonal Padé approximants for formal power series corresponding to an
indefinite Hamburger moment problem are represented as a ratio of the Lanzcos
polynomials of the 2-nd and the 1-st kind, [8]. In Theorem 5.4 we show that
the sub-diagonal Padé approximants of the corresponding formal power series is
a ratio of the generalized Stieltjes polynomials of the 2-nd and the 1-st
kind. In Theorem 5.6 convergence of Padé approximants is derived from the
classical results using the formula (1.29).

In Section 6 the results are illustrated by an example of indefinite moment
problem associated with Laguerre polynomials $L_{n}(z,\alpha)$ in the non-
classical case $\alpha<-1$.

##  2\. Preliminaries

###  2.1. Generalized Nevanlinna functions

A function $f\in\mathbf{N}_{\kappa}$ is said to belong to the class
$\mathbf{N}_{\kappa,-\ell}$
$(\kappa,\ell\in{\mathbb{Z}}_{+}:={\mathbb{N}}\cup\\{0\\})$ if $f$ admits the
asymptotic expansion (1.10) for some real numbers $s_{0},\dots,s_{\ell}$. Let
us also set

(2.1) $\mathbf{N}_{\kappa,-\infty}:=\bigcap_{n\geq 0}\mathbf{N}_{\kappa,-2n}.$

Every real polynomial $P(z)=p_{\nu}z^{\nu}+\ldots+p_{1}z+p_{0}$ of degree
$\nu$ belongs to a class $\mathbf{N}_{\kappa_{-}(P)}$, where the index
$\kappa_{-}(P)$ can be evaluated by (see [30, Lemma 3.5])

(2.2)
$\kappa_{-}(P)=\left\\{\begin{array}[]{cl}\left[\frac{\nu+1}{2}\right],&\mbox{
if }p_{\nu}<0;\mbox{ and }\nu\mbox{ is odd };\\\
\left[\frac{\nu}{2}\right],&\mbox{ otherwise }.\end{array}\right.$

Recall, that a function $f\in\mathbf{N}_{\kappa}$ is said to be from the
generalized Stieltjes class $\mathbf{N}_{\kappa}^{\pm k}$, if $z^{\pm 1}f(z)$
belongs to $\mathbf{N}_{k}$ $\left(\kappa,k\in\mathbb{Z}_{+}\right)$. Let us
collect some properties of generalized Nevanlinna functions, see [30], [9].

######  Proposition 2.1.

([30]) Let $\kappa,\kappa_{1},k\in{\mathbb{Z}}_{+}$. Then the following
statements hold:

  1. (i)

$f\in\mathbf{N}_{\kappa}\Longleftrightarrow-\frac{1}{f}\in\mathbf{N}_{\kappa}$;

  2. (ii)

$f\in\mathbf{N}_{\kappa}^{k}\Longleftrightarrow-\frac{1}{f}\in\mathbf{N}_{\kappa}^{-k}$;

  3. (iii)

$f\in\mathbf{N}_{\kappa}^{k}\Longleftrightarrow
zf(z)\in\mathbf{N}_{k}^{-\kappa}$;

  4. (iv)

if $f\in\mathbf{N}_{\kappa}$, $f_{1}\in\mathbf{N}_{\kappa_{1}}$ then
$f+f_{1}\in\mathbf{N}_{\kappa^{\prime}}$, where
$\kappa^{\prime}\leq\kappa+\kappa_{1}$. If, in addition, $f(iy)=o(y)$ as
$y\rightarrow\infty$ and $f_{1}$ is a polynomial, then
$f+f_{1}\in\mathbf{N}_{\kappa+\kappa_{1}}$;

  5. (v)

if a function $f\in\mathbf{N}_{\kappa}$ has an asymptotic expansion (1.5) for
evrey $n\in{\mathbb{N}}$, then there exists $\kappa^{\prime}\leq\kappa$, such
that $\\{s_{j}\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa^{\prime}}$.

The notions of generalized poles of non-positive type of a function
${f}\in{\bf N}_{\kappa}$ were introduced in [32]. The following definitions
are based on [34]. A point $\alpha\in{\mathbb{R}}$ is called a generalized
pole of non-positive type (GPNT) of the function ${f}\in{\bf N}_{\kappa}$ with
multiplicity $\kappa_{\alpha}({f})$ if

(2.3)
$-\infty<\lim_{z\widehat{\rightarrow}\alpha}(z-\alpha)^{2\kappa_{\alpha}+1}{f}(z)\leq
0,\quad
0<\lim_{z\widehat{\rightarrow}\alpha}(z-\alpha)^{2\kappa_{\alpha}-1}{f}(z)\leq\infty.$

Similarly, the point $\infty$ is called a generalized pole of $f$ of
nonpositive type (GPNT) with multiplicity $\kappa_{\infty}(f)$ if

(2.4)
$0\leq\lim_{z\widehat{\rightarrow}\infty}\frac{{f}(z)}{z^{2\kappa_{\infty}+1}}<\infty,\quad-\infty\leq\lim_{z\widehat{\rightarrow}\infty}\frac{{f}(z)}{z^{2\kappa_{\infty}-1}}<0.$

The following fundamental result was proved in [32, Theorem 3.5].

######  Proposition 2.2.

Let $f\in\mathbf{N}_{\kappa}$. Then the total multiplicity of the poles of $f$
in ${\mathbb{C}}_{+}$ and the generalized poles of negative type of $f$ in
${\mathbb{R}}\cup\\{\infty\\}$ is equal to $\kappa$.

###  2.2. Regular sequences and step–by–step algorithm

In the present paper we will consider so-called regular sequences s from
${\mathcal{H}}_{\kappa}^{k}$ introduced in [15].

######  Definition 2.3.

The sequence s is called regular (and is denoted as
$\textbf{s}\in{\mathcal{H}}_{\kappa}^{k,reg}$), if one of the following
equivalent conditions holds

  1. (i)

$P_{j}(0)\neq 0$ for every $j\in{\mathbb{N}}$;

  2. (ii)

$D_{{n}_{j}}^{+}:=\det S_{n_{j}}^{+}\neq 0$ for every $j\in{\mathbb{N}}$;

  3. (iii)

$D_{{n}_{j}-1}^{+}\neq 0$ for every $j\in{\mathbb{N}}$.

The following step–by–step algorithm of solving the indefinite Stieltjes
moment problem for regular sequences ${\bf s}\in\mathcal{H}_{\kappa}^{k,reg}$
was developed in [17]. For an indefinite Hamburger moment problem such an
algorithm was elaborated in [6, 7].

Let $f\in{\mathcal{M}}_{\kappa}^{k}({\bf s})$ and let $n_{1}$ be the first
normal index, i.e.

$s_{0}=\dots=s_{n_{1}-2}=0,\quad s_{n_{1}-1}\neq 0.$

By Proposition 2.1 (4) $-1/f\in\mathbf{N}_{\kappa}^{-k}$ and

(2.5) $-\frac{1}{f(z)}=P_{1}(z)+o(1),\quad{as}\quad
z\widehat{\rightarrow}\infty,$

where $P_{1}(z)$ is the Lanzcos polynomial of degree: $\deg P_{1}=n_{1}$.
Since ${\bf s}$ is regular, then $-l_{1}^{-1}:=P_{1}(0)\neq 0$ and hence (2.5)
can be rewritten as

(2.6)
$-\frac{1}{f(z)}=zm_{1}(z)-\frac{1}{g_{1}(z)}=zm_{1}(z)-\frac{1}{l_{1}+f_{1}(z)}.$

Let $\kappa_{1}\\!=\\!\kappa_{-}(zm_{1})$, $k_{1}\\!=\\!\kappa_{-}(m_{1})$,
$k_{1}^{+}=k_{1}+\kappa_{-}(zl_{1})$. Then by Proposition 2.1 (2), (4)

$zm_{1}\in\mathbf{N}_{\kappa_{1}}^{-k_{1}}\Rightarrow-\frac{1}{g_{1}}=-\frac{1}{f}-zm_{1}\in\mathbf{N}_{\kappa-\kappa_{1}}^{-(k-k_{1})}\Rightarrow
g_{1}\in\mathbf{N}_{\kappa-\kappa_{1}}^{k-k_{1}}\Rightarrow
f_{1}\in\mathbf{N}_{\kappa-\kappa_{1}}^{k-k_{1}^{+}}.$

The formula (2.6) yields the following representation of
$f\in{\mathcal{M}}_{\kappa}^{k}({\bf s})$:

(2.7) $f(z)=\displaystyle\frac{\displaystyle 1}{\displaystyle-
zm_{1}(z)+\displaystyle\frac{1}{\displaystyle l_{1}+f_{1}(z)}},$

where $f_{1}\in\mathbf{N}_{\kappa-\kappa_{1}}^{k-k_{1}^{+}}$ satisfies the
asymptotic expansion

(2.8)
$f_{1}(z)=-\frac{s^{(1)}_{0}}{z}-\frac{s^{(1)}_{1}}{z^{2}}-\cdots-\frac{s^{(1)}_{2n}}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right),\quad\quad
z\widehat{\rightarrow}\infty$

with some real numbers $s^{(1)}_{i}$, $i\in{\mathbb{Z}}_{+}$. Explicit
formulae for calculation of the sequence
$\mathbf{s}^{(1)}=\\{s^{(1)}_{i}\\}_{i=0}^{\infty}$ are presented in [17,
Remarks 3.4, 3.6], the polynomial $m_{1}(z)$ and the number $l_{1}$ can be
found by (see [17, (2.16), (3.27)]):

(2.9)
$m_{1}(z)=\frac{(-1)^{n_{1}+1}}{D_{n_{1}}}\begin{vmatrix}0&\ldots&0&s_{{n_{1}}-1}&s_{{n_{1}}}\\\
\vdots&&\ldots&\ldots&\vdots\\\
s_{{n_{1}}-1}&\ldots&\ldots&\ldots&s_{2{n_{1}}-2}\\\
1&z&\ldots&z^{{n_{1}}-2}&z^{{n_{1}}-1}\\\ \end{vmatrix}{,}$ (2.10)
$l_{1}=(-1)^{{n_{1}}+1}s_{{n_{1}}-1}\frac{D_{n_{1}}}{D_{n_{1}}^{+}},\quad(D_{\nu}:=\det
S_{\nu},\,D_{\nu}^{+}:=\det S_{\nu}^{+}).$

This completes the first step of the Schur algorithm.

Applying this algorithm repeatidly $N$ times one obtains a function
$f_{N}\in\mathbf{N}_{\kappa-\kappa_{N}}^{k-k_{N}^{+}}$, with $\kappa_{N}$ and
$k_{N}$ given by (1.18), connected with $f$ by the formula (1.19). Moreover,
the function $f_{N}$ satisfies the asymptotic expansion

(2.11)
$f_{N}(z)=-\frac{s^{(N)}_{0}}{z}-\frac{s^{(N)}_{1}}{z^{2}}-\cdots-\frac{s^{(N)}_{2n}}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right),\quad\quad
z\widehat{\rightarrow}\infty$

with some real $s^{(N)}_{i}$, $i\in{\mathbb{Z}}_{+}$.

######  Lemma 2.4.

([15]) Let ${\mathbf{s}}\in\mathcal{H}_{\kappa}^{k,reg}$. Then there exist
sequences of polynomials $m_{j}(z)$ and numbers ${l}_{j}$ such that the
$2j-$th convergent $\frac{u_{2j}}{v_{2j}}$ of the continued fraction

(2.12) $\displaystyle\frac{\displaystyle 1}{\displaystyle-
zm_{1}(z)+\displaystyle\frac{1}{\displaystyle
l_{1}+\dots\frac{1}{\displaystyle-zm_{j}(z)+\frac{1}{\displaystyle
l_{j}+\dots}}}}$

coincides with the $j-$th convergent of the $P-$fraction (1.14) corresponding
to the sequence ${\mathbf{s}}$. Let functions $f$ and $f_{N}$
($N\in{\mathbb{N}}$) be connected by (1.19) and let $\mathbf{s}^{(N)}$ is the
$N-$th induced sequence. Then:

(2.13) $f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s},\ell)\Longleftrightarrow
f_{N}\in{\mathcal{M}}_{\kappa-\kappa_{N}}^{k-k_{N}^{+}}(\mathbf{s}^{(N)},\ell-2n_{N}),$
(2.14) $f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s})\Longleftrightarrow
f_{N}\in{\mathcal{M}}_{\kappa-\kappa_{N}}^{k-k_{N}^{+}}(\mathbf{s}^{(N)}),$

where $\kappa_{N}$ and $k_{N}^{+}$ are given by (1.18).

