Chebyshev-Fibonacci polynomial relations using generating functions

Robert Frontczak111Statements and conclusions made in this article by R.
Frontczak are entirely those of the author. They do not necessarily reflect
the views of LBBW.
Landesbank Baden-Württemberg,
Stuttgart, Germany
<EMAIL_ADDRESS>

Taras Goy
Faculty of Mathematics and Computer Science,
Vasyl Stefanyk Precarpathian National University,
Ivano-Frankivsk, Ukraine
<EMAIL_ADDRESS>

###### Abstract

The main object of the paper is to reveal connections between Chebyshev
polynomials of the first and second kinds and Fibonacci polynomials introduced
by Catalan. This is achieved by relating the respective (ordinary and
exponential) generating functions to each other. As a consequence, we also
establish new combinatorial identities for balancing polynomials and Fibonacci
(Lucas) numbers.

## 1\. Introduction

For any integer $n\geq 0$, the Chebyshev polynomials $\\{T_{n}(x)\\}_{n\geq
0}$ of the first kind are defined by the second-order recurrence relation [14]

$T_{0}(x)=1,\quad T_{1}(x)=x,\quad T_{n+1}(x)=2xT_{n}(x)-T_{n-1}(x),$ (1)

while the Chebyshev polynomials $\\{U_{n}(x)\\}_{n\geq 0}$ of the second kind
are defined by

$U_{0}(x)=1,\quad U_{1}(x)=2x,\quad U_{n+1}(x)=2xU_{n}(x)-U_{n-1}(x).$ (2)

If we denote $\alpha(x)=x+\sqrt{x^{2}-1}$ and $\beta(x)=x-\sqrt{x^{2}-1}$,
then we have

$\displaystyle
T_{n}(x)=\frac{\alpha^{n}(x)+\beta^{n}(x)}{2}=\sum_{k=0}^{\lfloor{n}/{2}\rfloor}{n\choose
2k}(x^{2}-1)^{k}x^{n-2k},$ $\displaystyle
U_{n}(x)=\frac{\alpha^{n+1}(x)-\beta^{n+1}(x)}{2\sqrt{x^{2}-1}}=\sum_{k=0}^{\lfloor{n}/{2}\rfloor}{n+1\choose
2k+1}(x^{2}-1)^{k}x^{n-2k}.$

Fibonacci polynomials are polynomials that can be defined by Fibonacci-like
recursion relations. They were studied in 1883 by E. Catalan and E.
Jacobsthal. For example, Catalan studied the polynomials $F_{n}(x)$ defined by
the recurrence

$F_{n}(x)=xF_{n-1}(x)+F_{n-2}(x),\quad n\geq 2,$

with $F_{0}(x)=0$ and $F_{1}(x)=1$. A non-recursive expression for $F_{n}(x)$
is

$F_{n}(x)=\frac{\rho^{n}(x)-\sigma^{n}(x)}{\sqrt{x^{2}+4}}=\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}{n-k-1\choose
k}x^{n-2k-1},\quad n\geq 0,$

where $\rho(x)=\frac{x+\sqrt{x^{2}+4}}{2}$ and
$\sigma(x)=\frac{x-\sqrt{x^{2}+4}}{2}$.

Chebyshev and Fibonacci polynomials play an important role in applied
mathematics. They possess many interesting and unique properties. Excellent
sources are the textbooks [11, 14, 16], among others.

The most recent literature on Chebyshev and Fibonacci polynomials encompasses
the following articles. Kilic et al. [9] computed various types of power sums
for Chebyshev polynomials and deduced new connections between Chebyshev
polynomials and Fibonacci numbers. Kim et al. [10] recently derived new
expressions for sums of finite products of Chebychev and Fibonacci
polynomials. In [1], Abd-Elhammed et al. established new connection formulas
between Fibonacci polynomials and Chebyshev polynomials. These formulas are
expressed in terms of certain values of hypergeometric functions. Li and
Wenpeng [12], using the definitions and properties of Chebyshev polynomials,
studied the power sum problems involving Fibonacci polynomials and Lucas
polynomials and obtained some interesting divisibility properties. Li [13]
studied relationships between Chebyshev polynomials, Fibonacci polynomials,
and their derivatives, and got the formula for derivatives of Chebyshev
polynomials being represented by Chebyshev polynomials and Fibonacci
polynomials. Finally, we mention the paper by Flórez, McAnally and Mukherjee
[3] where many identities for generalized Fibonacci polynomials are derived.

The purpose of this paper is to obtain some identities involving Chebyshev
polynomials of the first and second kinds and Fibonacci polynomials. We
achieve this in a conventional manner by relating the respective (ordinary and
exponential) generating functions to each other, resulting in a range of
interesting functional equations. Our approach is in the spirit of [5, 6, 7].
Also, using simple connections between Chebyshev polynomials and balancing
polynomials we will be able to incorporate the later polynomial class into our
analysis. Some of the results of this paper were announced without proofs in
[8].

## 2\. Some generating functions

This section contains the generating functions that will be used later in this
article. We state the results without proofs as they can be derived without
much efforts. We recommend the article by Mező [15] for a comprehensive study
of generating functions for second-order sequences.

From (1) and (2) it can be shown that the ordinary generating functions for
Chebyshev polynomials $T_{n}(x)$, $U_{n}(x)$ and their odd and even indexed
companions are given by

$\displaystyle t(z,x)=\sum_{n\geq 0}T_{n}(x)z^{n}=\frac{1-xz}{1-2xz+z^{2}},$
(3) $\displaystyle t_{1}(z,x)=\sum_{n\geq
0}T_{2n+1}(x)z^{n}=\frac{x(1-z)}{1-(4x^{2}-2)z+z^{2}},$ (4) $\displaystyle
t_{2}(z,x)=\sum_{n\geq
0}T_{2n}(x)z^{n}=\frac{1-(2x^{2}-1)z}{1-(4x^{2}-2)z+z^{2}},$ (5)

and

$\displaystyle u(z,x)=\sum_{n\geq 0}U_{n}(x)z^{n}=\frac{1}{1-2xz+z^{2}},$ (6)
$\displaystyle u_{1}(z,x)=\sum_{n\geq
0}U_{2n+1}(x)z^{n}=\frac{2x}{1-(4x^{2}-2)z+z^{2}},$ (7) $\displaystyle
u_{2}(z,x)=\sum_{n\geq 0}U_{2n}(x)z^{n}=\frac{1+z}{1-(4x^{2}-2)z+z^{2}}.$ (8)

In addition, the corresponding exponential generating functions for these
polynomial sequences are given by

$\displaystyle\tau(z,x)=\sum_{n\geq
0}T_{n}(x)\frac{z^{n}}{n!}=e^{xz}\cosh\bigl{(}\sqrt{x^{2}-1}\,z\bigr{)},$ (9)
$\displaystyle\tau_{1}(z,x)=\sum_{n\geq 0}T_{2n+1}(x)\frac{z^{n}}{n!}$
$\displaystyle=e^{(2x^{2}-1)\,z}\Bigl{(}x\cosh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}+\sqrt{x^{2}-1}\sinh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}\Bigr{)},$
(10) $\displaystyle\tau_{2}(z,x)=\sum_{n\geq
0}T_{2n}(x)\frac{z^{n}}{n!}=e^{(2x^{2}-1)z}\cosh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)},$
(11)

and

$\displaystyle\omega(z,x)=\sum_{n\geq 0}U_{n}(x)\frac{z^{n}}{n!}$
$\displaystyle=\frac{e^{xz}}{\sqrt{x^{2}-1}}\Bigl{(}x\sinh\bigl{(}\sqrt{x^{2}-1}\,z\bigr{)}+\sqrt{x^{2}-1}\cosh\bigl{(}\sqrt{x^{2}-1}\,z\bigr{)}\Bigr{)},$
(12) $\displaystyle\omega_{1}(z,x)=\sum_{n\geq 0}U_{2n+1}(x)\frac{z^{n}}{n!}$
$\displaystyle=\frac{e^{(2x^{2}-1)z}}{\sqrt{x^{2}-1}}\Bigl{(}(2x^{2}-1)\sinh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}+2x\sqrt{x^{2}-1}\cosh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}\Bigr{)},$
(13) $\displaystyle\omega_{2}(z,x)=\sum_{n\geq 0}U_{2n}(x)\frac{z^{n}}{n!}$
$\displaystyle=\frac{e^{(2x^{2}-1)z}}{\sqrt{x^{2}-1}}\Bigl{(}x\sinh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}+\sqrt{x^{2}-1}\cosh\bigl{(}2x\sqrt{x^{2}-1}\,z\bigr{)}\Bigr{)}.$

Fibonacci polynomials $F_{n}(x)$, $F_{2n+1}(x)$ and $F_{2n}(x)$ have the
following ordinary generating functions

$\displaystyle f(z,x)=\sum_{n\geq 0}F_{n}(x)z^{n}=\frac{z}{1-xz-z^{2}},$ (14)
$\displaystyle f_{1}(z,x)=\sum_{n\geq
0}F_{2n+1}(x)z^{n}=\frac{1-z}{1-(x^{2}+2)z+z^{2}},$ (15) $\displaystyle
f_{2}(z,x)=\sum_{n\geq 0}F_{2n}(x)z^{n}=\frac{xz}{1-(x^{2}+2)z+z^{2}},$ (16)

while the exponential generating functions are

$\displaystyle\phi(z,x)=\sum_{n\geq
0}F_{n}(x)\frac{z^{n}}{n!}=\frac{2e^{\frac{xz}{2}}}{\sqrt{x^{2}+4}}\sinh\Bigl{(}\frac{\sqrt{x^{2}+4}}{2}\,z\Bigr{)},$
(17) $\displaystyle\phi_{1}(z,x)=\sum_{n\geq 0}F_{2n+1}(x)\frac{z^{n}}{n!}$
$\displaystyle=\frac{e^{\frac{x^{2}+2}{2}z}}{\sqrt{x^{2}+4}}\left(x\sinh\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\,z\Bigr{)}+\sqrt{x^{2}+4}\cosh\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\,z\Bigr{)}\right),$
(18) $\displaystyle\phi_{2}(z,x)=\sum_{n\geq
0}F_{2n}(x)\frac{z^{n}}{n!}=\frac{2e^{\frac{x^{2}+2}{2}z}}{\sqrt{x^{2}+4}}\sinh\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\,z\Bigr{)}.$
(19)

## 3\. Chebyshev-Fibonacci polynomial identities using ordinary generating
functions

In what follows, we will use the standard convention that
$\sum_{k=0}^{n}a_{k}=0$ for $n<0$.

######  Theorem 1.

For $n\geq 1$, the following polynomial identities hold:

$\displaystyle
F_{n}(x)=T_{n-1}(x)-\sum_{k=1}^{n-2}\bigl{(}xT_{n-1-k}(x)-2T_{n-2-k}(x)\bigr{)}F_{k}(x),$
(20) $\displaystyle
F_{n}(x)+xF_{n-1}(x)=U_{n-1}(x)-\sum_{k=1}^{n-2}\bigl{(}xU_{n-1-k}(x)-2U_{n-2-k}(x)\bigr{)}F_{k}(x).$
(21)

###### Proof.

To prove formula (20), observe that by (3) and (14), we obtain, respectively,

$\displaystyle 1-xz=\frac{1-xz+t(z,x)(xz-z^{2})}{t(z,x)},\qquad
1-xz=\frac{z+z^{2}f(z,x)}{f(z,x)},$

and thus

$(1-xz)f(z,x)-zt(z,x)=(2z^{2}-xz)t(z,x)f(z,x).$

Expanding both sides of the last equation as a power series in $z$ and using
the Cauchy product of two power series, we then obtain

$\displaystyle\sum_{n\geq 0}F_{n}(x)z^{n}-x\sum_{n\geq
0}F_{n}(x)z^{n+1}-\sum_{n\geq 0}T_{n}(x)z^{n+1}$ $\displaystyle=2\sum_{n\geq
0}\sum_{k=0}^{n}T_{n-k}(x)F_{k}(x)z^{n+2}-x\sum_{n\geq
0}\sum_{k=0}^{n}T_{n-k}(x)F_{k}(x)z^{n+1}$

or, equivalently,

$\displaystyle F_{0}(x)+F_{1}(x)z+\sum_{n\geq 2}F_{n}(x)z^{n}-x\sum_{n\geq
2}F_{n-1}(x)z^{n}-T_{0}(x)z-\sum_{n\geq 2}T_{n-1}(x)z^{n}$
$\displaystyle=2\sum_{n\geq
2}\sum_{k=0}^{n-2}T_{n-2-k}(x)F_{k}(x)z^{n}-x\sum_{n\geq
2}\sum_{k=0}^{n-1}T_{n-1-k}(x)F_{k}(x)z^{n},$ $\displaystyle\sum_{n\geq
2}\bigl{(}F_{n}(x)-xF_{n-1}(x)-T_{n-1}(x)\bigr{)}z^{n}$
$\displaystyle=\sum_{n\geq
2}\Bigl{(}2\sum_{k=0}^{n-2}T_{n-2-k}(x)F_{k}(x)-x\sum_{k=0}^{n-1}T_{n-1-k}(x)F_{k}(x)\Bigr{)}z^{n}.$

Comparing the coefficients on both sides, we have

$\displaystyle
F_{n}(x)-xF_{n-1}(x)-T_{n-1}(x)=\sum_{k=0}^{n-2}\bigl{(}2T_{n-2-k}(x)-xT_{n-1-k}(x)\bigr{)}F_{k}(x)-xT_{0}(x)F_{n-1}(x),$

as desired. The proof of (21) is very similar. From (6) and (14) the following
functional equation follows:

$\frac{1}{u(z,x)}=\frac{z}{f(z,x)}+z(2z-x)$

or, equivalently,

$f(z,x)-zu(z,x)=2z^{2}f(z,x)u(z,x)-xzf(z,x)u(z,x).$

The remainder of the proof is the same as above. ∎

In a similar manner, we use the generating functions (4), (7), (15), and (5),
(8), (16), respectively, to prove four additional relations between odd (even)
indexed Chebyshev and Fibonacci polynomials. These relations are contained in
the next theorem, those proofs we leave to the reader.

######  Theorem 2.

The following identities hold for $n\geq 1$

$\displaystyle
x\bigl{(}F_{2n+1}(x)-F_{2n-1}(x)\bigr{)}=T_{2n+1}(x)-T_{2n-1}(x)-(3x^{2}-4)\sum_{k=0}^{n-1}F_{2k+1}(x)T_{2(n-k)-1}(x),$
$\displaystyle
2xF_{2n+1}(x)=U_{2n+1}(x)-U_{2n-1}(x)-(3x^{2}-4)\sum_{k=0}^{n-1}F_{2k+1}(x)U_{2(n-k)-1}(x).$

The even indexed counterparts are given by

$\displaystyle
F_{2n}(x)-(2x^{2}-1)F_{2n-2}(x)=xT_{2n-2}(x)-(3x^{2}-4)\sum_{k=1}^{n-1}F_{2k}(x)T_{2(n-k-1)}(x),$
$\displaystyle
F_{2n}(x)+F_{2n-2}(x)=xU_{2n-2}(x)-(3x^{2}-4)\sum_{k=1}^{n-1}F_{2k}(x)U_{2(n-k-1)}(x).$

Next, we present a range of Chebyshev-Fibonacci identities with mixed indices.

######  Theorem 3.

For $n\geq 1$, we have

$\displaystyle xF_{n}(x)+(4x^{3}-x^{2}-3x)F_{n-1}(x)$
$\displaystyle=T_{2n-1}(x)-\sum_{k=1}^{n-2}\bigl{(}(4x^{2}-x-2)T_{2(n-k)-1}(x)-2T_{2(n-k)-3}(x)\bigr{)}F_{k}(x),$
$\displaystyle 2xF_{n}(x)+(8x^{3}-2x^{2}-4x)F_{n-1}(x)$
$\displaystyle=U_{2n-1}(x)-\sum_{k=1}^{n-2}\bigl{(}(4x^{2}-x-2)U_{2(n-k)-1}(x)-2U_{2(n-k)-3}(x)\bigr{)}F_{k}(x),$
$\displaystyle F_{n}(x)+(2x^{2}-x-1)F_{n-1}(x)$
$\displaystyle=T_{2n-2}(x)-\sum_{k=1}^{n-2}\bigl{(}(4x^{2}-x-2)T_{2(n-k-1)}(x)-2T_{2(n-k-2)}(x)\bigr{)}F_{k}(x),$
$\displaystyle F_{n}(x)+(4x^{2}-x-1)F_{n-1}(x)$
$\displaystyle=U_{2n-2}(x)-\sum_{k=1}^{n-2}\bigl{(}(4x^{2}-x-2)U_{2(n-k-1)}(x)-2U_{2(n-k-2)}(x)\bigr{)}F_{k}(x),$
$\displaystyle F_{2n+1}(x)-(2x^{2}-1)F_{2n-1}(x)$
$\displaystyle=T_{2n}(x)-T_{2n-2}(x)-(3x^{2}-4)\sum_{k=0}^{n-1}T_{2(n-k-1)}(x)F_{2k+1}(x),$
$\displaystyle F_{2n+1}(x)+F_{2n-1}(x)$
$\displaystyle=U_{2n}(x)-U_{2n-2}(x)-(3x^{2}-4)\sum_{k=0}^{n-1}U_{2(n-k-1)}(x)F_{2k+1}(x),$
$\displaystyle
x^{2}F_{2n-1}(x)=xT_{2n-1}(x)-(3x^{2}-4)\sum_{k=1}^{n-1}T_{2(n-k)-1}(x)F_{2k}(x),$
$\displaystyle
2xF_{2n}(x)=xU_{2n-1}(x)-(3x^{2}-4)\sum_{k=1}^{n-1}U_{2(n-k)-1}(x)F_{2k}(x),$
(22) $\displaystyle
F_{2n+1}(x)-xF_{2n-1}=T_{n}(x)-T_{n-1}(x)+(x^{2}-2x+2)\sum_{k=0}^{n-1}T_{n-1-k}(x)F_{2k+1}(x),$
$\displaystyle
F_{2n+1}(x)=U_{n}(x)-U_{n-1}(x)+(x^{2}-2x+2)\sum_{k=0}^{n-1}U_{n-1-k}(x)F_{2k+1}(x),$
$\displaystyle
F_{2n}(x)-xF_{2n-2}=xT_{n-1}(x)+(x^{2}-2x+2)\sum_{k=1}^{n-1}T_{n-1-k}(x)F_{2k}(x),$
$\displaystyle
F_{2n}(x)=xU_{n-1}(x)+(x^{2}-2x+2)\sum_{k=1}^{n-1}U_{n-1-k}(x)F_{2k}(x).$

###### Proof.

We will prove only (22), the others can be proved in a similar way. The
formula is essentially a consequence of the functional equation

$2xf_{2}(z,x)=xzu_{1}(z,x)-(3x^{2}-4)zu_{1}(z,x)f_{2}(z,x),$

which can be derived from (7) and (16). ∎

It is worth to note that our previous results can be used to establish
connections between Fibonacci polynomials and balancing and Lucas-balancing
polynomials, respectively. Recall that balancing polynomials $B_{n}(x)$ and
Lucas-balancing polynomials $C_{n}(x)$ are generalizations of balancing and
Lucas-balancing numbers. They are defined by the same recurrence [4]
$w_{n}(x)=6xw_{n-1}(x)-w_{n-2}(x)$, $n\geq 2,$ but with different initial
values $B_{0}(x)=0$, $B_{1}(x)=1$ and $C_{0}(x)=1$, $C_{1}(x)=3x$,
respectively. From the definitions (1) and (2), the following connections are
easily derived (see [4])

$\displaystyle B_{n}(x)=U_{n-1}(3x),\quad C_{n}=T_{n}(3x),\quad n\geq 1.$ (23)

In view of (23) and Theorems 1-3 relations between Fibonacci and balancing
(Lucas-balancing) polynomials are obvious. In the next statement we present
only a few of them.