In the case of a regular sequence $\mathbf{s}\in{\mathcal{H}}_{\kappa}^{k}$
the parameters ${l}_{j}$ and $m_{j}(z)$ in (2.12) can be calculated
recursively by the above Schur algorithm in terms of the sequence
${\mathbf{s}}$:

(2.15)
$m_{j}(z)=\frac{(-1)^{\nu+1}}{D^{(j-1)}_{\nu}}\begin{vmatrix}0&\ldots&0&s^{(j-1)}_{\nu-1}&s^{(j-1)}_{\nu}\\\
\vdots&&\ldots&\ldots&\vdots\\\
s^{(j-1)}_{\nu-1}&\ldots&\ldots&\ldots&s^{(j-1)}_{2\nu-2}\\\
1&z&\ldots&z^{\nu-2}&z^{\nu-1}\\\ \end{vmatrix},$

where $D^{(j)}_{\nu}:=\det S^{(j)}_{\nu}$, $\nu=n_{j}-n_{j-1}$ and

(2.16)
$l_{j}=(-1)^{\nu+1}\frac{D^{(j-1)}_{\nu}}{\left(D^{(j-1)}_{\nu}\right)^{+}}\quad(j=1,\dots,N-1).$

The $N-$th induced sequence $\mathbf{s}^{(N)}$ can be found as the sequence of
coefficients of the series expansion
$-\sum_{i=0}^{\infty}{s_{i}^{(N)}}{z^{-(i+1)}}$ corresponding to the continued
fraction

(2.17) $\frac{1}{\displaystyle-zm_{N+1}(z)+\frac{1}{\displaystyle
l_{N+1}+\frac{1}{\displaystyle-zm_{N+2}(z)+\frac{1}{\ddots}}}}.$

###  2.3. Generalized Stieltjes continued fractions

In [15] the expansion (2.12) of $f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s})$
into a generalized Stieltjes fraction for
$\textbf{s}\in{\mathcal{H}}_{\kappa}^{k,reg}$ was derived from the expansion
of its unwrapping transform $zf(z^{2})$ into the $P-$fraction.

######  Theorem 2.5.

([15]) Let ${\mathbf{s}}\in\mathcal{H}_{\kappa}^{k,reg}$ and let the
$P-$fraction (1.14) and the generalized $S-$fraction (2.12) correspond to the
sequence ${\mathbf{s}}$. Then the parameters ${l}_{j}$ and $m_{j}(z)$
($j\in\mathbb{Z}_{+}$) of the generalized $\mathbf{S}-$fraction (2.12) are
connected with the parameters $b_{j}$ and $a_{j}(z)$ ($j\in\mathbb{N}$) of the
$P-$fraction (1.14) by the equalities

(2.18) $b_{0}=\frac{1}{d_{1}},\quad
a_{0}(z)=\frac{1}{d_{1}}\left(zm_{1}(z)-\frac{1}{l_{1}}\right),$ (2.19)
$b_{j}=\frac{1}{l_{j}^{2}d_{j}d_{j+1}},\quad
a_{j}(z)=\frac{1}{d_{j+1}}\left(zm_{j+1}(z)-\left(\frac{1}{l_{j}}+\frac{1}{l_{j+1}}\right)\right),$

where $d_{j}$ is the leading coefficient of $m_{j}(z)$ $(j=1,\dots,N-1)$.

In the case when $m_{j}(z)\equiv m_{j}$ are constants the generalized
$S-$fraction reduces to the classical $S$-fraction (1.7) and the formulae
(2.18) and (2.19) coincide with the well known classical formulae from [40],
see also [1, Appendix, (3), (4)].

Conversely, $m_{j}$ and $l_{j}$ can be represented in terms of the
$P-$fractions, see [15, Corollary 4.1].

######  Corollary 2.6.

Let ${\mathbf{s}}\in\mathcal{H}_{\kappa}^{k,reg}$, let ${\mathbf{s}}$ be
associated with the $\mathbf{S}-$fraction (2.12), let $d_{i}$ be the leading
coefficient of the polynomial $m_{i}(z)$ and let

(2.20) $\widetilde{b}_{i}=b_{0}b_{1}\ldots b_{i},\quad i\in{\mathbb{Z}}_{+}.$

Then:

(2.21) $d_{i}=\frac{P_{i-1}^{2}(0)}{\widetilde{b}_{i-1}},\quad
l_{i}=-\frac{\widetilde{b}_{i-1}}{P_{i-1}(0)P_{i}(0)},\quad
m_{i}(z)=d_{i}\frac{a_{i-1}(z)-a_{i-1}(0)}{z},\quad i\in{\mathbb{N}}.$

######  Remark 2.7.

If ${\mathcal{N}}(\mathbf{s})={\mathbb{N}}$, then $\textup{deg}(m_{i})=0$ for
all $i\in{\mathbb{N}}$, and $d_{i}=m_{i}$, $l_{i}$ can be found by (2.21).
Alternatively, $m_{i}$, $l_{i}$ can be found by (1.6).

If $D_{i}^{+}>0$ for all $i\in{\mathbb{N}}$, then $\textup{deg}(m_{i})\leq 1$
for all $i\in{\mathbb{N}}$, and $m_{i}(z)=d_{i}z+m_{i}$, $l_{i}$ can be found
by (2.21). The parameters $d_{i}$, $m_{i}$ and $l_{i}$ can be also expressed
in terms of $D_{i}$ and $D_{i}^{+}$, see [32, Section 5.3], [15, Remark 4.1].

##  3\. Truncated indefinite moment problems

###  3.1. A system of difference equations and generalized Stieltjes
polynomials

Let us consider a system of difference equations associated with the continued
fraction (2.12)

(3.1) $\left\\{\begin{array}[]{l}y_{2j-1}-y_{2j-3}=-zm_{j}(z)y_{2j-2}{,}\\\
y_{2j}-y_{2j-2}=l_{j}y_{2j-1},\\\ \end{array}\right.\quad j\in{\mathbb{N}}.$

If the $j$–th convergent of this continued fraction is denoted by
$\frac{u_{j}}{v_{j}}$, then $u_{j}$, $v_{j}$ can be found as solutions of the
system (3.1) (see [42, Section 1]) subject to the following initial conditions

(3.2) $u_{-1}\equiv 1,\quad u_{0}\equiv 0;\qquad v_{-1}\equiv 0,\quad
v_{0}\equiv 1.$

The first two convergents of the continued fraction (2.12) take the form

$\frac{u_{1}}{v_{1}}=\frac{1}{-zm_{1}(z)}=T_{M_{1}}[\infty],\quad\frac{u_{2}}{v_{2}}=\frac{l_{1}}{-zl_{1}m_{1}(z)+1}=T_{M_{1}L_{1}}[0].$

######  Definition 3.1.

([16]) Let $\textbf{s}\in\mathcal{H}_{\kappa}^{k,reg}$. Define polynomials
$P^{+}_{i}(z)$, $Q_{i}^{+}(z)$ by

(3.3) $\begin{split}&P^{+}_{-1}(z)\equiv 0,\quad P^{+}_{0}(z)\equiv 1,\qquad
Q^{+}_{-1}(z)\equiv 1,\quad Q^{+}_{0}(z)\equiv 0,\\\
&P^{+}_{2i-1}(z)=-\frac{1}{\widetilde{b}_{i-1}}\begin{vmatrix}P_{{i}}(z)&P_{{i-1}}(z)\\\
P_{{i}}(0)&P_{{i-1}}(0)\\\ \end{vmatrix}\quad\mbox{and}\quad
P^{+}_{2i}(z)=\frac{P_{{i}}(z)}{P_{{i}}(0)},\\\
&Q_{2i-1}^{+}(z)=\frac{1}{\widetilde{b}_{i-1}}\begin{vmatrix}Q_{{i}}(z)&Q_{{i-1}}(z)\\\
P_{{i}}(0)&P_{{i-1}}(0)\\\ \end{vmatrix}\quad\mbox{and}\quad
Q^{+}_{2i}(z)=-\frac{Q_{{i}}(z)}{P_{{i}}(0)}.\end{split}$

The polynomials $P^{+}_{i}(z)$, $Q_{i}^{+}(z)$ are called _the generalized
Stieltjes polynomials_ corresponding to the sequence ${\mathbf{s}}$.

As was noticed in [16] the generalized Stieltjes polynomials coincide with the
solutions $u_{i}$ and $v_{i}$ of the system (3.1).

######  Proposition 3.2.

Let $\textbf{s}\in\mathcal{H}_{\kappa}^{k,reg}$ and let $P^{+}_{j}(z)$ and
$Q_{j}^{+}(z)$ be the generalized Stieltjes polynomials defined by (3.3). Then
the solutions $\\{u_{j}\\}_{j=0}^{N}$ and $\\{v_{j}\\}_{j=0}^{N}$ of the
system (3.1), (3.2) take the form

$u_{j}=Q_{j}^{+}(z),\quad v_{j}=P_{j}^{+}(z)\quad(j=-1,0,\dots,N).$

######  Remark 3.3.

The Stieltjes polynomials satisfy the following properties

(3.4) $P^{+}_{2i-1}(0)=0,\quad P^{+}_{2i-2}(0)=1\quad\mbox{and}\quad
Q^{+}_{2i-1}(0)=1.$

Obviously, by Definition 3.1

(3.5)
$P^{+}_{2i-1}(0)=-\frac{1}{\widetilde{b}_{i-1}}\begin{vmatrix}P_{{i}}(0)&P_{{i-1}}(0)\\\
P_{{i}}(0)&P_{{i-1}}(0)\\\ \end{vmatrix}=0\quad\mbox{and}\quad
P^{+}_{2i-2}(0)=\frac{P_{{i}}(0)}{P_{{i}}(0)}=1.$

By (3.7)

(3.6)
$Q^{+}_{2i-1}(0)=\frac{1}{\widetilde{b}_{i-1}}\begin{vmatrix}Q_{{i}}(0)&Q_{{i-1}}(0)\\\
P_{{i}}(0)&P_{{i-1}}(0)\\\
\end{vmatrix}=\frac{Q_{{i}}(0)P_{{i-1}}(0)-Q_{{i-1}}(0)P_{{i}}(0)}{\widetilde{b}_{i-1}}=1.$

Here we used the generalized Liouville-Ostrogradsky formula (see [17, (2.9)])

(3.7) $Q_{{i}}(z)P_{{i-1}}(z)-Q_{{i-1}}(z)P_{{i}}(z)=\widetilde{b}_{i-1},\quad
i\in{\mathbb{N}}.$

######  Lemma 3.4.

Let $P^{+}_{i}$ and $Q^{+}_{i}$ be the Stieltjes polynomials defined by (3.3).
Then

(3.8) $P^{+}_{2i}(z)Q^{+}_{2i-1}(z)-Q^{+}_{2i}(z)P^{+}_{2i-1}(z)=1.$

  * Proof.