######  Corollary 4.

For $n\geq 1$,

$\displaystyle
F_{n}(3x)=C_{n-1}(x)-\sum_{k=1}^{n-2}\bigl{(}3xC_{n-k-1}(x)-2C_{n-k-2}(x)\bigr{)}F_{k}(3x),$
$\displaystyle
F_{n}(3x)+3xF_{n-1}(3x)=B_{n}(x)-\sum_{k=1}^{n-2}\bigl{(}3xB_{n-k}(x)-2B_{n-k-1}(x)\bigr{)}F_{k}(3x),$
$\displaystyle
9x^{2}F_{2n}(3x)=C_{2n+1}(x)-C_{2n-1}(x)-(27x^{2}-4)\sum_{k=1}^{n-1}C_{2(n-k)-1}(x)F_{2k+1}(3x),$
$\displaystyle
6xF_{2n+1}(3x)=B_{2(n+1)}(x)-B_{2n}(x)-(27x^{2}-4)\sum_{k=0}^{n-1}B_{2(n-k)}(x)F_{2k+1}(3x),$
$\displaystyle
F_{2n}(3x)+F_{2n-2}(3x)=3xB_{2n-1}(x)-(27x^{2}-4)\sum_{k=1}^{n-1}B_{2(n-k)-1}(x)F_{2k}(3x),$
$\displaystyle
9x^{2}F_{2n-1}(3x)=3xC_{2n-1}(x)-(27x^{2}-4)\sum_{k=1}^{n-1}C_{2(n-k)-1}(x)F_{2k}(3x),$
$\displaystyle
6xF_{2n}(3x)=xB_{2n}(x)-(27x^{2}-4)\sum_{k=1}^{n-1}B_{2(n-k)}(x)F_{2k}(3x).$

## 4\. Chebyshev-Fibonacci polynomial identities via exponential generating
functions

Functional equations for exponential generating functions will yield
connections between Chebyshev and Fibonacci polynomials involving binomial
coefficients. Recall that we use the following notation

$\displaystyle\alpha(x)=x+\sqrt{x^{2}-1},\quad\beta(x)=x-\sqrt{x^{2}-1},\quad\rho(x)=\frac{x+\sqrt{x^{2}+4}}{2},\quad\sigma(x)=\frac{x-\sqrt{x^{2}+4}}{2}.$

######  Theorem 5.

For $n\geq 0$, the following identities hold

$\displaystyle\sum_{k=0}^{n-1}{n\choose
k}\bigl{(}\sqrt{x^{2}+4}\bigr{)}^{n-1-k}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{k}(x)$
$\displaystyle=\sum_{k=1}^{n-1}{n\choose
k}2^{k-1}\bigl{(}\sqrt{x^{2}-1}\bigr{)}^{n-k}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{k}(x),$
(24) $\displaystyle\sum_{k=0}^{n-1}{n\choose
k}\bigl{(}\sqrt{x^{2}+4}\bigr{)}^{n-1-k}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{k}(x)$
$\displaystyle=\sum_{k=1}^{n-1}{n\choose
k}2^{k-1}\bigl{(}\sqrt{x^{2}-1}\bigr{)}^{n-1-k}\bigl{(}\alpha(x)-(-1)^{n-k}\beta(x)\bigr{)}F_{k}(x).$
(25)

###### Proof.

To prove formula (24) we use the generating functions (9) and (17). They give
the functional equation

$2\tau\left(\frac{z}{2},x\right)\sinh\Bigl{(}\frac{\sqrt{x^{2}+4}}{2}z\Bigr{)}=\phi(z,x)\sqrt{x^{2}+4}\,\cosh\Bigl{(}\frac{\sqrt{x^{2}-1}}{2}z\Bigr{)}.$

From this equation we obtain

$\displaystyle\sum_{n\geq 0}\sum_{k=0}^{n}{n\choose
k}\frac{T_{k}(x)}{2^{k}}\left(\Bigl{(}\frac{\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k}-\Bigl{(}-\frac{\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k}\right)\frac{z^{n}}{n!}$
$\displaystyle=\frac{\sqrt{x^{2}+4}}{2}\sum_{n\geq 0}\sum_{k=1}^{n}{n\choose
k}F_{k}(x)\left(\Bigl{(}\frac{\sqrt{x^{2}-1}}{2}\Bigr{)}^{n-k}+\Bigl{(}-\frac{\sqrt{x^{2}-1}}{2}\Bigr{)}^{n-k}\right)\frac{z^{n}}{n!}.$

Comparing the coefficients of the both sides gives

$\displaystyle\sum_{k=0}^{n}{n\choose
k}\frac{T_{k}(x)}{2^{k}}\left(\Bigl{(}\frac{\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k}-\Bigl{(}-\frac{\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k}\right)$
$\displaystyle=\frac{\sqrt{x^{2}+4}}{2}\sum_{k=1}^{n}{n\choose
k}F_{k}(x)\left(\Bigl{(}\frac{\sqrt{x^{2}-1}}{2}\Bigr{)}^{n-k}+\Bigl{(}-\frac{\sqrt{x^{2}-1}}{2}\Bigr{)}^{n-k}\right).$

and after simplifications we get (24). The proof of (25) follows in a similar
way and is based on the functional equation

$\displaystyle
2\sqrt{x^{2}-1}\sinh\Bigl{(}\frac{\sqrt{x^{2}+4}}{2}z\Bigr{)}\omega\\!\left(\frac{z}{2},x\right)$
$\displaystyle=\sqrt{x^{2}+4}\left(x\sinh\Bigl{(}\frac{\sqrt{x^{2}-1}}{2}z\Bigr{)}+\sqrt{x^{2}-1}\cosh\Bigl{(}\frac{\sqrt{x^{2}-1}}{2}z\Bigr{)}\right)\phi(z,x),$

which we derive from generating functions (12) and (17). ∎

The next two theorems give us relations involving odd and even indexed
Chebyshev and Fibonacci polynomial sequences.

######  Theorem 6.

For $n\geq 0$, the following formulas hold

$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{3}-x)}{x^{2}+2}\Bigr{)}^{n-1-k}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}T_{2k+1}(x)$
$\displaystyle=\bigl{(}2x\sqrt{x^{2}-1}\bigl{)}^{n-1}\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{2x^{2}-1}{(x^{3}+2x)\sqrt{x^{2}-1}}\Bigr{)}^{k-1}\bigl{(}\alpha(x)+(-1)^{n-k}\beta(x)\bigr{)}F_{2k+1}(x)$

and

$\displaystyle\frac{1}{2}\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{3}-x)}{x^{2}+2}\Bigr{)}^{n-1-k}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}U_{2k+1}(x)$
$\displaystyle=\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-2}\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{2x^{2}-1}{(x^{3}+2x)\sqrt{x^{2}-1}}\Bigr{)}^{k-1}$
$\displaystyle\times\bigl{(}\alpha^{2}(x)-(-1)^{n-k}\beta^{2}(x)\bigr{)}F_{2k+1}(x).$

###### Proof.

The stated formulas follow from the functional equations

$\displaystyle\left(x\sinh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}+\sqrt{x^{2}+4}\cosh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}\right)\tau_{1}(z,x)$
$\displaystyle=\sqrt{x^{2}+4}\left(x\cosh\bigl{(}2x\sqrt{x^{2}-1}z\bigr{)}+\sqrt{x^{2}-1}\sinh\bigl{(}2x\sqrt{x^{2}-1}z\bigr{)}\right)\phi_{1}\left(\frac{4x^{2}-2}{x^{2}+2}z,x\right)$

and

$\displaystyle
2\left(x\sinh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}+\sqrt{x^{2}+4}\cosh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}\right)\omega_{1}(z,x)$
$\displaystyle=\sqrt{\frac{x^{2}+4}{x^{2}-1}}\left(\alpha^{2}(x)e^{2x\sqrt{x^{2}-1}z}-\beta^{2}(x)e^{-2x\sqrt{x^{2}-1}z}\right)\phi_{1}\left(\frac{4x^{2}-2}{x^{2}+2}z,x\right),$

that one can obtain from (10), (18) and (13), (18), respectively. ∎

######  Theorem 7.

For $n\geq 0$, the following formulas hold

$\displaystyle x\sum_{k=0}^{n-1}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{3}-x)}{x^{2}+2}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{2k}(x)$
$\displaystyle=\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-1}\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{2x^{2}-1}{(x^{3}+2x)\sqrt{x^{2}-1}}\Bigr{)}^{k-1}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{2k}(x)$

and

$\displaystyle\frac{1}{2}\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{3}-x)}{x^{2}+2}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{2k}(x)$
$\displaystyle=\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-2}\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{2x^{2}-1}{(x^{3}+2x)\sqrt{x^{2}-1}}\Bigr{)}^{k-1}\bigl{(}\alpha(x)-(-1)^{n-k}\beta(x)\bigr{)}F_{2k}(x).$

###### Proof.

Generating functions (11), (19) and (13), (19), respectively, yield

$\displaystyle
2\sinh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}\tau_{2}(z,x)=\sqrt{x^{2}+4}\cosh\bigl{(}2x\sqrt{x^{2}-1}z\bigr{)}\phi_{2}\left(\frac{4x^{2}-2}{x^{2}+2}z,x\right)$

and

$\displaystyle\sqrt{x^{2}-1}\sinh\Bigl{(}\frac{(2x^{3}-x)\sqrt{x^{2}+4}}{x^{2}+2}z\Bigr{)}\omega_{2}(z,x)$
$\displaystyle=2\sqrt{x^{2}+4}\left(x\sinh\bigl{(}2x\sqrt{x^{2}-1}z\bigr{)}+\sqrt{x^{2}-1}\cosh\bigl{(}2x\sqrt{x^{2}-1}z\bigr{)}\right)\phi_{2}\left(\frac{4x^{2}-2}{x^{2}+2}z,x\right).$

The results follow from writing in terms of power series and collecting terms.
∎

The last theorem contains additional relations for Chebyshev and Fibonacci
polynomials that we found.

######  Theorem 8.

For $n\geq 0$, the following formulas hold

$\displaystyle\sum_{k=0}^{n-1}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{2}-1)}{x}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{2k}(x)$
$\displaystyle=\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{4x^{2}-2}{x}\Bigr{)}^{k-1}\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-k}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{k}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{2}-1)}{x}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{2k}(x)$
$\displaystyle=2x\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{4x^{2}-2}{x}\Bigr{)}^{k-1}\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-k-1}\bigl{(}\alpha(x)-(-1)^{n-k}\beta(x)\bigr{)}F_{k}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{2}-1)}{x}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{2k+1}(x)$
$\displaystyle=\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{4x^{2}-2}{x}\Bigr{)}^{k-1}\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-k}\bigl{(}\alpha(x)+(-1)^{n-k}\beta(x)\bigr{)}F_{k}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{\sqrt{x^{2}+4}(2x^{2}-1)}{x}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{2k+1}(x)$
$\displaystyle=x\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{4x^{2}-2}{x}\Bigr{)}^{k-1}\bigl{(}2x\sqrt{x^{2}-1}\bigr{)}^{n-k-1}\bigl{(}\alpha^{2}(x)-(-1)^{n-k}\beta^{2}(x)\bigr{)}F_{k}(x),$
$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{2x}\Bigr{)}^{k}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}T_{k}(x)$
$\displaystyle=\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{(x^{2}+2)\sqrt{x^{2}-1}}{2x}\Bigr{)}^{n-k}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{2k+1}(x),$
$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{2x}\Bigr{)}^{k-1}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{x}\Bigr{)}^{n-k-1}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}U_{k}(x)$
$\displaystyle=\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{(x^{2}+2)\sqrt{x^{2}-1}}{2x}\Bigr{)}^{n-k-1}\bigl{(}\alpha(x)-(-1)^{n-k}\beta(x)\bigr{)}F_{2k+1}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{2x}\Bigr{)}^{k}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{k}(x)$
$\displaystyle=\sum_{k=1}^{n}{n\choose
k}\Bigl{(}\frac{(x^{2}+2)\sqrt{x^{2}-1}}{2x}\Bigr{)}^{n-k}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{2k}(x),$
$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{2x}\Bigr{)}^{k-1}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{k}(x)$
$\displaystyle=\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{(x^{2}+2)\sqrt{x^{2}-1}}{2x}\Bigr{)}^{n-k-1}\bigl{(}\alpha(x)+(-1)^{n-k}\beta(x)\bigr{)}F_{2k}(x),$
$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{4x^{2}-2}\Bigr{)}^{k}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k}\bigl{(}1-(-1)^{n-k}\bigr{)}T_{2k+1}(x)$
$\displaystyle=\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x(x^{2}+2)\sqrt{x^{2}-1}}{2x^{2}-1}\Bigr{)}^{n-k}\bigl{(}\alpha(x)+(-1)^{n-k}\beta(x)\bigr{)}F_{2k}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{4x^{2}-2}\Bigr{)}^{k-1}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}1-(-1)^{n-k}\bigr{)}U_{2k+1}(x)$
$\displaystyle=2\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x(x^{2}+2)\sqrt{x^{2}-1}}{2x^{2}-1}\Bigr{)}^{n-k-1}\bigl{(}\alpha^{2}(x)-(-1)^{n-k}\beta^{2}(x)\bigr{)}F_{2k}(x),$
$\displaystyle x\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{4x^{2}-2}\Bigr{)}^{k}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}T_{2k}(x)$
$\displaystyle=\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x(x^{2}+2)\sqrt{x^{2}-1}}{2x^{2}-1}\Bigr{)}^{n-k}\bigl{(}1+(-1)^{n-k}\bigr{)}F_{2k+1}(x),$
$\displaystyle\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x^{2}+2}{4x^{2}-2}\Bigr{)}^{k-1}\Bigl{(}\frac{x\sqrt{x^{2}+4}}{2}\Bigr{)}^{n-k-1}\bigl{(}\rho(x)-(-1)^{n-k}\sigma(x)\bigr{)}U_{2k}(x)$
$\displaystyle=2\sum_{k=0}^{n}{n\choose
k}\Bigl{(}\frac{x(x^{2}+2)\sqrt{x^{2}-1}}{2x^{2}-1}\Bigr{)}^{n-k-1}\bigl{(}\alpha(x)-(-1)^{n-k}\beta(x)\bigr{)}F_{2k+1}(x).$

## 5\. Concluding comments: Fibonacci and Lucas identities implied by
Chebyshev-Fibonacci identities

The polynomial relations derived in this paper imply many Fibonacci and Lucas
identities, some of which are certainly known but some of which could turn out
to be new. These identities come from the various links between Chebyshev
polynomials and Fibonacci (Lucas) numbers. In [2] and [17] many such links are
listed. Among the various connections we have

$\displaystyle T_{n}\Bigl{(}\frac{3}{2}\Bigr{)}=\frac{1}{2}L_{2n},\qquad
U_{n}\Bigl{(}\frac{3}{2}\Bigr{)}=F_{2n+2},$ (26) $\displaystyle
T_{n}\Bigl{(}\frac{i}{2}\Bigr{)}=\frac{i^{n}}{2}L_{n},\qquad
U_{n}\Bigl{(}\frac{i}{2}\Bigr{)}=i^{n}F_{n+1},$ $\displaystyle
T_{2n}\Bigl{(}\frac{\sqrt{5}}{2}\Bigr{)}=\frac{1}{2}L_{2n},\qquad
U_{2n}\Bigl{(}\frac{\sqrt{5}}{2}\Bigr{)}=L_{2n+1},$ $\displaystyle
T_{2n+1}\Bigl{(}\frac{\sqrt{5}}{2}\Bigr{)}=\frac{\sqrt{5}}{2}F_{2n+1},\qquad
U_{2n+1}\Bigl{(}\frac{\sqrt{5}}{2}\Bigr{)}=\sqrt{5}F_{2n+2}.$

Using (26), for instance, from Theorems 1–3, we can immediately obtain new
families of Fibonacci and Lucas identities. In the next statement, we state
some examples.

######  Corollary 9.

For $n\geq 1$, we have the following identities:

$\displaystyle 15F_{2(2n-1)}=5\cdot 4^{n}-20\cdot
4^{-n}+11\sum_{k=1}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}F_{2(2n-2k-1)},$
$\displaystyle
15F_{4n}=12\bigl{(}4^{n}-4^{-n}\bigr{)}+11\sum_{k=1}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}F_{4(n-k)},$
$\displaystyle
15F_{2n}=4\bigl{(}4^{n}-4^{-n}\bigr{)}-5\sum_{k=1}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}F_{2(n-k)},$
$\displaystyle 15L_{4(n-1)}=4^{n}+104\cdot
4^{-n}+11\sum_{k=1}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}L_{4(n-k-1)},$
$\displaystyle 5L_{2(2n-1)}=9\cdot 4^{n}+36\cdot
4^{-n}+11\sum_{k=1}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}L_{2(2n-2k-1)},$
$\displaystyle 3L_{2n-2}=4^{n}+8\cdot
4^{-n}-\sum_{k=0}^{n-1}\bigl{(}4^{k}-4^{-k}\bigr{)}L_{2(n-k-1)}.$

To give another example, observe that from

$T_{n}(-\sqrt{5})=\left.\begin{cases}\frac{1}{2}L_{3n},&\mbox{$n$ even,}\\\
-\frac{\sqrt{5}}{2}F_{3n},&\mbox{$n$ odd},\end{cases}\right.\qquad
U_{n}(-\sqrt{5})=\left.\begin{cases}\frac{1}{4}L_{3n+3},&\mbox{$n$ even,}\\\
-\frac{\sqrt{5}}{4}F_{3n+3},&\mbox{$n$ odd},\end{cases}\right.$

and

$F_{n}(-\sqrt{5})=\left.\begin{cases}-\frac{\sqrt{5}}{3}F_{2n},&\mbox{$n$
even,}\\\ \frac{1}{3}L_{2n},&\mbox{$n$ odd},\end{cases}\right.$

from Theorem 2 we get the next summation identities.

######  Corollary 10.

Let $n\geq 0$. Then

$\displaystyle 11\sum_{k=1}^{n}F_{4k}L_{6(n-k)}=3L_{6n}-2F_{4n+4}+18F_{4n},$
$\displaystyle
11\sum_{k=1}^{n}F_{4k}L_{6(n-k)+3}=3L_{6n+3}-4F_{4n+4}-4F_{4n},$
$\displaystyle
11\sum_{k=0}^{n}L_{4k+2}F_{6(n-k)+3}=3(F_{6n+9}-F_{6n+3})-2(L_{4n+6}-L_{4n+2}),$
$\displaystyle
11\sum_{k=0}^{n}L_{4k+2}F_{6(n-k)+6}=3(F_{6n+12}-F_{6n+6})-8L_{4n+6}.$

Finally, with $F_{n}(4)=F_{3n}/2$, we get from Theorem 1 the following.

######  Corollary 11.

Let $n\geq 1$. Then

$\displaystyle
F_{3n}=2T_{n-1}(4)-\sum_{k=1}^{n-2}\bigl{(}4T_{n-1-k}(4)-2T_{n-2-k}(4)\bigr{)}F_{3k},$
$\displaystyle
F_{3n}+4F_{3n-3}=2U_{n-1}(4)-\sum_{k=1}^{n-2}\bigl{(}4U_{n-1-k}(4)-2U_{n-2-k}(4)\bigr{)}F_{3k},$

with

$T_{n}(4)=4^{n}\sum_{j=0}^{\lfloor{n/2}\rfloor}{n\choose
2j}\Big{(}\frac{15}{16}\Big{)}^{j}\quad\mbox{and}\quad
U_{n}(4)=4^{n}\sum_{j=0}^{\lfloor{n/2}\rfloor}{n+1\choose
2j+1}\Big{(}\frac{15}{16}\Big{)}^{j}.$

More experiments in this direction are left for a personal study.

## References

  * [1] W. M. Abd-Elhameed, Y. H. Youssri, N. El-Sissi, and M. Sadek, New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials, Ramanujan J. 42 (2017), 347-361.
  * [2] D. Castellanos, A generalization of Binet’s formula and some of its consequences, Fibonacci Quart. 27 (1989), 424–438.
  * [3] R. Flórez, N. McAnally, and A. Mukherjee, Identities for the generalized Fibonacci polynomial, Integers 18B (2018), #A2, 13 pages.
  * [4] R. Frontczak, On balancing polynomials, Appl. Math. Sci. 13 (2019), no. 2, 57–66.
  * [5] R. Frontczak, Relations for generalized Fibonacci and Tribonacci sequences, Notes Number Theory Discrete Math. 25 (2019), no. 1, 178–192.
  * [6] R. Frontczak, Some Fibonacci-Lucas-Tribonacci-Lucas identities, Fibonacci Quart. 56 (2018), no. 3, 263–274.
  * [7] R. Frontczak and T. Goy, Mersenne-Horadam identities using generating functions, Carpathian Math. Publ. 12 (2020), no. 1, 34–45.
  * [8] R. Frontczak and T. Goy, Some relations between Chebyshev and Fibonacci polynomials, Proc. of the Scientific Conf. “Modern Problem of Probability Theory and Mathematical Analysis” 2020 (Vorokhta, Ukraine), pp. 86–87.
  * [9] E. Kilic, S. Koparal, and N. Ömür, Powers sums of the first and second kinds of Chebyshev polynomials, Iran. J. Sci. Technol. Trans. Sci. 44 (2020), 425–435.
  * [10] T. Kim, D. S. Kim, D. V. Dolgy, and J.-W. Park, Sum of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials, J. Inequal. Appl. 2018 (2018), Article 148.
  * [11] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, Hoboken, New Jersey, 2017.
  * [12] C. Li and Z. Wenpeng, Chebyshev polynomials and their some interesting applications, Adv. Differ. Equ. 2017 (2017), Article 303.
  * [13] Y. Li, On Chebyshev polynomials, Fibonacci polynomials, and their derivatives, J. Appl. Math. 2014 (2014), Article ID 451953.
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  * [15] I. Mező, Several generating functions for second-order recurrence sequences, J. Integer Seq. 12 (2009), Article 09.3.7.
  * [16] T. J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley-Interscience, New York, 1990.
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2010 Mathematics Subject Classification: 11B37, 11B39.