By Definition 3.1 and (3.7) we obtain

$\begin{split}P^{+}_{2i}(z)Q^{+}_{2i-1}(z)-Q^{+}_{2i}(z)P^{+}_{2i-1}(z)=&\frac{P_{{i}}(z)}{\widetilde{b}_{i-1}P_{{i}}(0)}(Q_{{i}}(z)P_{{i-1}}(0)-P_{{i}}(0)Q_{{i-1}}(z))-\\\
&-\frac{Q_{{i}}(z)}{\widetilde{b}_{i-1}P_{{i}}(0)}(P_{{i}}(z)P_{{i-1}}(0)-P_{{i}}(0)P_{{i-1}}(z))\\\
=&\frac{Q_{{i}}(z)P_{{i-1}}(z)-P_{{i}}(z)Q_{{i-1}}(z)}{\widetilde{b}_{i-1}}=1\end{split}$

This completes the proof. $\Box$

######  Lemma 3.5.

Let $\textbf{s}\in\mathcal{H}_{\kappa}^{k,reg}$ and let $P_{i}(z)$ and
$Q_{i}(z)$ ($i\in{\mathbb{Z}}_{+}$) be Lanczos polynomials of the first and
second kind and let ${l}_{j}$ and $m_{j}(z)$ ($j\in\mathbb{N}$) be parameters
of the generalized $\mathbf{S}-$fraction (2.12). Then:

  1. (i)

The constants $l_{i}$ can be calculated by

(3.9) $l_{i}=-\frac{Q_{i}(0)}{P_{i}(0)}+\frac{Q_{i-1}(0)}{P_{i-1}(0)},\quad
i\in{\mathbb{N}};$

  2. (ii)

For every $N\in{\mathbb{N}}$ the following formulas hold

(3.10)
$\sum_{i=1}^{N}l_{i}=-\frac{Q_{N}(0)}{P_{N}(0)},\qquad\sum_{i=1}^{N}d_{i}=\sum_{i=0}^{N-1}|P^{2}_{i}(0)|\widetilde{b}_{i}^{-1};$
(3.11) $\sum_{i=1}^{N}m_{i}(0)=-P^{+^{{}^{\prime}}}_{2N-1}(0).$

  * Proof.

1) Let $Q^{+}_{i}(z)$ be Stieltjes polynomials defined by (3.3). Substituting
in (3.1) $y_{j}=Q_{j}^{+}$ and $z=0$, we obtain

$Q^{+}_{2i}(0)-Q^{+}_{2i-2}(0)=l_{i}Q^{+}_{2i-1}(0).$

By Definition 3.1 and by the generalized Liouville-Ostrogradsky formula (3.7)

$Q^{+}_{2i-1}(0)=\frac{1}{\widetilde{b}_{i-1}}(Q_{{i}}(0)P_{{i-1}}(0)-Q_{{i-1}}(0)P_{{i}}(0)=1.$

This implies (3.9).

2) Summing the equalities (3.9) for $i=1,\dots,N$ one obtains the first
equality in (3.10). The second equality in (3.10) is implied by the relation
(see [15, Corollary 4.1])

(3.12) $d_{i+1}=|P^{2}_{i}(0)|\widetilde{b}_{i}^{-1}.$

3) Differentiating the first equality in (3.1) one obtains

$P^{+^{{}^{\prime}}}_{2i-1}(z)=-m_{i}(z)P^{+}_{2i-2}(z)-z(m_{i}(z)P^{+}_{2i-2}(z))^{\prime}+P^{+^{{}^{\prime}}}_{2i-3}(z).$

Substituting $z=0$ and using the equality $P_{2i-2}^{+}(0)=1$, one obtains

$m_{i}(0)=\,P^{+^{{}^{\prime}}}_{2i-3}(0)-P^{+^{{}^{\prime}}}_{2i-1}(0).$

Hence

$\sum_{i=1}^{N}m_{i}(0)=P^{+^{{}^{\prime}}}_{-1}(0)-P^{+^{{}^{\prime}}}_{1}(0)+\ldots+P^{+^{{}^{\prime}}}_{2N-3}(0)-P^{+^{{}^{\prime}}}_{2N-1}(0)=-P^{+^{{}^{\prime}}}_{2N-1}(0).$

This proves (3.11). $\Box$

###  3.2. The class $\mathcal{U}_{\kappa}^{k}(J)$ and linear fractional
transformations

Let $J$ and ${\mathcal{Z}}$ be the $2\times 2$ matrices

$J=\begin{pmatrix}0&-i\\\ i&0\\\
\end{pmatrix}\quad\mbox{and}\quad{\mathcal{Z}}=\begin{pmatrix}z&0\\\ 0&1\\\
\end{pmatrix}.$

######  Definition 3.6.

Let $W(z)$ be a $2\times 2$ matrix valued function meromorphic in
$\mathbb{C}_{+}$ and let ${\mbox{\gothic\char 104\relax}}_{W}^{+}$ be the
domain of holomorphy of $W$ in $\mathbb{C}_{+}$, $\kappa\in{\mathbb{Z}}_{+}$.
Then $W(z)$ is called a generalized $J$-inner matrix valued function from the
class ${\mathcal{U}}_{\kappa}(J)$, if:

  1. (i)

the kernel

(3.13)
${\mathsf{K}}_{\omega}^{W}(z)=\frac{J-W(z)JW(\omega)^{*}}{-i(z-\bar{\omega})}$

has $\kappa$ negative squares in ${\mbox{\gothic\char
104\relax}}_{W}^{+}\times{\mbox{\gothic\char 104\relax}}_{W}^{+}$;

  2. (ii)

$J-W(\mu)JW(\mu)^{*}=0$ for a.e. $\mu\in\mathbb{R}$.

A matrix valued function $W\in{\mathcal{U}}_{\kappa}(J)$ is said to belong to
the class ${\mathcal{U}}_{\kappa}^{k}(J)$, $\kappa,k\in{\mathbb{Z}}_{+}$, if

(3.14) ${\mathcal{Z}}W{\mathcal{Z}}^{-1}\in{\mathcal{U}}_{k}(J).$

Consider the linear fractional transformation

(3.15) $T_{W}[\tau]=(w_{11}\tau(z)+w_{12})(w_{21}\tau(z)+w_{22})^{-1}$

associated with the matrix valued function $W(z)=(w_{i,j}(z))_{i,j=1}^{2}$.
The linear fractional transformation associated with the product $W_{1}W_{2}$
of two matrix valued functions $W_{1}(z)$ and $W_{2}(z)$, coincides with the
composition $T_{W_{1}}\circ T_{W_{2}}$.

The following statement is an easy corollary of Definition 3.6.

######  Lemma 3.7.

Let $W\in{\mathcal{U}}_{\kappa_{1}}^{k_{1}}(J)$ and
$\tau\in{\mathbf{N}}_{\kappa_{2}}^{k_{2}}$,
$\kappa_{1},\kappa_{2},k_{1},k_{2}\in{\mathbb{Z}}_{+}$. Then
$T_{W}[\tau]\in{\mathbf{N}}_{\kappa^{\prime}}^{k^{\prime}}$, where
$\kappa^{\prime}\leq{\kappa_{1}+\kappa_{2}}$, $k^{\prime}\leq k_{1}+k_{2}$.

  * Proof.

The proof of the inclusion $T_{W}[\tau]\in{\mathbf{N}}_{\kappa^{\prime}}$ for
some $\kappa^{\prime}\leq{\kappa_{1}+\kappa_{2}}$ is similar to that in [13,
Lemma 3.4].

Let us denote $f(z)=T_{W}(z)[\tau(z)]$. Then

$zf(z)=T_{{\mathcal{Z}}W(z){\mathcal{Z}}^{-1}}[z\tau(z)].$

Since ${\mathcal{Z}}W{\mathcal{Z}}^{-1}\in{\mathcal{U}}_{k_{1}}$ and
$z\tau\in{\mathbf{N}}_{k_{2}}$ then $f\in{\mathbf{N}}_{k^{\prime}}$ for some
$k^{\prime}\leq k_{1}+k_{2}$. This proves that
$T_{W}[\tau]\in{\mathbf{N}}_{\kappa^{\prime}}^{k^{\prime}}$ for some
$\kappa^{\prime}\leq{\kappa_{1}+\kappa_{2}}$, $k^{\prime}\leq k_{1}+k_{2}$.
$\Box$

In the present paper two special types of matrix valued functions from
${\mathcal{U}}_{\kappa}^{k}(J)$ play an important role (see [17, Lemma 2.11,
Lemma 2.12]).

######  Lemma 3.8.

Let $m(z)$ be a real polynomial such that $\kappa_{-}(zm(z))=\kappa_{1}$,
$\kappa_{-}(m(z))=k_{1}$, let $M(z)$ be a $2\times 2$ matrix valued function

(3.16) $M(z)=\begin{pmatrix}1&0\\\ -zm(z)&1\\\ \end{pmatrix}$

and let $\tau$ be a meromorphic function, such that

(3.17) $\tau(z)^{-1}=o(z)\mbox{ as }z\widehat{\rightarrow}\infty.$

Then $M\in{\mathcal{U}}_{\kappa_{1}}^{k_{1}}(J)$ and the following
equivalences hold:

(3.18) $\tau\in{\mathbf{N}}_{\kappa_{2}}\Longleftrightarrow
T_{M}[\tau]\in{\mathbf{N}}_{\kappa_{1}+\kappa_{2}},$ (3.19)
$\tau\in{\mathbf{N}}_{\kappa_{2}}^{k_{2}}\Longleftrightarrow
T_{M}[\tau]\in{\mathbf{N}}_{\kappa_{1}+\kappa_{2}}^{k_{1}+k_{2}}.$

######  Lemma 3.9.

Let $l(z)$ be a real polynomial such that $\kappa_{-}(l(z))=\kappa_{1}$,
$\kappa_{-}(zl(z))=k_{1}$, let $L(z)$ be the $2\times 2$ matrix valued
function

(3.20) $L(z)=\begin{pmatrix}1&l(z)\\\ 0&1\\\ \end{pmatrix}$

and let $\phi$ be a meromorphic function, such that

(3.21) $\phi(z)=o(1)\mbox{ as }z\widehat{\rightarrow}\infty.$

Then $L\in{\mathcal{U}}_{\kappa_{1}}^{k_{1}}(J)$ and the following
equivalences hold:

(3.22) $\phi\in{\mathbf{N}}_{\kappa_{2}}\Longleftrightarrow
T_{L}[\phi]\in{\mathbf{N}}_{\kappa_{1}+\kappa_{2}},$ (3.23)
$\phi\in{\mathbf{N}}_{\kappa_{2}}^{k_{2}}\Longleftrightarrow
T_{L}[\phi]\in{\mathbf{N}}_{\kappa_{1}+\kappa_{2}}^{k_{1}+k_{2}}.$

###  3.3. Truncated indefinite moment problem and resolvent matrices

As was mentioned in Theorem 1.2 the matrix valued function $W_{2N}(z)$ defined
by (1.26) is the resolvent matrix of a truncated even indefinite moment
problem $MP_{\kappa}^{k}(\textbf{s},2N-1)$ with $\kappa\geq\kappa_{N}$ and
$k\geq k_{N}^{+}$, where $\kappa_{N}$ and $k_{N}^{+}$ are defined by (1.18).
The resolvent matrix $W_{2N}(z)$ provides an example of generalized $J-$inner
matrix valued function from the class
${\mathcal{U}}_{\kappa_{N}}^{k_{N}^{+}}(J)$.

######  Theorem 3.10.

([17]) Let
${\mathbf{s}}=\\{s_{i}\\}_{i=0}^{\infty}\in\mathcal{H}_{\kappa}^{k,reg}$, let
$P_{{j}}^{+}(z)$ and $Q_{{j}}^{+}(z)$ be generalized Stieltjes polynomials of
the first and second kind, let $m_{i}(z)$, $l_{i}$ be defined by (2.15) and
(2.16) and let $W_{2N}(z)$, $M_{i}(z)$ and $L_{i}$ be given by (1.26) and
(1.28). Then:

  1. (i)

The matrix valued function $W_{2N}(z)$ admits the factorization

(3.24) $W_{2N}(z)=M_{1}(z)L_{1}\dots M_{N}(z)L_{N};$

  2. (ii)

$W_{2N}\in{\mathcal{U}}_{\kappa_{N}}^{k_{N}^{+}}(J)$.

  * Proof.