_Keywords:_ Chebyshev polynomials, Fibonacci polynomials, Fibonacci numbers,
Lucas numbers, balancing polynomials, generating function.


# Invariant expectation values in the sampling of discrete frequency
distributions

Paolo Rossi  Dipartimento di Fisica dell’Università di Pisa and I.N.F.N.,
Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy

(May 2, 2013)

###### Abstract

The general relationship between an arbitrary frequency distribution and the
expectation value of the frequency distributions of its samples is discussed.
A wide set of measurable quantities (“invariant moments”) whose expectation
value does not in general depend on the size of the sample is constructed and
illustrated by applying the results to Ewens sampling formula. Invariant
moments are especially useful in the sampling of systems characterized by the
absence of an intrinsic scale. Distribution functions that may parametrize the
samples of scale-free distributions are considered and their invariant
expectation values are computed. The conditions under which the scaling limit
of such distributions may exist are described.

###### pacs:

02.50.Ng, 89.20-a, 89.75.Da

##  I Introduction

In many interesting physical, biological and social phenomena, whenever no
intrinsic scale for the relevant variables is present, the emergence of
”scaling laws” is phenomenologically observed Newman . However, strictly
speaking, a power law is not a proper way of fitting empirical data, since no
choice of the exponent can keep the higher moments of a power law distribution
from diverging, while every phenomenological distribution leads to finite
values for all moments. This is not just a technicality: it is rather a
reflection of the fact that the long tail of a power law distribution is in
practice cutoffed by the existence of some ”hidden” scale, irrelevant in the
scaling region, but eventually forcing some upper limit on the variables
describing the system. It would therefore be convenient to parametrize the
data by means of more regular distribution functions, sufficiently damped for
very large values of the variables, but admitting power law distributions as
regular limits when the control parameter implementing the cutoff is sent to
its limiting value.

A related issue concerns the effects of sampling, which may be non trivial
even when we restrict our attention to the expectation values of the sampled
variables. On average sampling does not affect the distributions of individual
objects belonging to different kinds, but when we consider frequency
distributions (that is the number of kinds that are represented $k$ times in a
given population) we cannot in general expect that the frequency distribution
in the samples be the same as in the original population, even after averaging
on many different samples, basically because the cutoff induced by sampling
acts differently (and in general nontrivially) at different scales. It is
therefore quite important to be able to extract from the frequency
distribution of the samples some information reflecting directly some
intrinsic property of the underlying distribution.

Our purpose is therefore threefold. First we want to discuss the general
relationship existing between some arbitrary frequency distribution and the
expectation value of the frequency distributions of its samples, and construct
observables whose expectation values turn out to be independent of the sample
size, and therefore coinciding with the value taken by the same observables in
the full distribution.

Moreover we want to study classes of distributions whose samples preserve the
functional dependence on the parameters present in the original distribution,
establishing the connection between the (a priori unknown) values of the
parameters of the distribution and the (empirically measured) parameters of
the sample distributions.

Finally we want to study the scaling limit of these distributions (when it
exists), in order to explore the possibility of their use for the
phenomenological description of systems that are theoretically expected to
show scaling in the limit when all empirical cutoffs (including those induced
by sampling) are going to disappear.

In Section II we establish the notation and the general framework of our
analysis.

In Section III we construct a wide set of combinations of expectation values
that do not depend on the sample size.

In Section IV we apply our approach to the popular Ewens sampling formula,
showing that its features are consistent with the general pattern and
computing its invariant expectation values.

In Section V we consider the limiting case of a small sampling applied to a
large population.

In Section VI we focus on the case when the original population and its
samples are sufficiently large in comparison with typical frequency values,
finding a useful mathematical relationship between the generating function of
the expectation values of the sample distributions and the generating function
of the original distribution.

In Section VII we analyze a class of distributions (the so-called negative
binomial distributions) admitting a scaling limit and enjoying the property
that the distribution of expectation values of the samples has the same
mathematical form as the original distribution. We also compute in a closed
form the values ot the basic invariant expectation values for these
distributions.

Finally in Section VIII we analyze the scaling limit itself and discuss the
conditions under which one may expect this limit to be a sensible description
of the original system.

Appendices are devoted to the proofs of some mathematical results and to
discussing the issue of correlation between random samples.

##  II The general framework

We are considering a set of $N$ objects (“individuals”) belonging to $S$
different kinds (“species”), and we assume that the set contains $\hat{N}_{a}$
objects of the $a$-th kind, subject to the constraint $\sum_{a}\hat{N}_{a}=N$.

A sample is a set of $n$ objects, containing $\hat{n}_{a}$ objects of the
$a$-th kind, subject to the constraint $\sum_{a}\hat{n}_{a}=n$.

The probability $P_{\\{\hat{n}_{a}\\}}$ of extracting a specific sample
$\\{\hat{n}_{a}\\}$ from a given set $\\{\hat{N}_{a}\\}$ is obtained from the
multivariate hypergeometric distribution

$P_{\\{\hat{n}_{a}\\}}={N\choose
n}^{-1}\prod_{a=1}^{S}{\hat{N}_{a}\choose\hat{n}_{a}}.$

We can easily compute the relevant expectation values, obtaining in particular

$\langle\hat{n}_{a}\rangle=\hat{N}_{a}{n\over
N}=n\,\hat{p}_{a},\qquad\qquad\langle\hat{n}_{a}^{2}\rangle-\langle\hat{n}_{a}\rangle^{2}={N-n\over
N-1}n\hat{p}_{a}(1-\hat{p}_{a})$

where $\hat{p}_{a}\equiv{\bar{N}_{a}/N}$ is the probability of extracting an
object of the $a$-th kind in a single extraction.

It may be useful to consider also the limit of small samples
$\hat{n}_{a}\ll\hat{N}_{a}$. In this limit the probability of a specific
sample is well approximated by the multinomial distribution

$P_{\\{\hat{n}_{a}\\}}=n!\prod_{a=1}^{S}{1\over\hat{n}_{a}!}(\hat{p}_{a})^{\hat{n}_{a}}.$

A frequency distribution is a set of values $\\{N_{k}\\}$, where $N_{k}$ is
the number of kinds such that for each of them there are $k$ objects in the
original set. According to the definition, the following conditions must be
satisfied:

$\sum_{k=1}^{N}N_{k}=S,\qquad\qquad\qquad\sum_{k=1}^{N}k\,N_{k}=N.$

The frequency distribution of a sample is a set of values $\\{n_{l}\\}$,
satisfying the conditions

$\sum_{l=0}^{n}n_{l}=S,\qquad\qquad\qquad\sum_{l=1}^{n}l\,n_{l}=n.$

Notice that the frequency distribution of a sample formally includes the
(unobservable) value $n_{0}$, corresponding to the number of kinds, present in
the original set, which are not represented in the sample.

It is in principle possible to compute the probability of any sample
distribution $\\{n_{l}\\}$ as a function of a given set $\\{N_{k}\\}$. To this
purpose it is convenient to define the intermediate variables $N_{kl}$,
representing the (random) number of kinds characterized by $k$ objects in the
original set and by $l$ ($l\leq k$) objects in the sample. The variables
$N_{kl}$ are strongly constrained, since they must satisfy all the conditions:

$\sum_{l=0}^{n}N_{kl}=N_{k},\qquad\qquad\qquad\sum_{k=1}^{N}N_{kl}=n_{l}.$

The probability $P_{\\{N_{kl}\\}}$ of a specific configuration
${\\{N_{kl}\\}}$ follows from the general probability formula Zelterman :

$P_{\\{N_{kl}\\}}={N\choose
n}^{-1}\prod_{k=1}^{N}\biggl{[}N_{k}!\prod_{l=0}^{k}{1\over N_{kl}!}{k\choose
l}^{N_{kl}}\biggr{]},$

subject to the constraint $\sum_{l=0}^{n}N_{kl}=N_{k}$.

The probability $P_{\\{n_{l}\\}}$ of finding a frequency distribution
$\\{n_{l}\\}$ in a sample is obtained by summing the probabilities
$P_{\\{N_{kl}\\}}$ over all configurations satisfiying the constraint
$\sum_{k=1}^{N}N_{kl}=n_{l}$. The corresponding multivariate generating
function can be defined as

$\varepsilon^{(n)}(\\{t_{l}\\})\equiv\sum_{\\{n_{l}\\}}P_{\\{n_{l}\\}}\prod_{l=0}^{n}t_{l}^{n_{l}}=\sum_{\\{N_{kl}\\}}P_{\\{N_{kl}\\}}\prod_{k=1}^{N}\prod_{l=0}^{k}t_{l}^{N_{kl}}.$

It is also possible (and it will be quite convenient) to define a cumulative
generating function for the probability of finding the frequency distributions
$P_{\\{n_{l}\\}}$ for samples of all possible sizes :

$E(x;\\{t_{l}\\})\equiv\sum_{n=0}^{N}{N\choose
n}\varepsilon^{(n)}(\\{t_{l}\\})x^{n}=\sum_{\\{N_{kl}\\}}\prod_{k=1}^{N}\Bigl{(}N_{k}!\prod_{l=0}^{k}{1\over
N_{kl}!}\Bigl{[}{k\choose
l}x^{l}t_{l}\Bigr{]}^{N_{kl}}\Bigr{)}=\prod_{k=1}^{N}\Bigl{[}\sum_{l=0}^{k}{k\choose
l}x^{l}t_{l}\Bigr{]}^{N_{k}},$

where we used the explicit expression of $P_{\\{N_{kl}\\}}$ and all the
relevant constraints.

The expectation values $\langle n_{l}\rangle$ can be computed starting from
the above expressions and from the relationship

$\langle n_{l}\rangle=\sum_{k=1}^{N}\langle
N_{kl}\rangle=\sum_{k=1}^{N}\sum_{\\{N_{jm}\\}}N_{kl}P_{\\{N_{jm}\\}}.$

Straightforward manipulations lead to the results Zelterman

$\langle N_{kl}\rangle=N_{k}{{k\choose l}{N-k\choose n-l}\over{N\choose
n}},\qquad\qquad\langle n_{l}\rangle={\sum_{k=1}^{N}N_{k}{k\choose
l}{N-k\choose n-l}\over{N\choose n}}.$

It is easy to check that the following relationships are satisfied:

$\sum_{l=0}^{n}\langle
n_{l}\rangle=\sum_{k=1}^{N}N_{k}=S,\qquad\qquad\sum_{l=0}^{n}l\,\langle
n_{l}\rangle=\Bigl{(}\sum_{k=1}^{N}k\,N_{k}\Bigr{)}{n\over N}=n.$

In order to fully appreciate the relevance of considerations based on the
expectation values we must evaluate the weight of the fluctuations. Taking
second derivatives of the generating function $E(x;t_{l})$ one obtains:

$\langle n_{l}^{2}\rangle-\langle
n_{l}\rangle^{2}=\sum_{k,k^{\prime}}N_{k}N_{k^{\prime}}{k\choose
l}{k^{\prime}\choose l}\Biggl{[}{{N-k-k^{\prime}\choose n-2l}\over{N\choose
n}}-{{N-k\choose n-l}\over{N\choose n}}{{N-k^{\prime}\choose
n-l}\over{N\choose n}}\Biggr{]}+\sum_{k}N_{k}{k\choose l}\Biggl{[}{{N-k\choose
n-l}\over{N\choose n}}-{k\choose l}{{N-2k\choose n-2l}\over{N\choose
n}}\Biggr{]}.$

Notice that in the large $N$ limit the term quadratic in $N_{k}$ is depressed
by a power of $1/N$. This observation suggests that very important limits of
the above results may be obtained when considering large populations.

##  III Invariant expectation values

It is very important to be able to define a set of expectation values that are
independent of the size of the sample, and therefore may reflect very directly
the properties of the original frequency distribution.

Let’s consider the following combinations of expectation values:

$\langle m_{\\{p_{i}\\}}^{(n)}\rangle\equiv{n\choose
P}^{-1}\sum_{\\{q_{i}\\}}\prod_{i=1}^{I}\Bigl{[}{q_{i}\choose
p_{i}}{\partial\over\partial
t_{q_{i}}}\Bigr{]}\varepsilon^{(n)}(\\{t_{l}\\}|_{\\{t_{l}=1\\}},$

where $p_{i}$ are $I$ arbitrary positive integer numbers, subject only to the
constraint that $P\equiv\sum_{i}p_{i}\leq n$.

The definition of the quantities appearing in the r.h.s. implies that the
derivatives with respect to $t_{q_{i}}$ are the joint factorial moments of the
distribution; therefore we are dealing with weighted combinations of joint
factorial moments. When some of the indices $p_{i}$ are equal to one, the
expectation values may be expressed in terms of a combination of lower rank
moments ($I^{\prime}\langle I$).

It is possible to recognize that the quantities $\langle
m_{\\{p_{i}\\}}^{(P)}\rangle$ are related (up to a trivial combinatorial
factor taking into account the existence of $n_{p}$ coincident values of the
indices $p_{i}$) to the probability of finding the configurations
$\\{p_{i}\\}$ in the sample containing $P$ elements, and are therefore
strictly connected with the probabilities $P_{\\{n_{p}\\}}$ .

Exploiting the properties of the generating function $E(x;\\{t_{l}\\})$ we
prove in Appendix A that

$\langle m_{\\{p_{i}\\}}^{(n)}\rangle=\langle
m_{\\{p_{i}\\}}^{(N)}\rangle\equiv M_{\\{p_{i}\\}}$

for all sets $\\{p_{i}\\}$ such that $P\leq n$. Hence the expectation values
of the nontrivial invariant moments $m_{\\{p_{i}\\}}$ evaluated for samples of
arbitrary size $n\geq P$, coincide with the moments $M_{\\{p_{i}\\}}$ of the
original frequency distribution. If the original set was generated by a random
process, also the $M_{\\{p_{i}\\}}$ will be expectation values. Recalling that
$\varepsilon^{(N)}(\\{t_{l}\\}\equiv\prod_{k}t_{k}^{N_{k}}$ we may now
generate a representation of all $P_{\\{n_{p}\\}}$ in terms of $N_{k}$,
without making use of the coefficients $N_{kl}$.

The properties of the binomial coefficients make it possible to invert the
relationship between invariant moments and joint factorial moments, thus
finding that

$\Bigl{[}\prod_{i=1}^{I}{\partial\over\partial
t_{q_{i}}}\Bigr{]}\varepsilon^{(n)}(\\{t_{l}\\}|_{\\{t_{l}=1\\}}=\sum_{\\{p_{i}\\}}\prod_{i=1}^{I}(-1)^{p_{i}-q_{i}}{p_{i}\choose
q_{i}}\langle{n\choose
P}m_{\\{p_{i}\\}}^{(n)}\rangle=\sum_{\\{p_{i}\\}}\prod_{i=1}^{I}(-1)^{p_{i}-q_{i}}{p_{i}\choose
q_{i}}{n\choose P}M_{\\{p_{i}\\}}.$

The basic invariant moments are

$m_{p}^{(n)}={n\choose p}^{-1}\sum_{q=p}^{n}{q\choose p}n_{q}.$

According to the inversion formula

$\langle n_{l}\rangle={\partial\varepsilon^{(n)}\over\partial
t_{l}}|_{\\{t_{m}=1\\}}=\sum_{p=l}^{n}(-1)^{p-l}{p\choose l}{n\choose
p}\langle m_{p}^{(n)}\rangle=\sum_{p=l}^{n}(-1)^{p-l}{p\choose l}{n\choose
p}M_{p}.$

One may define generating functions for the expectation values of $n_{l}$ and
of the basic invariant moments:

$f^{(n)}(t)\equiv\sum_{l=0}^{n}\langle n_{l}\rangle t^{l},\qquad\qquad
g^{(n)}(z)\equiv f^{(n)}\bigl{(}1+{z\over n}\bigr{)}=\sum_{p=0}^{n}{n\choose
p}M_{p}\Bigl{(}{z\over n}\Bigr{)}^{p}.$

Notice that a special case of the above formula is

$F(t)\equiv\sum_{k=0}^{N}N_{k}t^{k},\qquad\qquad G(z)\equiv F\bigl{(}1+{z\over
N}\bigr{)}=\sum_{p=0}^{N}{N\choose p}M_{p}\Bigl{(}{z\over N}\Bigr{)}^{p}.$

It is immediate to recognize that $g^{(n)}(0)=G(0)=M_{0}\equiv S$, and
${dg\over dz}^{(n)}(0)={dG\over dz}(0)=M_{1}\equiv 1$.

##  IV Application to Ewens sampling formula

The multivariate Ewens distribution Ewens ; Karlin , called in genetics the
Ewens sampling formula, describes a specific probability for the partition of
$n$ into parts, and found its main applications in the context of the neutral
theory of evolution and in the unified neutral theory of biodiversity Hubbell
; Rosindell . Since Ewens formula and its possibile generalizations have been
the subject of a wide and still growing literature Griffiths ; Etienne ;
Lessard ; Lambert , it may be interesting to apply the results presented in
Section III to this specific instance. In our notation Ewens probability
distribution takes the form

$P_{\\{n_{l}\\}}={1\over\aleph_{n}}\prod_{l=1}^{n}{1\over
n_{l}!}\Bigl{(}{\theta\over
l}\Bigr{)}^{n_{l}},\qquad\qquad\qquad\aleph_{n}\equiv{\Gamma(\theta+n)\over
n!\,\Gamma(\theta)},$

where $0<\theta<\infty$ and $\sum l\,n_{l}=n$.

Joint factorial moments of the Ewens distribution are easily computed Johnson
and one can show that

$\prod_{i=1}^{I}\Bigl{[}{\partial\over\partial
t_{q_{i}}}\Bigr{]}\sum_{\\{n_{l}\\}}P_{\\{n_{l}\\}}\prod_{l=0}^{n}t_{l}^{n_{l}}={\aleph_{n-Q}\over\aleph_{n}}\prod_{i}\Bigl{(}{\theta\over
q_{i}}\Bigr{)},$

where $Q\equiv\sum_{i}q_{i}\leq n$.

We are then left with the task of computing the summations appearing in the
equation

$\langle m_{\\{p_{i}\\}}^{(n)}\rangle={n\choose
P}^{-1}\sum_{\\{q_{i}\\}}\prod_{i=1}^{I}{q_{i}\choose
p_{i}}{\aleph_{n-Q}\over\aleph_{n}}\prod_{i}\Bigl{(}{\theta\over
q_{i}}\Bigr{)}={P!\,(n-P)!\,\Gamma(\theta)\over\Gamma(\theta+n)}\prod_{i=1}^{I}\Bigl{(}{\theta\over
p_{i}}\Bigr{)}\sum_{\\{q_{i}\\}}\prod_{i=1}^{I}{q_{i}-1\choose
p_{i}-1}{\Gamma(\theta+n-Q)\over\Gamma(\theta)\,(n-Q)!}.$

We prove in Appendix B that

$\sum_{\\{q_{i}\geq p_{i}\\}}\prod_{i=1}^{I}{q_{i}-1\choose
p_{i}-1}{\Gamma(\theta+n-Q)\over\Gamma(\theta)\,(n-Q)!}={\Gamma(\theta+n)\over\Gamma(\theta+P)\,(n-P)!},$

hence

$\langle
m_{\\{p_{i}\\}}^{(n)}\rangle={P!\,\Gamma(\theta)\over\Gamma(\theta+P)}\prod_{i=1}^{I}\Bigl{(}{\theta\over
p_{i}}\Bigr{)}\equiv{1\over\aleph_{P}}\prod_{i=1}^{I}\Bigl{(}{\theta\over
p_{i}}\Bigr{)},$

showing explicitly that the expectation values of the invariant moments of the
Ewens distribution are independent of the sample size and related to the
probability of the configuration $\\{p_{i}\\}$ in the sampling of $P$
elements.

We stress that invariant moments, because of their independence from the size
of the sample, may become a highly valuable tool in testing the applicability
of Ewens distribution (and of the conceptual assumptions underlying its
derivation) to the interpretation of actual empirical data.