It follows from (3.1) that

(3.25) $W_{2N}(z)=W_{2N-2}(z)M_{N}(z)L_{N}.$

Applying (3.25) $N$ times one obtains (3.24).

By [17, Theorem 4.3 (4)]

$\begin{split}&\kappa_{N}=\sum\limits_{j=1}^{N}\kappa_{-}(zm_{j}),\quad
k_{N}^{+}=\sum\limits_{j=1}^{N}k_{-}(m_{j})+\sum\limits_{j=1}^{N}\kappa_{-}(zl_{j}).\end{split}$

Since by Lemmata 3.8, 3.9
$M_{i}\in{\mathcal{U}}_{\kappa_{-}(zm_{i})}^{\kappa_{-}(m_{i})}$,
$L_{i}\in{\mathcal{U}}_{0}^{\kappa_{-}(zl_{i})}$, $i\in{\mathbb{N}}$, one
obtains from the factorization formula (3.24)
$W_{2N}\in{\mathcal{U}}_{\kappa^{\prime}}^{k^{\prime}}(J)$, where

(3.26) $\kappa^{\prime}\leq\kappa_{N},\quad k^{\prime}\leq k_{N}^{+}.$

On the other hand it follows from Lemma 3.7 that
$f:=T_{W_{2N}[0]}\in\mathbf{N}_{\kappa_{N}}^{k_{N}^{+}}$. Therefore, by Lemma
3.7

(3.27) $\kappa_{N}\leq\kappa^{\prime},\quad k_{N}^{+}\leq k^{\prime}.$

The statement (ii) follows from (3.26) and (3.27). $\Box$

Let us formulate an analog of Theorems 1.2 and 3.10 for the odd truncated
indefinite moment problem (see [17]).

######  Theorem 3.11.

Let
${\mathbf{s}}=\\{s_{i}\\}_{i=0}^{2n_{N}-2}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
let $P_{j}^{+}(z)$ and $Q_{j}^{+}(z)$ $(0\leq j\leq 2N-1)$ be generalized
Stieltjes polynomials, let $M_{i}(z)$ and $L_{i}$ be given by (1.28) and let
the matrix valued function $W_{2N-1}$ be defined by

(3.28) $W_{2N-1}(z)=\begin{pmatrix}Q^{+}_{2N-1}(z)&Q^{+}_{2N-2}(z)\\\
P^{+}_{2N-1}(z)&P^{+}_{2N-2}(z)\\\ \end{pmatrix}.$

Then:

  1. (i)

The odd moment problem $MP_{\kappa}^{k}(\textbf{s},2n_{N}-2)$ is solvable, if
and only if

(3.29) $\kappa_{N}:=\nu_{-}(S_{n_{N}})\leq\kappa\quad\mbox{and}\quad
k_{N}:=\nu_{-}(S_{n_{N}-1}^{+})\leq k.$

  2. (ii)

$f\in\mathcal{M}_{\kappa}^{k}(\textbf{s},2n_{N}-2)$ if and only if $f$ admits
the representation

(3.30)
$f(z)=T_{W_{2N-1}(z)}[\tau(z)]=\frac{Q^{+}_{2N-1}(z)\tau(z)+Q^{+}_{2N-2}(z)}{P^{+}_{2N-1}(z)\tau(z)+P^{+}_{2N-2}(z)},$

where $\tau\in N^{k-k_{N}}_{\kappa-\kappa_{N}}$ and ${\tau(z)}^{-1}=o(z)$ as
$z\widehat{\rightarrow}\infty.$

  3. (iii)

The matrix valued function $W_{2N-1}(z)$ belongs to the class
${\mathcal{U}}_{\kappa_{N}}^{k_{N}}(J)$ and admits the factorization

(3.31) $W_{2N-1}(z)=M_{1}(z)L_{1}\dots L_{N-1}M_{N}(z).$

Substituting in (3.30) $\tau(z)=\infty$ one obtains from Theorem 3.11 the
following:

######  Corollary 3.12.

The function $\frac{Q^{+}_{2N-1}(z)}{P^{+}_{2N-1}(z)}$ belongs to the class
${\mathbf{N}}_{\kappa_{N}}^{k_{N}}$.

Similarly, substituting in (1.23) $\tau(z)\equiv 0$ one obtains from Theorem
1.2.

######  Corollary 3.13.

The function $\frac{Q^{+}_{2N}(z)}{P^{+}_{2N}(z)}$ belongs to the class
${\mathbf{N}}_{\kappa_{N}}^{k_{N}^{+}}$.

###  3.4. Two lemmata about linear fractional transformations $T_{M}$ and
$T_{L}$

The statements of Lemmata 3.8 and 3.9 fail to hold if the conditions

(3.32) $\tau(z)^{-1}=o(z)\mbox{ as }z\widehat{\rightarrow}\infty,$ (3.33)
$\phi(z)=o(1)\quad\mbox{ as }z\widehat{\rightarrow}\infty$

are not satisfied. In these cases the number of moments interpolating by the
linear fractional transformations $T_{W}[\tau]$ can be reduced and also their
indices can decrease. We will start with the linear fractional transformations
$T_{M}$ in the simplest cases when $m$ is a positive constant and hence
$M\in{\mathcal{U}}_{0}^{0}(J)$.

######  Lemma 3.14.

Let $M(z)$ be a $2\times 2$ matrix valued function

(3.34) $M(z)=\begin{pmatrix}1&0\\\ -zm&1\\\ \end{pmatrix},{\quad m>0,}$

let $\tau\in{\mathbf{N}}_{\kappa}^{k}$ be a function from
${\mathbf{N}}_{\kappa}^{k}$, such that $\tau(z)^{-1}\neq o(z)\mbox{ as
}z\widehat{\rightarrow}\infty$ and let $\phi=T_{M}[\tau]$. Then:

  1. ( i)

either $\phi\in{\mathbf{N}}_{\kappa}^{k}$ and $\phi(z)=o(1)\mbox{ as
}z\widehat{\rightarrow}\infty$,

  2. ( ii)

or $\phi\in{\mathbf{N}}_{\kappa-1}^{k}$.

  * Proof.

1. Verification of ${(i)}$ for $\tau\in{\mathbf{N}}_{\kappa}^{k}$, such that$:$

(3.35) $\lim_{z\widehat{\rightarrow}\infty}\frac{-1}{z\tau(z)}=-\infty.$

By Lemma 2.1 $-\tau^{-1}\in{\mathbf{N}}_{\kappa}^{-k}$. If (3.35) holds then
$mz-\tau(z)^{-1}$ has GPNT (generalized pole of negative type) at $\infty$ of
the same multiplicity as $-\tau(z)^{-1}$, i.e.

(3.36)
${\kappa_{\infty}\left(mz-\frac{1}{\tau(z)}\right)=\kappa_{\infty}\left(-\frac{1}{\tau(z)}\right).}$

Since also

(3.37)
$\kappa_{\infty}\left(m-\frac{1}{z\tau(z)}\right)=\kappa_{\infty}\left(-\frac{1}{z\tau(z)}\right).$

one obtains by Theorem 2.2 $mz-\tau(z)^{-1}\in{\mathbf{N}}_{\kappa}^{-k}$.
Hence $\phi(z)=\frac{-1}{mz-\tau(z)^{-1}}\in{\mathbf{N}}_{\kappa}^{k}$ by
Lemma 2.1 and, moreover, $\phi(z)=o(1)$ as $z\widehat{\rightarrow}\infty$,
since

$\lim_{z\widehat{\rightarrow}\infty}\phi(z)=\lim_{z\widehat{\rightarrow}\infty}\frac{1/z}{m-(z\tau(z))^{-1}}=0.$

2. Verification of $({i})$ for $\tau\in{\mathbf{N}}_{\kappa}^{k}$, such that$:$

(3.38) $\lim_{z\widehat{\rightarrow}\infty}\frac{-1}{z\tau(z)}=a$

and $a\geq 0$. In this case $\kappa_{\infty}(-\tau(z)^{-1})=0$ and
$-\tau(z)^{-1}$ admits the representation

(3.39) $-\tau(z)^{-1}=az-\tau_{1}(z)^{-1},$

where $\lim\limits_{z\widehat{\rightarrow}\infty}\frac{-1}{z\tau_{1}(z)}=0$.
Then the function $mz-\tau(z)^{-1}=(m+a)z-\tau_{1}(z)^{-1}$ has no GPNT at
$\infty$ since

$\lim_{z\widehat{\rightarrow}\infty}\left(m-\frac{1}{z\tau(z)}\right)=m+a>0.$

Then (3.36) and (3.37) hold and by Theorem 2.2
$mz-\tau(z)^{-1}\in{\mathbf{N}}_{\kappa}^{-k}$. Moreover,

(3.40)
$\lim_{z\widehat{\rightarrow}\infty}\phi(z)=\lim_{z\widehat{\rightarrow}\infty}\frac{1/z}{m+a-(z\tau(z))^{-1}}=0.$

and hence $({i})$ holds.

3. Verification of $({i})$ for $\tau\in{\mathbf{N}}_{\kappa}^{k}$, such that (3.38) holds and $a<-m$ $:$ In this case

$\lim_{z\widehat{\rightarrow}\infty}\left(m-\frac{1}{z\tau(z)}\right)=m+a<0$

and hence

$\kappa_{\infty}\left(mz-\frac{1}{\tau(z)}\right)=\kappa_{\infty}\left(-\frac{1}{\tau(z)}\right)=1.$

Since also (3.37) holds then by Theorem 2.2
$mz-\tau(z)^{-1}\in{\mathbf{N}}_{\kappa}^{-k}$, and by (3.40) $\phi(z)=o(1)$
as $z\widehat{\rightarrow}\infty$. Hence (1) holds.

4. Verification of $({ii})$ for $\tau\in{\mathbf{N}}_{\kappa}^{k}$, such that (3.38) holds and $a\in[-m,0)$ $:$ In this case

$\kappa_{\infty}\left(mz-\frac{1}{\tau(z)}\right)=0,\quad\kappa_{\infty}\left(-\frac{1}{\tau(z)}\right)=1$

and since (3.37) holds then $mz-\tau(z)^{-1}\in{\mathbf{N}}_{\kappa-1}^{-k}$
by Theorem 2.2. Therefore,
$\phi(z)=\frac{-1}{mz-\tau(z)^{-1}}\in{\mathbf{N}}_{\kappa-1}^{k}$ by Lemma
2.1. This proves $({ii})$. $\Box$

######  Lemma 3.15.

Let $L$ be the $2\times 2$ matrix

(3.41) $L=\begin{pmatrix}1&l\\\ 0&1\\\ \end{pmatrix},\quad\mbox{ where }l>0,$

let $\phi$ be a function from ${\mathbf{N}}_{\kappa}^{k}$, such that

(3.42) $\phi(z)\neq o(1){\quad\mbox{ as }z\widehat{\rightarrow}\infty}$

and let $\tau=T_{L}[\phi]=l+\phi$. Then:

  1. ( i)

either $\tau\in{\mathbf{N}}_{\kappa}^{k}$ and $\tau(z)^{-1}=o(z)\mbox{ as
}z\widehat{\rightarrow}\infty$,

  2. ( ii)

or $\tau\in{\mathbf{N}}_{\kappa}^{k-1}$.

  * Proof.

1. Verification of $({i})$ for $\phi\in{\mathbf{N}}_{\kappa}^{k}$, such that $z\phi(z)$ has a GPNT at $\infty$ and$:$

(3.43) $\lim_{z\widehat{\rightarrow}\infty}\phi(z)=-\infty.$

The function $z\tau(z)=lz+z\phi(z)$ has GPNT at $\infty$ of the same
multiplicity as $z\phi(z)$ and by Theorem 2.2 $z\tau(z)\in{\mathbf{N}}_{k}$.
Hence $\tau\in{\mathbf{N}}_{\kappa}^{k}$ and the equality

$\lim_{z\widehat{\rightarrow}\infty}\frac{1}{z\tau(z)}=\lim_{z\widehat{\rightarrow}\infty}\frac{1}{lz+z\phi(z)}{=0}$

implies that $\tau(z)^{-1}=o(z)$ as $z\widehat{\rightarrow}\infty$. Thus (1)
holds.