##  V Large population and small samples

A significant simplification occurs when $N\rightarrow\infty$ while all other
variables are kept finite. Setting $\tilde{x}\equiv Nx$ and $t_{0}=1$ in the
cumulative generating function and taking the large $N$ limit we obtain

$\tilde{E}(\tilde{x};\\{t_{l}\\})\equiv
1+\sum_{n=1}^{\infty}\tilde{\varepsilon}^{(n)}(\\{t_{l}\\}){\tilde{x}^{n}\over
n!}=\prod_{k=1}^{\infty}\Bigl{[}1+\sum_{l=1}^{k}{k\choose
l}\Bigl{(}{\tilde{x}\over
N}\Bigl{)}^{l}t_{l}\Bigr{]}^{N_{k}}\rightarrow\prod_{k=1}^{\infty}\Bigl{[}1+\sum_{l=1}^{\infty}\Bigl{(}{k\,\tilde{x}\over
N}\Bigr{)}^{l}{t_{l}\over l!}\Bigr{]}^{N_{k}}.$

Let’s now define (for $n,l$ different from zero) the following set of
coefficients:

$c^{(n)}(\\{n_{l}\\})\equiv(-1)^{s-1}(s-1)!\prod_{l=1}^{n}{1\over
n_{l}!}\Bigl{(}{1\over l!}\Bigr{)}^{n_{l}},$

where $s=\sum_{l}n_{l}$ and $n=\sum_{l}l\,n_{l}$, and notice that the
definition of $\tilde{E}$ implies that

$\ln\tilde{E}(\tilde{x};\\{t_{l}\\})\equiv\sum_{n=1}^{\infty}\Bigl{[}\sum_{\\{n_{l}\\}}c^{(n)}(\\{n_{l}\\})\prod_{l=1}^{n}\Bigl{(}\tilde{\varepsilon}^{(l)}(\\{t_{l}\\})\Bigr{)}^{n_{l}}\Bigr{]}\tilde{x}^{n}.$

It is also possible to recognize that, under the same assumptions,

$\ln\tilde{E}(\tilde{x};\\{t_{l}\\})=\sum_{n=1}^{\infty}\Bigl{[}\sum_{\\{n_{l}\\}}c^{(n)}(\\{n_{l}\\})\prod_{l=1}^{n}t_{l}^{n_{l}}\Bigr{]}\tilde{m}_{n}^{(N)}x^{n},$

where we introduced the large $N$ limit of the basic invariant moments:
$\tilde{m}_{p}^{(N)}\equiv\sum_{q}N_{q}(q/N)^{p}$.

Comparing the two results we conclude that, for each value of $n\rangle 0$,

$\sum_{\\{n_{l}\\}}c^{(n)}(\\{n_{l}\\})\prod_{l=1}^{n}\Bigl{(}\tilde{\varepsilon}^{(l)}(\\{t_{l}\\})\Bigr{)}^{n_{l}}=\Bigl{[}\sum_{\\{n_{l}\\}}c^{(n)}(\\{n_{l}\\})\prod_{l=1}^{n}t_{l}^{n_{l}}\Bigr{]}\tilde{m}_{n}^{(N)}.$

These equations allow in principle for the recursive determination of all
$\tilde{\varepsilon}^{(n)}(\\{t_{l}\\}$ in terms of
$\\{\tilde{m}_{p}^{(N)}\\}$ (with $p\leq n$), starting from the initial
condition $\tilde{\varepsilon}^{(1)}=t_{1}$. Higher rank invariant moments
($I\rangle 1$) in the large $N$ limit become polynomials in the basic moments.
However one must keep in mind that, when the set $\\{N_{k}\\}$ not fixed, but
generated by a probability distribution (as in the case of the Ewens formula),
the expectation values of the products of basic moments appearing in the
l.h.s. do not coincide with the products of the expectation values.

##  VI Large populations and large samples

When $k,l\ll N,n$ one may systematically exploit the property that, for small
$a$ and $b$,

${N-a\choose
n-b}\rightarrow{\rho^{b}(1-\rho)^{b-a}\over\rho^{n}(1-\rho)^{N-n}}\qquad\qquad\qquad\rho\equiv{n\over
N}.$

Expressing $N$ and $n$ in terms of $N_{kl}$ one may then obtain

$P_{\\{N_{kl}\\}}\rightarrow\prod_{k=1}^{N}\biggl{[}N_{k}!\prod_{l=0}^{k}{1\over
N_{kl}!}P_{kl}^{N_{kl}}\biggr{]}\qquad\qquad\qquad P_{kl}\equiv{k\choose
l}\rho^{l}(1-\rho)^{k-l}.$

As shown in Appendix C, when computing expectation values with the above
probability distribution, the constraint $\sum l\,n_{l}=n$ becomes irrelevant
in the large $N$ limit, and expectation values of products of $N_{kl}$ with
different values of the index $k$ factorize into products of expectation
values computed for each separate value of $k$. We can therefore compute
directly the generating function for a fixed sample size, generalizing the
multivariate multinomial distribution:

$\varepsilon^{(n)}(\\{t_{l}\\})=\prod_{k=1}^{N}\sum_{\\{N_{kl}\\}}\biggl{[}N_{k}!\prod_{l=0}^{k}{1\over
N_{kl}!}(P_{kl}\,t_{l})^{N_{kl}}\biggr{]}=\prod_{k=1}^{N}\Bigl{[}\sum_{l=0}^{k}P_{kl}\,t_{l}\Bigr{]}^{N_{k}}.$

A consistency check is easily obtained by observing that
$\varepsilon^{(n)}({1})=1$, because of the property that
$\sum_{l=0}^{k}P_{kl}=1$.

The expectation value of $n_{l}$ turns out to be:

$\langle n_{l}\rangle=\sum_{k=1}^{N}N_{k}P_{kl},$

and one may check that the conditions on $\sum_{l}\langle n_{l}\rangle$ and on
$\sum_{l}l\,\langle n_{l}\rangle$ are still satisfied.

We can also estimate the behavior of fluctuations when $k,l\ll N,n$:

$\langle n_{l}^{2}\rangle-\langle
n_{l}\rangle^{2}=\sum_{k=1}^{N}N_{k}P_{kl}(1-P_{kl}).$

The above expression is always smaller than $\langle n_{l}\rangle$ and as a
consequence fluctuations become unimportant for sufficiently large values of
$\langle n_{l}\rangle$.

In the same limit we may derive a notable relationship between the generating
function of the original frequency distribution and the generating function of
the expectation values of its samples. In fact we may recognize that for
sufficiently large $N$ and $n$

$g^{(n)}(z)=\sum_{p=0}^{\infty}M_{p}{z^{p}\over p!}=G(z).$

Since in general $f^{(n)}(t)=g^{(n)}\bigl{(}n(t-1)\bigr{)}$ and
$F(t)=G\bigl{(}N(t-1)\bigr{)}$, it is then easy to check that in the limit
under consideration

$f^{(n)}(t)=F(1-\rho+\rho\,t),$

As a direct consequence of these results, whenever the (size-independent)
function $\gamma(z)\equiv G(z)-G(0)$ can be cast into a form exhibiting no
explicit parametric dependence on $N$, the expectation values $\langle
n_{l}\rangle$ can be obtained from $N_{k}$ by the replacement $N\rightarrow
n$.

Notice that in the limit $k,l\ll N,n$ the definition of the basic invariant
moments $m_{p}^{(n)}$ simplifies to

$m_{p}^{(n)}\rightarrow{p!\over n^{p}}\sum_{l=p}^{n}n_{l}{l\choose p}.$

It is worth analyzing in this limit the explicit expressions of the second
basic invariant moment:

$M_{2}\rightarrow{1\over
N^{2}}\sum_{k=1}^{N}k(k-1)N_{k}=\sum_{a=1}^{S}\bigl{(}{\hat{N}_{a}\over
N}\bigr{)}^{2}-{1\over N}=\sum_{a=1}^{S}\langle\bigl{(}{\hat{n}_{a}\over
n}\bigr{)}^{2}\rangle-{1\over n}\equiv{1\over\alpha}.$

As shown in Appendix D the above results may be used also in order to
parametrize the expected correlation between samples under the assumption of
independent random sampling.

##  VII A class of distributions and its properties

As mentioned in the Introduction, distributions found in samples may often
correspond to systems whose asymptotic ($N\rightarrow\infty$) distribution is
expected to obey a scaling law. However the exponent of the scaling law will
in general be nontrivial, in contrast with the prediction offered by the
simplest neutral models. An example of empirical and theoretical evidence for
nontriviality is offered by surname frequency distributions (see Ref. Rossi
for a recent review), recalling that surnames are expected to mimick the
behavior of selectively neutral alleles. it is therefore especially
interesting to consider parametrizations that may reflect notriviality of
exponents, ad in particular the class of negative binomial distributions Hilbe
, which can be obtained starting from the generating function

$F_{c}(t)={N\over x}{(1-x)^{1-c}\over
c}\bigl{[}1-(1-xt)^{c}\bigr{]}=\sum_{k=1}^{\infty}{N\over
x}{(1-x)^{1-c}\over\Gamma(1-c)}{\Gamma(k-c)\over k!}(xt)^{k},$

where $0\langle x\langle 1$ and the parameter c is assumed to vary in the
range $0\leq c\langle 1$.

The asymptotic behaviour of the distribution for large $k$ is easily obtained
by observing that in this limit

${\Gamma(k-c)\over k!}\rightarrow{1\over k^{1+c}},\qquad\qquad
N_{k}\rightarrow{N\over x}{(1-x)^{1-c}\over\Gamma(1-c)}{x^{k}\over k^{1+c}}.$

We can now compute the generating function for the expectation values of the
samples according to the general rule previously discussed, and obtain

$f_{c}(t)=f_{c}(0)+{n\over y}{(1-y)^{1-c}\over
c}\bigl{[}1-(1-yt)^{c}\bigr{]},$

where we have defined $y={\rho x\over 1-x+\rho x}$.

The distribution of the samples has the same form as the original
distribution, once the replacements $N\rightarrow n$ and $x\rightarrow y$ have
been performed, and therefore we obtain the asymptotic behaviour

$n_{l}\rightarrow{n\over y}{(1-y)^{1-c}\over\Gamma(1-c)}{y^{k}\over k^{1+c}}.$

It is possible to define a combination of parameters independent of the
dimension of the sample:

$\beta=N{1-x\over x}=n{1-y\over y}.$

It is useful to represent $x$ and $y$ in a form showing explicitly their
dependence on the dimension of the sample and on the invariant parameter
$\beta$:

$x={N\over\beta+N},\qquad\qquad\qquad y={n\over\beta+n}.$

It is now possible to evaluate the expectation value of the invariant moments
from the expression

$\gamma_{c}(z)\equiv G_{c}(z)-G_{c}(0)={\beta\over
c}\Bigl{[}1-\Bigl{(}1+{z\over\beta}\Bigr{)}^{c}\Bigr{]},$

showing no explicit parametric dependence on $N$; we therefore obtain (for
$p\neq 0$)

$M_{p}=\beta^{1-p}{\Gamma(p-c)\over\Gamma(1-c)}\rightarrow{\beta^{1-p}\over\Gamma(1-c)}{p!\over
p^{1+c}},\qquad\qquad\alpha\equiv{1\over M_{2}}={\beta\over 1-c}.$

The limit of the above results when $c\rightarrow 0$ is smooth, and it
corresponds to Fisher distribution Fisher , such that

$F_{0}(t)=-\beta\ln(1-xt),\qquad\qquad\qquad N_{k}=\beta{x^{k}\over k},$

and

$f_{0}(t)=\beta\ln\bigl{(}{\beta+N\over\beta+n}\bigr{)}-\beta\ln(1-yt),\qquad\qquad
n_{l}=\beta{y^{l}\over l}.$

The generating function of the invariant moments is obtained from
$\gamma_{0}(z)=-\beta\ln(1+z/\beta)$, and as a consequence the expected values
of the invariant moments ($p\neq 0$) are exactly $M_{p}=(p-1)!\,\beta^{1-p}$.

Notice in particular the relationship $\beta=\alpha$, peculiar to Fisher
distribution. By comparing these results with the large $\theta$ limit of the
invariant moments of Ewens distribution we may easily check Watterson’s
Watterson  and Hubbell’s Hubbell  observation that in this limit $\theta$ is
strictly connected to Fisher’s $\alpha$.

##  VIII The scaling limit

Let’s now consider very large systems, and assume that we can gather
information only through the sampling of $n$ objects belonging to the system,
with $n$ large but not necessarily comparable to $N$.

The analysis of the invariant moments may then allow us to check the
applicability of a phenomenological description of the samples based on some
distribution falling into the classes discussed in the previous Sections. In
the case of a positive response to the check it is then possible to find
numerical estimates of the parameter $\beta$ and of the exponent $c$. Such
estimates are clearly meaningful only if $\beta$ does not turn out to be
significantly greater than $n$.

Under these assumptions, we can infer a description of the original system,
and in the case $N\rangle\rangle n$ such a description will correspond to
computing the limit $x\rightarrow 1$ of the previous results. As a
consequence, at least for observable (i.e. not too large) values of $k$, the
original distribution is expected to be well described by the scaling form

$N_{k}\rightarrow{N^{c}\beta^{1-c}\over\Gamma(1-c)}{1\over k^{1+c}}.$

##  Appendix A Expectation values of the invariant moments

Let’s consider the cumulative generating function for the expectation value of
a given invariant moment (setting $P=\sum p_{i}$) for samples of all possible
sizes:

$\sum_{n=0}^{N}{N\choose n}{n\choose P}\langle m_{\\{p_{i}\\}}^{(n)}\rangle
x^{n}\equiv\sum_{\\{q_{i}\\}}\prod_{i=1}^{I}\Bigl{[}{q_{i}\choose
p_{i}}{\partial\over\partial
t_{q_{i}}}\Bigr{]}E(x;\\{t_{l}\\})|_{\\{t_{l}=1\\}}.$

Let’s now observe that, due to the property that

$\ln E(x;\\{t_{l}\\})=\sum_{k=1}^{N}N_{k}\ln\Bigl{[}\sum_{l=0}^{k}{k\choose
l}t_{l}x^{l}\Bigr{]},$

the derivatives appearing in the above defined cumulative generating function,
computed at ${t_{l}=1}$, can be expressed as summations (over $\\{k_{i}\\}$
indices) of products of $N_{k_{i}}$ times a universal $x$-dependent factor

$\Bigl{[}\prod_{i=1}^{I}{k_{i}\choose q_{i}}\Bigr{]}x^{Q}(1+x)^{N-K},$

where $Q=\sum_{i}q_{i}$ and $K=\sum_{i}k_{i}$. However the following identity
holds:

${q_{i}\choose p_{i}}{k_{i}\choose q_{i}}={k_{i}\choose
p_{i}}{k_{i}-p_{i}\choose q_{i}-p_{i}}.$

As a consequence the cumulative generating function is proportional to the
factor

$\Bigl{[}\prod_{i=1}^{I}{k_{i}\choose
p_{i}}\Bigr{]}x^{P}(1+x)^{N-K}\sum_{\\{q_{i}\\}}\Bigl{[}\prod_{i=1}^{I}{k_{i}-p_{i}\choose
q_{i}-p_{i}}x^{Q-P}\Bigr{]}=\Bigl{[}\prod_{i=1}^{I}{k_{i}\choose
p_{i}}\Bigr{]}x^{P}(1+x)^{N-P}.$

The summations over the indices $\\{k_{i}\\}$ may now be formally performed,
and, by matching the coefficient of $x^{N}$ in the two sides of the equation,
the result can be easily recognized to coincide with the expected value of the
invariant moment computed for the original distribution times the
combinatorial factor ${N\choose P}$.

In conclusion we find

$\sum_{n=0}^{N}{N\choose n}{n\choose P}\langle m_{\\{p_{i}\\}}^{(n)}\rangle
x^{n}={N\choose P}\langle m_{\\{p_{i}\\}}^{(N)}\rangle x^{P}(1+x)^{N-P}.$

Expanding the r.h.s. in powers of $x$ and noticing that ${N\choose n}{n\choose
P}={N\choose P}{N-P\choose n-P}$ we finally obtain

$\sum_{n=0}^{N}{N\choose P}{N-P\choose n-P}\langle
m_{\\{p_{i}\\}}^{(n)}\rangle x^{n}=\sum_{n=0}^{N}{N\choose P}{N-P\choose
n-P}\langle m_{\\{p_{i}\\}}^{(N)}\rangle x^{n},$

implying immediately that, as long as $n\geq P$,

$\langle m_{\\{p_{i}\\}}^{(n)}\rangle=\langle
m_{\\{p_{i}\\}}^{(N)}\rangle\equiv M_{\\{p_{i}\\}}.$

##  Appendix B Proof of a combinatorial formula

The identity

$\sum_{\\{q_{i}\geq p_{i}\\}}\prod_{i=1}^{I}{q_{i}-1\choose
p_{i}-1}{\Gamma(\theta+n-Q)\over\Gamma(\theta)\,(n-Q)!}={\Gamma(\theta+n)\over\Gamma(\theta+P)\,(n-P)!},$

where $Q=\sum_{i}q_{i}$ and $P=\sum_{i}P_{i}$, can be proven by recalling that
for real numbers $\alpha$ and positive integers $q\geq p$

$(1-x)^{-\alpha}=\sum_{m=0}^{\infty}{\Gamma(\alpha+m)\over\Gamma(\alpha)\,m!}x^{m},\qquad\qquad(1-x)^{-p}=\sum_{q=p}^{\infty}{q-1\choose
p-1}x^{q-p}.$

Hence expanding in powers of $x$ the two sides of the identity

$\bigl{[}\prod_{i=1}^{I}(1-x)^{-p_{i}}\bigr{]}(1-x)^{-\theta}=(1-x)^{-(\theta+P)}$

and exchanging the order of summations in the l.h.s. we get

$\sum_{\\{q_{i}\geq p_{i}\\}}\prod_{i=1}^{I}{q_{i}-1\choose
p_{i}-1}{\Gamma(\theta+m+P-Q)\over\Gamma(\theta)\,(m+P-Q)!}={\Gamma(\theta+P+m)\over\Gamma(\theta+P)\,m!}.$

The desired result is then obtained by setting $m=n-P$.

##  Appendix C Fluctuations of the sample size in the unconstrained large N
limit

The multivariate generating function $\varepsilon(\\{t_{l}\\})$ was computed
in Section VI after relaxing the constraint $\sum_{l}l\,n_{l}=n$. In order to
show that the constraint is automatically satisfied in the large $N$ limit we
construct a generating function for the expectation value of the powers of
$\nu=\sum_{l}l\,n_{l}$ in the large $N$ distribution:

$\eta(w)\equiv\sum_{\nu}P_{(\nu)}w^{\nu}=\sum_{\\{N_{kl}\\}}P_{\\{N_{kl}\\}}w^{\sum
l\,n_{l}}=\prod_{k=1}^{N}\Bigl{(}\sum_{\\{N_{kl}\\}}N_{k}!\prod_{l=0}^{k}{1\over
N_{kl}!}(P_{kl}w^{l})^{N_{kl}}\Bigr{)},$

and applying the multinomial formula we obtain

$\eta(w)=\prod_{k=1}^{N}\bigl{(}\sum_{l=0}^{k}P_{kl}w^{l}\bigr{)}^{N_{k}}=\prod_{k=1}^{N}(1-\rho+\rho\,w)^{kN_{k}}=((1-\rho+\rho\,w)^{N}.$

Expanding the result in powers of $w$ we immediately obtain
$P_{(\nu)}=P_{N\nu}$.

Since $\nu$ is distributed according to the binomial distribution, the
relevant expectation values are

$\langle{\nu\over N}\rangle=\rho\equiv{n\over
N},\qquad\qquad\langle{\nu^{2}\over N^{2}}\rangle-\langle{\nu\over
N}\rangle^{2}={1\over N}\rho(1-\rho).$

Hence fluctuations of $\nu/N$ around $\rho$ vanish like $1/N$ in the large $N$
limit.

##  Appendix D Correlation between samples

An important test of randomness in sampling is offered by the measure of the
correlation between two different samples. Let’s consider two random samples,
characterized by the sets of values $\\{\hat{n}_{a}\\}$ and
$\\{\hat{m}_{a}\\}$ and by their sizes $n$ and $m$. The index $a$ labels
different kinds, as in Section II. The correlation between the two samples is

$C={\sum_{a=1}^{S}\hat{n}_{a}\hat{m}_{a}\over\sqrt{\sum_{a=1}^{S}\hat{n}_{a}^{2}}\sqrt{\sum_{a=1}^{S}\hat{m}_{a}^{2}}}.$

Replacing $\hat{n}_{a}$ and $\hat{n}_{a}^{2}$ with their expectation values,
computed in Section II, we obtain (in the large $N$ limit)

$\sum_{a=1}^{S}\langle\hat{n}_{a}\rangle\langle\hat{m}_{a}\rangle=nm\sum_{a=1}^{S}\hat{p}_{a}^{2},$
$\sum_{a=1}^{S}\langle\hat{n}_{a}^{2}\rangle=n^{2}\bigl{(}\sum_{a=1}^{S}\hat{p}_{a}^{2}+{1\over
n}-{1\over
N}\bigr{)},\qquad\qquad\sum_{a=1}^{S}\langle\hat{m}_{a}^{2}\rangle=m^{2}\bigl{(}\sum_{a=1}^{S}\hat{p}_{a}^{2}+{1\over
m}-{1\over N}\bigr{)}.$

By making use of the results presented in Section VI we can now express the
expected value of the correlation between samples in the form

$\langle C\rangle={{1\over\alpha}+{1\over N}\over\sqrt{{1\over\alpha}+{1\over
n}}\sqrt{{1\over\alpha}+{1\over m}}}.$

For samples of equal size $n$ the expected value of the correlation takes the
form $\langle C\rangle={n\over\alpha+n}{\alpha+N\over N}$.

###### Acknowledgements.

I am indebted to Sergio Caracciolo for an important observation on the rôle of
fluctuations. I am also indebted to Roberto Dvornicich, Steve Shore and Ettore
Vicari for critical reading of the manuscript.

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