2. Verification of $({i})$ for $\phi\in{\mathbf{N}}_{\kappa}^{k}$, such that $\lim\limits_{z\widehat{\rightarrow}\infty}\phi(z)=a$ and $a>0$$:$ In this case $z\phi(z)$ admits the representation

(3.44) $z\phi(z)=az+z\phi_{1}(z),$

where $\lim\limits_{z\widehat{\rightarrow}\infty}\phi_{1}(z)=0$ and
$\phi_{1}\in{\mathbf{N}}_{\kappa}^{k}$ by Theorem 2.2. The function
$z\tau(z)=lz+z\phi(z)$ has no GPNT at $\infty$ and therefore
$\tau\in{\mathbf{N}}_{\kappa}^{k}$. Moreover,

(3.45)
$\lim_{z\widehat{\rightarrow}\infty}\frac{1}{z\tau(z)}=\lim_{z\widehat{\rightarrow}\infty}\frac{1}{(l+a)z+z\phi(z)}=0.$

and hence $({i})$ holds.

3. If $\phi\in{\mathbf{N}}_{\kappa}^{k}$, $\lim\limits_{z\widehat{\rightarrow}\infty}\phi(z)=a$ and $a<-l$, then also $({i})$ holds$:$ In this case

$\kappa_{\infty}(z\phi)=1\quad\mbox{and}\quad\kappa_{\infty}(z\tau)=1,$

that is the functions $z\phi(z)$ and $z\tau(z)$ have GPNT at $\infty$ of the
same multiplicity. By Theorem 2.2 $\tau\in{\mathbf{N}}_{\kappa}^{k}$ and by
(3.45) $\tau(z)^{-1}=o(z)$ as $z\widehat{\rightarrow}\infty$. Hence $({i})$
holds.

4. If $\lim\limits_{z\widehat{\rightarrow}\infty}\phi(z)=a$ and $a\in[-l,0)$ then $({ii})$ holds$:$ In this case

$\kappa_{\infty}(z\phi)=1\quad\mbox{and}\quad\kappa_{\infty}(z\tau)=0,$

and hence $z\tau\in{\mathbf{N}}_{k-1}$ by Theorem 2.2. Thus
$\tau\in{\mathbf{N}}_{\kappa}^{k-1}$. $\Box$

######  Corollary 3.16.

Let a matrix valued function $W_{2n}(z)$ of the form (1.26) belong to the
class ${\mathcal{U}}_{0}^{0}(J)$, let $\phi\in{\mathbf{N}}_{\kappa}^{k}$,
(3.33) fails to hold, let $n\geq\mbox{min }\\{\kappa,k\\}$ and let
$f=T_{W_{2n}}[\phi]$. Then there exists $r\in{\mathbb{Z}}_{+}$
$(r\leq\mbox{min }\\{\kappa,k\\})$, such that :

  1. ( i)

either $f\in{\mathcal{M}}_{\kappa-r}^{k-r}(\mathbf{s},2(n-r)-2)$;

  2. ( ii)

or $f\in{\mathcal{M}}_{\kappa-r}^{k-r-1}(\mathbf{s},2(n-r)-3)$.

In particular, $f(z)=o(1)$ as $z\widehat{\rightarrow}\infty$.

  * Proof.

By Theorem 3.11 the matrix valued function $W_{2n}(z)$ admits the
factorization

(3.46) $W_{2n}(z)=M_{1}(z)L_{1}\dots M_{n}(z)L_{n}.$

Let us denote $\tau_{1}:=T_{L_{n}}[\phi]$. By Lemma 3.15 there are three
possibilities

    1. (a1)

either $\tau_{1}\in\mathbf{N}_{\kappa}^{k}$ and $\tau_{1}^{-1}(z)=o(z)$;

    2. (a2)

or $\tau_{1}\in\mathbf{N}_{\kappa}^{k-1}$ and $\tau_{1}^{-1}(z)=o(z)$;

    3. (a3)

or $\tau_{1}\in\mathbf{N}_{\kappa}^{k-1}$ and $\tau_{1}^{-1}(z)\neq o(z)$.

In the case (a1) one obtains by Theorem 3.11
$f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s},2n-2)$, which gives (i) with $r=0$.

In the case (a2) by Theorem 3.11
$f\in{\mathcal{M}}_{\kappa}^{k-1}(\mathbf{s},2n-2)\subset{\mathcal{M}}_{\kappa}^{k-1}(\mathbf{s},2n-3)$,
which gives ( ii) with $r=0$.

In the case (a3) we apply Lemma 3.14 to the function
$\phi_{1}(z)=T_{M_{n}}[\tau_{1}(z)]$ and then again there are three
possibilities:

    1. (b1)

$\phi_{1}:=T_{M_{n}}[\tau]\in\mathbf{N}_{\kappa}^{k-1}$ and
$\phi_{1}(z)=o(1)$;

    2. (b2)

$\phi_{1}:=T_{M_{n}}[\tau]\in\mathbf{N}_{\kappa-1}^{k-1}$ and
$\phi_{1}(z)=o(1)$;

    3. (b3)

$\phi_{1}:=T_{M_{n}}[\tau]\in\mathbf{N}_{\kappa-1}^{k-1}$ and $\phi_{1}(z)\neq
o(1)$.

In the case (b1) by Theorem 1.2 one obtains
$f\\!\\!\in\\!\\!{\mathcal{M}}_{\kappa}^{k\\!-\\!1}(\mathbf{s},2n\\!-\\!3)$
which gives (2) with $r\\!=\\!0$.

In the case (b2) by Theorem 1.2
$f\in{\mathcal{M}}_{\kappa-1}^{k-1}(\mathbf{s},2n-3)\subset{\mathcal{M}}_{\kappa-1}^{k-1}(\mathbf{s},2n-4)$
which gives ( i) with $r=1$.

In the case (b3) one should continue this process based on Lemma 3.14, Lemma
3.15, Theorem 1.2 and Theorem 3.11.

If, for instance, $k\leq\kappa$ and if the process will not stop until the
step $r=k$, then when applying Lemma 3.15 to
$\phi_{k}\in{\mathbf{N}}_{\kappa-k}^{0}$ one obtains only one possibility:
$\tau_{k+1}\in{\mathbf{N}}_{\kappa-k}^{0}$ and $\tau_{k+1}(z)^{-1}=o(z)$ as
$z\widehat{\to}\infty$. Then the process stops and
$f\in{\mathcal{M}}_{\kappa-k}^{0}(\mathbf{s},2(n-k)-2)$ by Theorem 3.11.

Similarly, one can treat the case $\kappa<k$ by using Lemma 3.14 and Theorem
1.2. $\Box$

##  4\. Full indefinite moment problem $MP_{\kappa}^{k}(\textbf{s})$.

###  4.1. Indefinite moment problem $MP_{\kappa}^{k}(\textbf{s})$ for
${\mathbf{s}}\in{\mathcal{H}}_{0}^{0}$.

Recall, that the moment problem $MP_{\kappa}^{k}({\mathbf{s}})$ is called
indeterminate if it has more then one solution. A sequence s is called
nondegenerate, if

(4.1) $\mbox{there is $N\in{\mathbb{N}}$},\,\mbox{ such that }\,\quad
D_{n}\neq 0,\quad D_{n}^{+}\neq 0\quad\mbox{for all \,\, $n\geq N$.}$

In this subsection we consider the indefinite moment problem
$MP_{\kappa}^{k}(\textbf{s})$ for a nondegenerate sequence
${\mathbf{s}}\in{\mathcal{H}}_{0}^{0}$. As is known, see [1, Theorem 0.5], for
a nondegenerate sequence ${\mathbf{s}}\in{\mathcal{H}}_{0}^{0}$ the
corresponding moment problem $MP_{0}^{0}(\textbf{s})$ is indeterminate, if and
only if

(4.2)
$\sum_{i=1}^{\infty}m_{i}<\infty\quad\mbox{and}\quad\sum_{i=1}^{\infty}l_{i}<\infty.$

######  Theorem 4.1.

Let s be a nondegenerate sequence from ${\mathcal{H}}_{0}^{0}$ and let (4.2)
holds. Then:

  1. ( i)

The moment problem $MP_{\kappa}^{k}(\textbf{s})$ is solvable and indeterminate
for any pair of $\kappa,k\in{\mathbb{Z}}_{+}$.

  2. ( ii)

The sequence of resolvent matrices $W_{2n}(z)$ converges to an entire matrix
valued function $W_{\infty}^{+}(z)=(w_{ij}^{+}(z))_{i,j=1}^{2}$ of order
$\leq$ 1/2.

  3. ( iii)

The formula

(4.3)
$f(z)=\frac{w_{11}^{+}(z)\tau(z)+w_{12}^{+}(z)}{w_{21}^{+}(z)\tau(z)+w_{22}^{+}(z)}$

establishes a one-to-one correspondence between the class
${\mathcal{M}}_{\kappa}^{k}(\textbf{s})$ and the set of functions
$\tau\in\mathbf{N}_{\kappa}^{k}$.

  * Proof.

1. Verification of ( ii): Due to [27, Appendix II.13] the Stieltjes moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$ is indeterminate if and only if (4.2) holds and in this case the sequence $W_{2N}(z)$ (see (1.26)) of its resolvent matrices converges to an entire matrix valued function $W_{\infty}^{+}(z)=(w_{ij}^{+}(z))_{i,j=1}^{2}$ of order $\leq$ 1/2.

2. Verification of the implication $f\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s})\Longrightarrow\tau\in\mathbf{N}_{\kappa}^{k}$ $:$ Since $f\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s})$ then $f\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s},2n-1)$ for every $n\in{\mathbb{N}}$. By Theorem 1.2 there exists a sequence of functions $\phi_{N}(z)\in\mathbf{N}_{\kappa}^{k}$ such that (3.33) holds and

(4.4) $f(z)=T_{W_{2N}(z)}[\phi_{N}(z)]\quad\textup{for every
$N\in{\mathbb{N}}$.}$

It follows from (4.4) that

$-(Q^{+}_{2N-1}(z)-f(z)P^{+}_{2N-1}(z))\phi_{N}(z)=Q^{+}_{2N}(z)-f(z)P^{+}_{2N}(z).$

Notice that $Q^{+}_{2N-1}(z)-f(z)P^{+}_{2N-1}(z)\not\equiv 0$, since otherwise
we would have $Q^{+}_{2N}(z)-f(z)P^{+}_{2N}(z)\equiv 0$, which contradicts the
generalized Liouville-Ostrogradsky identity (3.7).

As was mentioned above the matrix valued functions $W_{{2N}}(z)$ converges
locally uniformly in ${\mathbb{C}}$ to
$W_{\infty}^{+}(z)=(w_{ij}^{+}(z))_{i,j=1}^{2}$, in particular,

$w_{11}^{+}(z)=\lim_{N\to\infty}Q^{+}_{{2N}-1}(z),\quad
w_{21}^{+}(z)=\lim_{N\to\infty}P^{+}_{{2N}-1}(z).$

By Corollary 3.12
$\frac{Q^{+}_{{2N}-1}}{P^{+}_{{2N}-1}}\in\mathbf{N}_{0}^{0}=S$. Hence the
limiting function $\frac{w_{11}^{+}(z)}{w_{21}^{+}(z)}$ belongs to
$\mathbf{N}_{0}^{0}=S$ and it corresponds to $\phi=\infty$ in the linear
fractional transformation $f=T_{W_{\infty}^{+}}[\tau]$. Apart from this case
the function

(4.5)
$w_{11}^{+}(z)-f(z)w_{21}^{+}(z)=\lim_{N\to\infty}(Q^{+}_{{2N}-1}(z)-f(z)P^{+}_{{2N}-1}(z))$

is not identically equal to 0. Let $\Omega$ be the open set of points in
${\mathbb{C}}_{+}$ such that $w_{11}^{+}(z)-f(z)w_{21}^{+}(z)\neq 0$. Then for
every point $z\in\Omega$ the sequence of functions

$\phi_{N}(z):=\frac{Q^{+}_{{2N}}(z)-f(z)P^{+}_{{2N}}(z)}{Q^{+}_{{2N}-1}(z)-f(z)P^{+}_{{2N}-1}(z)}$

is correctly defined in a neighborhood of $z$ and converges locally uniformly
in $\Omega$ to a function $\phi(z)$. Since
$\phi_{N}\in\mathbf{N}_{\kappa}^{k}$ then
$\phi\in\mathbf{N}_{\kappa^{\prime}}^{k^{\prime}}$ with
$\kappa^{\prime}\leq\kappa$ and $k^{\prime}\leq k$. It follows from (4.4) that

(4.6) $f(z)=T_{W_{\infty}^{+}(z)}[\phi(z)]$

and by Lemma 3.7 $f\in\mathbf{N}_{\kappa^{\prime\prime}}^{k^{\prime\prime}}$
with $\kappa^{\prime\prime}\leq\kappa^{\prime}\leq\kappa$ and
$k^{\prime\prime}\leq k^{\prime}\leq k$. Since $f\in\mathbf{N}_{\kappa}^{k}$
this implies $\kappa^{\prime\prime}=\kappa^{\prime}=\kappa$ and
$k^{\prime\prime}=k^{\prime}=k$ and hence $\tau\in\mathbf{N}_{\kappa}^{k}$.

3. Proof of the fact that $T_{W_{\infty}^{+}}[\tau]$ satisfies (3.33) for every $\tau\in\mathbf{N}_{\kappa}^{k}$, $\kappa,k\in{\mathbb{Z}}_{+}$. Application of the Schur algorithm to the Stieltjes moment problem $\mathcal{M}_{\kappa}^{k}({\mathbf{s}})$ gives on the $N-$th step an induced sequence ${\mathbf{s}^{(N)}}$, which can be found as the sequence of coefficients of the series expansion $-{\sum\limits_{i=0}^{\infty}{s_{i}^{(N)}}{z^{-(i+1)}}}$ corresponding to the continued fraction (2.17). Denote by $W^{(N)}_{2j}(z)$ the resolvent matrix of the corresponding Stieltjes moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}},2j-1)$.

(4.7) $W^{(N)}_{2j}(z)=\begin{pmatrix}Q^{N,+}_{2j-1}(z)&Q^{N,+}_{2j}(z)\\\
P^{N,+}_{2j-1}(z)&P^{N,+}_{2j}(z)\\\ \end{pmatrix},\quad j\in{\mathbb{N}}.$

Then for every $j\in{\mathbb{N}}$ the resolvent matrix $W_{2j}(z)$ admits the
factorization (see [17, Proposition 4.7])

(4.8) $W_{2j}(z)=W_{{2N}}(z)W_{2(j-N)}^{(N)}(z).$

Taking the limit as $j\to\infty$ one obtains the following factorization of
the entire matrix valued function $W_{\infty}^{+}(z)$

(4.9) $W_{\infty}^{+}(z)=W_{{2N}}(z)W_{\infty}^{(N)}(z),$

where $W_{\infty}^{(N)}(z)$ is the resolvent matrix of the induced Stieltjes
moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$. Then
$f=T_{W_{\infty}^{+}}[\tau]$ admits the reresentation

(4.10)
$f=T_{W_{{2N}}}[\phi_{N}],\quad\mbox{where}\quad\phi_{N}=T_{W_{\infty}^{(N)}}[\tau]\in\mathbf{N}_{\kappa^{\prime}}^{k^{\prime}},\,\,\mbox{with
$\kappa^{\prime}\leq\kappa$ and $k^{\prime}\leq k$}.$

Since ${W_{{2N}}}\in{\mathcal{U}}_{0}^{0}$ by Corollary 3.16 one obtains
$f=o(1)$, if $N>\mbox{min }\\{\kappa,k\\}$.

4. Verification of the implication $\tau\in\mathbf{N}_{\kappa}^{k}\Longrightarrow f=T_{W_{\infty}^{+}}[\tau]\in{\mathcal{M}}_{\kappa}^{k}(\textbf{s})$: Due to the above item ( iii) the function $\phi_{N}$ from (4.10) satisfies the condition (3.33).

Next By Theorem 1.2
$f\in{\mathcal{M}}_{\kappa^{\prime}}^{k^{\prime}}(\mathbf{s},2N-1)$. Since $N$
can be chosen arbitrarily large
$f\in{\mathcal{M}}_{\kappa^{\prime}}^{k^{\prime}}(\mathbf{s})$. Now it follows
from item $\mathbf{2}$ that $\tau\in\mathbf{N}_{\kappa^{\prime}}^{k^{\prime}}$
and hence $\kappa^{\prime}=\kappa$ and $k^{\prime}=k$.

5. Verification of ( i): Solvability of the indefinite Stieltjes moment problem $MP_{\kappa}^{k}(\textbf{s})$ for any pair of $\kappa,k\in{\mathbb{Z}}_{+}$ follows from item $\mathbf{4}$. The formula (4.3) gives two different solutions of the problem $MP_{\kappa}^{k}(\textbf{s})$ for different parameter functions $\tau_{1},\tau_{2}\in\mathbf{N}_{\kappa}^{k}$ and thus the problem $MP_{\kappa}^{k}(\textbf{s})$ is indeterminate. $\Box$

###  4.2. Indefinite moment problem $MP_{\kappa}^{k}(\textbf{s})$, general
case.

######  Theorem 4.2.

Let s be a nondegenerate sequence from
${\mathcal{H}}_{\kappa_{0}}^{k_{0},reg}$, $\kappa_{0},k_{0}\in{\mathbb{N}}$
and let ${l}_{j}$ and $m_{j}(z)$ ($j\in\mathbb{N}$) be parameters of the
generalized $\mathbf{S}-$fraction (2.12). Then the moment problem
$MP_{\kappa_{0}}^{k_{0}}(\textbf{s})$ is indeterminate, if and only if

(4.11)
$\sum_{i=1}^{\infty}m_{i}(0)<\infty\quad\mbox{and}\quad\sum_{i=1}^{\infty}l_{i}<\infty.$

If (4.11) holds, then:

  1. (i)

The sequence of resolvent matrices $W_{2n}(z)$ converges to an entire matrix
valued function $W_{\infty}^{+}(z)=(w_{ij}^{+}(z))_{i,j=1}^{2}$ of order
$\leq$ 1/2.

  2. (ii)

The moment problem $MP_{\kappa}^{k}(\textbf{s})$ is solvable, if and only if

(4.12) $\kappa_{0}\leq\kappa,\quad\mbox{and}\quad k_{0}\leq k.$

  3. (iii)

The formula

(4.13)
$f(z)=\frac{w_{11}^{+}(z)\tau(z)+w_{12}^{+}(z)}{w_{21}^{+}(z)\tau(z)+w_{22}^{+}(z)}$

establishes a one-to-one correspondence between the class
${\mathcal{M}}_{\kappa}^{k}(\textbf{s})$ and the set of functions
$\tau\in\mathbf{N}_{\kappa-\kappa_{0}}^{k-k_{0}}$.

  * Proof.

1. Redaction of the indefinite moment problem $MP_{\kappa}^{k}(\textbf{s})$ to a classical one: Let us choose $N$ big enough, so that

(4.14)
$\nu_{-}(S_{j})=\kappa_{0}=\nu_{-}(S_{n_{N}}),\quad\nu_{-}(S_{j}^{(1)})=k_{0}=\nu_{-}(S_{n_{N}}^{(1)})\quad\mbox{for
all}\quad j\geq n_{N}.$

Then the induced sequence ${\mathbf{s}^{(N)}}$ which arises on the $N-$th step
of the Schur algorithm (see Section 2.2) belongs to the class
${\mathcal{H}}_{0}^{0}$. The corresponding Stieltjes moment problem
$\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$ is classical. By Lemma 2.4

(4.15) $f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s})\Longleftrightarrow
f_{N}=T_{W_{2n_{N}}}[f]\in{\mathcal{M}}_{\kappa-\kappa_{0}}^{k-k_{0}}(\mathbf{s}^{(N)}).$

In particular,

(4.16)
$f\in{\mathcal{M}}_{\kappa_{0}}^{k_{0}}(\mathbf{s})\Longleftrightarrow{f_{N}=T_{W_{2n_{N}}}[f]\in{\mathcal{M}}_{0}^{0}(\mathbf{s}^{(N)}).}$

Hence the problem $\mathcal{M}_{\kappa_{0}}^{k_{0}}({\mathbf{s}^{(N)}})$ is
indeterminate if and only if the Stieltjes moment problem
$\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$ is indeterminate. The latter is
equivalent to (4.11).

2. Verification of (i): Let $P^{N,+}_{j}(z)$ and $Q^{N,+}_{j}(z)$ be generalized Stieltjes polynomials associated with the moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$ and let $W^{(N)}_{2j}$ (see (4.7)). If (4.11) holds, then the Stieltjes moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$ is indeterminate and the sequence $W^{(N)}_{2j}$ converges to an entire matrix valued function $W^{(N)}_{\infty}(z)=(w_{ij}^{(N)}(z))_{i,j=1}^{2}$ of order $\leq$ 1/2.

The resolvent matrices $W_{2j}(z)$ of the indefinite moment problem
$MP_{\kappa_{0}}^{k_{0}}(\textbf{s},2j-1)$ are connected with the resolvent
matrices $W^{(N)}_{2(j-N)}(z)$ of the induced moment problem
$MP_{0}^{0}(\textbf{s}^{(N)},2(j-N)-1)$ by the formula (4.8). Therefore, the
sequence of matrix valued functions $W_{2n-1}(z)$ also converges to an entire
matrix valued function $W_{\infty}^{+}(z)=(w_{ij}^{+}(z))_{i,j=1}^{2}$ of
order $\leq$ 1/2, which is connected with $W_{\infty}^{(N)}(z)$ by the formula
(4.9).

3. Verification of (ii) and (iii) : By Proposition 2.1 (v) the inequalities (4.12) are necessary fo solvability of the problem ${\mathcal{M}}_{\kappa}^{k}(\textbf{s})$.

Now assume that (4.12) holds, let
$\tau\in\mathbf{N}_{\kappa-\kappa_{0}}^{k-k_{0}}$ and let
$f=T_{W_{\infty}^{+}}[\tau]$. Then by Theorem 4.1

(4.17)
$f_{N}=T_{W_{\infty}^{(N)}}[\tau]\in{\mathcal{M}}_{\kappa-\kappa_{0}}^{k-k_{0}}(\mathbf{s}^{(N)}).$

By (4.9) and Lemma 2.4

(4.18) $f=T_{W_{2N}}[f_{N}]\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s}).$

Conversely, let $f\in{\mathcal{M}}_{\kappa}^{k}(\mathbf{s})$. Then by Lemma
2.4 there is a function
$f_{N}\in{\mathcal{M}}_{\kappa-\kappa_{0}}^{k-k_{0}}(\mathbf{s}^{(N)})$ such
that $f=T_{W_{2N}}[f_{N}]$. Hence by Theorem 4.1 there exists a function
$\tau\in\mathbf{N}_{\kappa-\kappa_{0}}^{k-k_{0}}$, such that
$f_{N}=T_{W_{\infty}^{(N)}}[\tau]$. Therefore, $f=T_{W_{\infty}^{+}}[\tau]$
for $\tau\in\mathbf{N}_{\kappa-\kappa_{0}}^{k-k_{0}}$. $\Box$

######  Proposition 4.3.

Let
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
let $P_{i}(z)$ and $Q_{i}(z)$ ($i\in{\mathbb{Z}}_{+}$) be Lanczos polynomials
of the first and second kind and let ${l}_{j}$ and $m_{j}(z)$
($j\in\mathbb{N}$) be parameters of the generalized $\mathbf{S}-$fraction
(2.12). Assume additionally that

(4.19) $l_{j}>0\quad\mbox{for all\,\, $j\in{\mathbb{N}}$}.$

Then the following statements are equivalent:

  1. (i)

the moment problem $MP_{\kappa}(\textbf{s})$ is indeterminate;

  2. (ii)

the following series converge

(4.20)
$\sum_{i=0}^{\infty}|P_{i}(0)|^{2}\widetilde{b}_{i}^{-1}<\infty\quad\mbox{ and
}\quad\sum_{i=0}^{\infty}|Q_{i}(0)|^{2}\widetilde{b}_{i}^{-1}<\infty;$

  3. (iii)

the following series converges

(4.21) $\sum_{i=1}^{\infty}(l_{1}+l_{2}+\cdots+l_{i})^{2}d_{i+1}(0)<\infty.$

  * Proof.

By [8, 31] $(i)$ and $(ii)$ are equivalent.

Let us show, that $(ii)$ and $(iii)$ are equivalent. By the first equality in
(3.10)

$Q_{i}(0)=-(l_{1}+l_{2}+\cdots+l_{i})P_{i}(0).$

and hence by (3.11)

$\sum_{i=0}^{N}|Q_{i}(0)|^{2}\widetilde{b}_{i}^{-1}=\sum_{i=0}^{N}(l_{1}+l_{2}+\cdots+l_{i})^{2}|P_{i}(0)|^{2}\widetilde{b}_{i}^{-1}=\sum_{i=1}^{N}(l_{1}+l_{2}+\cdots+l_{i})^{2}d_{i+1}(0).$

This proves the implication $(ii)\Rightarrow(iii)$.

Conversely, if
$\sum\limits_{i=0}^{\infty}|Q_{i}(0)|^{2}\widetilde{b}_{i}^{-1}<\infty$, then
the convergence of the series
$\sum\limits_{i=0}^{\infty}|P_{i}(0)|^{2}\widetilde{b}_{i}^{-1}$ follows from
the inequality

$l_{1}^{2}\sum_{i=0}^{N}|P_{i}(0)|^{2}\widetilde{b}_{i}^{-1}\leq\sum_{i=0}^{N}(l_{1}+l_{2}+\cdots+l_{i})^{2}|P_{i}(0)|^{2}\widetilde{b}_{i}^{-1}{=\sum\limits_{i=0}^{\infty}|Q_{i}(0)|^{2}\widetilde{b}_{i}^{-1}}.$

This proves the implication $(iii)\Rightarrow(ii)$. $\Box$

######  Remark 4.4.

In the case $\kappa=k=0$ the above criterion for the Hamburger moment problem
to be indeterminate is well known, see [1, Appendix, Theorem 0.5]. When
treating system (3.1) as a Stieltjes string with masses $m_{j}$ and lengthes
$l_{j}$ one can consider series (4.21) as ”the moment of inertia” of the
string.

##  5\. Padé approximants

######  Definition 5.1.

([3]) The $[L/M]$ Padé approximant for a formal power series

(5.1) $-\sum_{j=0}^{\infty}\frac{s_{j}}{z^{j+1}}$

is a ratio

(5.2) $f^{[L/M]}({z})=\frac{A^{[L/M]}(1/{z})}{B^{[L/M]}(1/{z})}$

of polynomials $A^{[L/M]}$, $B^{[L/M]}$ of formal degree $L$, $M$,
respectively, such that $B^{[L/M]}(0)\neq 0$ and

(5.3)
$f^{[L/M]}(z)+\sum_{j=0}^{L+M}\frac{s_{j}}{z^{j+1}}=O\left(\frac{1}{z^{L+M+1}}\right),\quad
z\widehat{\to}\infty.$

The $[n/n]$ Padé approximant is called diagonal and the $[n/n-1]$ Padé
approximant is called subdiagonal.

######  Remark 5.2.

Notice, that for diagonal Padé approximants the representation (5.2) is
equivalent to the representation

$f^{[n/n]}({z})=\frac{z^{n}A^{[n/n]}(1/{z})}{z^{n}B^{[n/n]}(1/{z})}$

as a ratio, where the numerator $z^{n}A^{[n/n]}(1/{z})$ is a polynomial of
formal degree $n$ and the denominator $z^{n}B^{[n/n]}(1/{z})$ is a polynomial
of exact degree $n$. For subdiagonal Padé approximants the representation
(5.2) is equivalent to the representation

$f^{[n/n-1]}({z})=\frac{z^{n}A^{[n/n-1]}(1/{z})}{z^{n}B^{[n/n-1]}(1/{z})}$

as a ratio, where the numerator is a polynomial of formal degree $n$ and the
denominator is a polynomial of exact degree $n$ vanishing at $0$.

Explicit formula for diagonal Padé approximants for sequences
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}$
was found in [7], in the classical case $\textbf{s}\in{\mathcal{H}}_{0}$ see
[3, 38]. In this section we will formulate the corresponding statements for
sequences
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$.

######  Proposition 5.3.

Let
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
$\kappa,k\in{\mathbb{Z}}_{+}$. Then the $[n/n]$ Padé approximant for a formal
power series (5.1) exists if $n\in{\mathcal{N}}({\bf s})$ and

(5.4)
$f^{[n_{j}/n_{j}]}(z)=-\frac{Q_{j}(z)}{P_{j}(z)}=\frac{Q_{2j}^{+}(z)}{P_{2j}^{+}(z)},\quad
j\in{\mathbb{N}}.$

  * Proof.

We present a proof of this statement from [17] for the convenience of the
reader. It follows from (3.3) and Theorem 3.11 that the function

$-\frac{Q_{j}(z)}{P_{j}(z)}=\frac{Q_{2j}^{+}(z)}{P_{2j}^{+}(z)}=T_{W_{2j}(z)}[0]$

belongs to ${\mathcal{M}}({\bf s},2n_{j}-1)$. Therefore, the function
$-\frac{Q_{j}(z)}{P_{j}(z)}$ has the asymptotic

(5.5)
$-\frac{Q_{j}(z)}{P_{j}(z)}=-\frac{s_{0}}{z}-\dots-\frac{s_{2n_{j}-1}}{z^{2n_{j}}}+O\left(\frac{1}{z^{2n_{j}+1}}\right),\quad
z\widehat{\to}\infty.$

Since $Q_{j}(z)$ is a polynomial of degree $n_{j}-n_{1}<n_{j}$ and $P_{j}(z)$
is a polynomial of exact degree $n_{j}$ the function
$-\frac{Q_{j}(z)}{P_{j}(z)}$ is the $[n_{j}/n_{j}]$ Padé approximant for the
formal power series (5.1) due to Remark 5.2 and (5.5). $\Box$

In the following proposition stated in [16] without proof it is shown that the
sub-diagonal Padé approximants can be calculated in terms of generalized
Stieltjes polynomials.

######  Proposition 5.4.

Let
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
$\kappa,k\in{\mathbb{Z}}_{+}$. Then the $[n_{j}/n_{j}-1]$ Padé approximants
for the formal power series (5.1) exists and has the form

(5.6) $f^{[n_{j}/n_{j}-1]}(z)=\frac{Q_{2j-1}^{+}(z)}{P_{2j-1}^{+}(z)},\quad
j\in{\mathbb{N}}.$

  * Proof.

It follows from (3.3) that

$\frac{Q_{2j-1}^{+}(z)}{P_{2j-1}^{+}(z)}=T_{W_{2j-1}(z)}[\infty].$

By Theorem 3.11 the function $\frac{Q_{2j-1}^{+}(z)}{P_{2j-1}^{+}(z)}$ belongs
to ${\mathcal{M}}({\bf s},2n_{j}-2)$, and hence it has the asymptotic

(5.7)
$\frac{Q_{2j-1}^{+}(z)}{P_{2j-1}^{+}(z)}=-\frac{s_{0}}{z}-\dots-\frac{s_{2n_{j}-2}}{z^{2n_{j}-1}}+O\left(\frac{1}{z^{2n_{j}+1}}\right),\quad
z\widehat{\to}\infty.$

Since $Q_{2j-1}^{+}(z)$ is a polynomial of degree $n_{j}-n_{1}<n_{j}$ and
$P_{2j-1}^{+}(z)$ is a polynomial of exact degree $n_{j}$ vanishing at $0$ the
function $-\frac{Q_{2j-1}^{+}(z)}{P_{2j-1}^{+}(z)}$ is the $[n_{j}/n_{j}-1]$
Padé approximant for the formal power series (5.1) due to Remark 5.2 and
(5.7). $\Box$

######  Lemma 5.5.

Let
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
$\kappa,k\in{\mathbb{Z}}_{+}$, let $W_{2N}(z)$ be given by (1.26),
$N\in{\mathbb{N}}$ and let $\mathbf{s}^{(N)}$ be the induced sequence defined
in Lemma 2.4, and let

(5.8) $-\sum_{j=0}^{\infty}{s_{j}^{(N)}}{z^{-(j+1)}}$

be the corresponding formal power series. Then:

  1. (i)

the diagonal Padé approximants for the formal power series (5.1) are connected
with diagonal $g^{[n/n]}$ Padé approximants for the power series (5.8) by the
formula

(5.9)
$f^{[n_{j}/n_{j}]}(z)=T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)],\quad
j>N,\,j\in{\mathbb{N}}.$

  2. (ii)

the subdiagonal Padé approximants for the power series (5.1) are connected
with subdiagonal $g^{[n/n-1]}$ Padé approximants for the power series (5.8) by

(5.10)
$f^{[n_{j}/n_{j}-1]}(z)=T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}-1]}(z)],\quad
j>N,\,j\in{\mathbb{N}}.$

  * Proof.

Consider the induced moment problem
$MP_{\kappa-\kappa_{N}}^{k-k_{N}^{+}}(\mathbf{s}^{(N)},2(n_{j}-n_{N})-1)$ and
let

(5.11)
$W_{2(j-N)}^{(N)}(z)=\left(\begin{array}[]{cc}Q^{N,+}_{2(j-N)-1}(z)&Q^{N,+}_{2(j-N)}(z)\\\
P^{N,+}_{2(j-N)-1}(z)&P^{N,+}_{2(j-N)}(z)\\\ \end{array}\right)$

be the resolvent matrix of this moment problem. Then the matrices $W_{2j}(z)$
and $W_{2(j-N)}^{(N)}(z)$ are connected by (see (4.8))

(5.12) $W_{2j}(z)=W_{2N}(z)W_{2(j-N)}^{(N)}(z).$

Similarly, the resolvent matrix

(5.13)
$W_{2(j-N)-1}^{(N)}(z)=\left(\begin{array}[]{cc}Q^{N,+}_{2(j-N)-1}(z)&Q^{N,+}_{2(j-N)-2}(z)\\\
P^{N,+}_{2(j-N)-1}(z)&P^{N,+}_{2(j-N)-2}(z)\\\ \end{array}\right)$

of the moment problem
$MP_{\kappa-\kappa_{N}}^{k-k_{N}}(\mathbf{s}^{(N)},2(n_{j}-n_{N})-1)$ is
connected with the matrix $W_{2j-1}(z)$ by

(5.14) $W_{2j-1}(z)=W_{2N}(z)W_{2(j-N)-1}^{(N)}(z).$

By Proposition 5.3 diagonal Padé approximants
$g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)$ for the formal power series (5.8) are given
by

(5.15)
$g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)=\frac{Q_{2(n_{j}-n_{N})}^{N,+}(z)}{P_{2(n_{j}-n_{N})}^{N,+}(z)}=T_{W_{2(j-N)}^{(N)}(z)}[0],\quad
j\in{\mathbb{N}}.$

It follows from the factorization formula (5.12) and (5.15) that

$T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)]=T_{W_{2j}(z)}[0].$

Hence by Proposition 5.3 $T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)]$
coincides with the diagonal Padé approximants $f^{[n_{j}/n_{j}]}(z)$ for the
formal power series (5.1).

By Proposition 5.4 subdiagonal Padé approximants
$g^{[n_{j}-n_{N}/n_{j}-n_{N}-1]}(z)$ for the formal power series (5.8) are
given by

(5.16)
$g^{[n_{j}-n_{N}/n_{j}-n_{N}-1]}(z)=\frac{Q_{2(n_{j}-n_{N})-1}^{N,+}(z)}{P_{2(n_{j}-n_{N})-1}^{N,+}(z)}=T_{W_{2(j-N)-1}^{(N)}(z)}[\infty],\quad
j\in{\mathbb{N}}.$

It follows from the factorization formula (5.14) and (5.16) that

$T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}]}(z)]=T_{W_{2j}(z)}[\infty].$

Hence by Proposition 5.4 $T_{W_{2N}(z)}[g^{[n_{j}-n_{N}/n_{j}-n_{N}-1]}(z)]$
coincides with the subdiagonal Padé approximants $f^{[n_{j}/n_{j}-1]}(z)$ for
the formal power series (5.1). $\Box$

######  Theorem 5.6.

Let
$\textbf{s}=\left\\{s_{j}\right\\}_{j=0}^{\infty}\in{\mathcal{H}}_{\kappa}^{k,reg}$,
$\kappa,k\in{\mathbb{Z}}_{+}$. Then:

  1. (i)

If the problem $MP_{\kappa}^{k}(\mathbf{s})$ is determinate, then diagonal and
subdiagonal Padé approximants converge to the unique solution of
$MP_{\kappa}^{k}(\mathbf{s})$ locally uniformly on
${\mathbb{C}}\setminus{\mathbb{R}}_{+}$.

  2. (ii)

If the problem $MP_{\kappa}^{k}(\mathbf{s})$ is indeterminate, then the
sequence $f^{[n/n-1]}$ of subdiagonal Padé approximants converges locally
uniformly on ${\mathbb{C}}\setminus{\mathbb{R}}_{+}$, while the sequence
$f^{[n/n]}$ of diagonal Padé approximants is not convergent but precompact in
the topology of locally uniform convergence.

  * Proof.

Let us choose $N$ big enough, so that

(5.17)
$\nu_{-}(S_{j})=\kappa=\nu_{-}(S_{n_{N}}),\quad\nu_{-}(S_{j}^{(1)})=k=\nu_{-}(S_{n_{N}}^{(1)})\quad\mbox{for
all}\quad j\geq n_{N}.$

Then the induced sequence ${\mathbf{s}^{(N)}}$ belongs to the class
${\mathcal{H}}_{0}^{0}$ and by Lemma 5.5 the problem of convergence of
diagonal and subdiagonal Padé approximants is reduced to the corresponding
problem for diagonal and subdiagonal Padé approximants for the series
$-\sum\limits_{j=0}^{\infty}{s_{j}^{(N)}}{z^{-(j+1)}}$ corresponding to the
classical Stieltjes moment problem $\mathcal{M}_{0}^{0}({\mathbf{s}^{(N)}})$.
The stated results for classical Stieltjes moment problems were proved in [38,
Theorems 5.30, 5.31]. $\Box$

##  6\. Example. Laguerre polynomials

The monic Laguerre polynomials
$\widetilde{L}_{n}(z,\alpha):=(-1)^{n}z^{-\alpha}e^{z}(z^{\alpha+n}e^{-z})^{(n)}$
are solutions of the three–term difference equation (see [39])

(6.1)
$\begin{split}y_{n+1}(z)+(2n+\alpha+1-z)y_{n}(z)+(n+\alpha)ny_{n-1}(z),\quad
n\in\mathbb{N}.\end{split}$

subject to the initial conditions $\widetilde{L}_{-1}(z,\alpha)\equiv 0$,
$\widetilde{L}_{0}(z,\alpha)\equiv 1$.

If $\alpha>-1$, then the polynomials
$\\{\widetilde{L}_{n}(z,\alpha)\\}_{n=0}^{\infty}$ are orthogonal in the
Hilbert space $L_{2}(\mathbb{R}_{+},w_{\alpha})$, with the weight function
$w_{\alpha}(z)=z^{\alpha}e^{-z}$.

Here we consider the case when $\alpha<-1$ and $\alpha$ is not a negative
integer. The case when $\alpha$ is a negative integer was treated in [35]. If
$-k-1<\alpha<-k$, $k\in{\mathbb{N}}$, then $\widetilde{L}_{n}(z,\alpha)$ are
orthogonal polynomials with respect to the indefinite inner product (see [11],
[37])

(6.2) $\left\langle
f,g\right\rangle_{\alpha}=\int\limits_{0}^{\infty}x^{\alpha}\left(e^{-x}f\overline{g}-\sum_{j=0}^{k-1}(e^{-x}f\overline{g})^{(j)}(0)\frac{x^{j}}{j!}\right)dx.$

Polynomials $\widetilde{Q}_{n}(x,\alpha)$ of the second kind are defined as
solutions of (6.1) subject to the initial conditions
$\widetilde{Q}_{-1}(z,\alpha)\equiv-1$ and $\widetilde{Q}_{0}(z,\alpha)\equiv
0$.

The $P$ – fraction corresponding to the system (6.1) has the form (1.14) with
the atoms $(a_{n},b_{n})$ given by

(6.3) $b_{0}=\Gamma(1+\alpha),\quad b_{n}=n(n+\alpha)\quad\mbox{and}\quad
a_{n-1}(z)=z-2n-\alpha+1,\quad n\in\mathbb{N}.$

The moments $s_{n}$ for all $n\in{\mathbb{N}}$ are defined by

(6.4) $s_{n}=\mathfrak{S}(z^{n})=\Gamma(n+\alpha+1).$

By the Buslaev formula, see [5, formula (13)] the determinants of $S_{n}$ and
$S_{n}^{+}$ take the form

$D_{n}=b_{0}^{n}b_{1}^{n-1}\dots
b_{n-1}=\prod_{j=1}^{n}(j-2)!\Gamma(\alpha+j),\quad
D_{n}^{+}=\prod_{j=1}^{n}(j-2)!\Gamma(\alpha+j+1),\quad n\in{\mathbb{N}}.$

Therefore, the sequence $\textbf{s}=\\{s_{n}\\}_{n=0}^{\infty}$ is regular and
the set of its normal indices coincides with ${\mathbb{N}}$ and

$\textbf{s}\in\mathcal{H}_{k}^{k-1},\,\mbox{ if
}\,-2k<\alpha<-2k+1,\quad\textbf{s}\in\mathcal{H}_{k}^{k},\,\mbox{ if
}\,-2k-1<\alpha<-2k;$

Next, it follows from (1.6) that the parameters of the generalized S-fraction
take the form

$l_{n}:=\frac{D_{n}^{2}}{D_{n}^{+}D_{n-1}^{+}}=\frac{(n-1)!\Gamma^{2}(1+\alpha)}{\Gamma(1+\alpha+n)},\,m_{n}:=\frac{(D_{n-1}^{+})^{2}}{D_{n}D_{n-1}}=\frac{\Gamma(\alpha+n)}{(n-1)!\Gamma^{2}(1+\alpha)}\quad(n\in\mathbb{N}).$

The monic Laguerre polynomials $\widetilde{L}_{n}(z,\alpha)$ can be calculated
by (see [39],[41])

(6.5)
$\widetilde{L}_{n}(z,\alpha)=\sum_{k=0}^{n}\binom{n}{k}\frac{\Gamma(n+\alpha+1)}{\Gamma(k+\alpha+1)}(-1)^{n+k}z^{k}.$

Then using the formula

${Q}_{n}(z,\alpha)=\mathfrak{S}_{t}\left(\frac{\widetilde{L}_{n}(z,\alpha)-\widetilde{L}_{n}(t,\alpha)}{z-t}\right)$

one can find the Lanczos polynomials of the second kind:

(6.6)
$\begin{split}{Q}_{n}(z,\alpha)&=\mathfrak{S}_{t}\left(\sum_{k=0}^{n}\binom{n}{k}\frac{\Gamma(n+\alpha+1)}{\Gamma(k+\alpha+1)}(-1)^{n+k}\frac{z^{k}-t^{k}}{z-t}\right)=\\\
&=\sum_{k=1}^{n}z^{k-1}\Gamma(\alpha+n+1)\sum_{j=0}^{n-k}(-1)^{n+k+j}\binom{n}{k+j}\frac{\Gamma(\alpha+j+1)}{\Gamma(\alpha+k+j+1)}.\end{split}$

Due to (3.3) the Stieltjes polynomials of the first and second kind are
calculated by

$\begin{split}&P^{+}_{2n}(z,\alpha)=\Gamma(1+\alpha)\sum_{i=0}^{n}\binom{n}{i}\frac{(-1)^{i}z^{i}}{\Gamma(1+\alpha+i)},\\\
&P_{2n-1}^{+}(z,\alpha)=-\frac{1}{(n-1)!}\prod_{j=1}^{n}(j+n)\left(\sum_{i=0}^{n-1}\frac{(-1)^{i-1}(n-1)!i\,z^{i}}{i!(n-i)!\Gamma(1+\alpha+i)}+\frac{(-1)^{n-1}z^{n}}{\Gamma(1+\alpha+n)}\right),\\\
&Q^{+}_{2n}(z,\alpha)=\sum_{i=1}^{n}z^{i-1}\Gamma(1+\alpha)\sum_{j=0}^{n-i}(-1)^{i+j+1}\binom{n}{i+j}\frac{\Gamma(1+\alpha+n)}{\Gamma(1+\alpha+i+j+1)},\\\
&Q^{+}_{2n-1}(z,\alpha)=\Gamma(1+\alpha+n)\sum_{i=1}^{n}z^{i-1}\sum_{j=0}^{n-i}\frac{(-1)^{i+j-1}\Gamma(1+\alpha+j)(n-1)!}{i!(n-j-i)!\Gamma(1+\alpha+i+j)}.\end{split}$

The set of solutions of the truncated moment problem
$MP_{\kappa}^{k}(\mathbf{s},2n-1)$ is described by the formula (1.23).

Notice that the full moment problem is determinate and in the case $\alpha>-1$
its unique solution of $MP_{0}^{0}(\mathbf{s})$ is given by

$f(z)=\int_{0}^{\infty}\frac{x^{\alpha}e^{-x}}{x-z}dx=z^{\frac{\alpha-1}{2}}e^{-\frac{z}{2}}W_{\frac{-\alpha-1}{2},\frac{\alpha}{2}}(-z),$

where $W_{\alpha,\beta}$ is the Whitteker function, see [26, 9.222].

If $-2k-1<\alpha<-2k$, $k\in{\mathbb{N}}$, then the solution $f(z)$ of
$MP_{k}^{k}(\mathbf{s})$ can be found by the formula

(6.7)
$f(z)=\mathfrak{S}_{x}\left(\frac{e^{-x}x^{\alpha}}{x-z}\right)=\int_{0}^{\infty}x^{\alpha}\left(h(x,z)-\sum_{j=0}^{2k-1}\frac{h_{x}^{(j)}(0,z)}{j!}x^{j}\right)dx,$

where $h(x,z)=\frac{e^{-x}}{x-z}$. Similar formula holds also for the case
$-2k<\alpha<-2k+1$. In particular, if $-2<\alpha<-1$, then

$f(z)=I(z)+I^{\prime}(z)\in MP_{1}^{0}(\mathbf{s}),\quad\mbox{ where
}\,I(z)=\frac{z^{\frac{\alpha}{2}}e^{-\frac{z}{2}}}{\alpha+1}W_{\frac{-\alpha-2}{2},\frac{\alpha+1}{2}}(-z).$

By Propositions 5.3, 5.4 the Padé approximants of $f$ take the form (see (5.4)
and (5.6)):

$f^{[n/n]}(z)=\frac{Q_{2n}^{+}(z,\alpha)}{P_{2n}^{+}(z,\alpha)},\quad
f^{[n/n-1]}(z)=\frac{Q_{2n-1}^{+}(z,\alpha)}{P_{2n-1}^{+}(z,\alpha)},\quad
n\in{\mathbb{N}}.$

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