Identities Arising from a Certain Family of Special Combinatorial Numbers and Polynomials

Abstract

The results obtained in this paper extend several identities and interpolation formulas previously established for the numbers $y_{9,n}(\lambda ;a)$ to the higher-order case. We derive several explicit formulas, identities, and recurrence-type relations, and establish connections with well-known families of special numbers and polynomials, including the Apostol–Euler, Apostol–Bernoulli, Frobenius–Euler, Fubini, and Stirling numbers. We also define an interpolation function for the higher-order numbers and show that its values at negative integers recover these numbers up to an explicit normalization factor. Furthermore, we obtain a residue-class decomposition of this interpolation function and express it in terms of the structured generalized hypergeometric Hurwitz–Lerch-type specialization arising from the residue-class decomposition. Some special cases and structural properties of this zeta-type function, including its relation to the classical Lerch transcendent and a differential identity, are also investigated. As applications, we show that for $\lambda <-1$ the interpolation function governs the moments of a zero-truncated negative binomial distribution naturally attached to the higher-order numbers, and we derive the asymptotic behavior of these numbers from the dominant singularity of the generating function.

1. Introduction

Generating functions constitute one of the principal tools for studying special sequences and polynomial families. Besides encoding an entire sequence in a single analytic expression, they facilitate the derivation of convolution identities, recurrence relations, explicit representations, and connections among different classes of special numbers. This approach has been used extensively in combinatorics, analytic number theory, complex analysis, and the theory of special functions. Classical examples include the Bernoulli, Euler, Frobenius–Euler, Fubini, and Stirling numbers, together with their Apostol-type extensions [1,2,3,4,5].

Another important aspect of this subject is the construction of analytic functions that interpolate discrete families of numbers and polynomials. Such functions often assume, at nonpositive integers, values determined by the coefficients of the corresponding generating functions. Consequently, interpolation formulas provide a natural link between combinatorial sequences and zeta-type functions, including the Hurwitz, Lerch, and Hurwitz–Lerch zeta functions [6,7,8,9,10].

A family relevant to the present investigation was introduced in [11] through the generating function

$${\displaystyle \frac{2}{a^t+\lambda }}.$$

Its coefficients are denoted by $y_{9,n}(\lambda ;a)$, while the corresponding polynomials are obtained by multiplying this generating function by ${(1+t)}^x$. Suitable choices of a and $\lambda$ connect these coefficients with several familiar sequences. For example, the specialization $a=e$ and $\lambda =-2$ produces the Fubini numbers, up to a constant factor. The same generating function can also be related to the families $Y_n(u;a)$, as well as to Apostol–Euler- and Frobenius–Euler-type numbers.

The purpose of this paper is to extend this construction by introducing the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ and their polynomial counterparts $y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)$. The additional parameter $\alpha$ provides a unified setting in which the original family is recovered when $\alpha =1$. Within this framework, we establish coefficient formulas, convolution identities, recurrence relations, and representations involving classical special numbers and polynomials. In particular, connections with the Apostol–Euler, Apostol–Bernoulli, Frobenius–Euler, Fubini, and Stirling families are obtained.

We also associate an interpolation function with the higher-order numbers. Its values at negative integers reproduce $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$, subject to an explicit normalization. A decomposition over residue classes then leads to a generalized hypergeometric Hurwitz–Lerch-type zeta function. Several limiting cases and relations with known zeta functions are derived. Thus, the present study develops both the algebraic and analytic aspects of the higher-order family. As concrete applications, in Section 4 we show that for $\lambda <-1$ the interpolation function $Z_y$ governs the moments of a zero-truncated negative binomial distribution naturally attached to the higher-order numbers, and we derive the asymptotic behavior of these numbers from the dominant singularity of the generating function. Section 5 outlines four computational directions that follow from these results, including numerical evaluation, parameter-dependent nonlinear equations, high-dimensional parameter regimes, and combinatorial representations.

For completeness, we next collect the notation and the specific generating functions required in the subsequent sections. Throughout the paper, $\mathbb{N}$, ${\mathbb{N}}_0$, $\mathbb{Z}$, ${\mathbb{Z}}^{+}$, ${\mathbb{Z}}^{-}$, ${\mathbb{Z}}_0^{-}$, $\mathbb{R}$, ${\mathbb{R}}^{+}$, and $\mathbb{C}$ denote the sets of positive integers, nonnegative integers, integers, positive integers, negative integers, nonpositive integers, real numbers, positive real numbers, and complex numbers, respectively.

Throughout this paper, for $z^w=\exp \left(w\ln z\right)$, we assume that $\ln z$ denotes the principal branch of the many-valued function $\ln z$ with the imaginary part $\Im \left(\ln z\right)$ constrained by $-\pi <\Im \left(\ln z\right)\le \pi$—see, for details, [12,13]. We also assume that $0^0=1$ and $0^n=0$ for $n\in \mathbb{N}.$

We emphasize that the results of Section 2 are primarily algebraic consequences of the higher-order generating functions, obtained through binomial expansions, Cauchy products, and coefficient comparisons. The main novelty of the paper lies in Section 3. Although generalized hypergeometric extensions of the Hurwitz–Lerch function are known in the literature [13], the function introduced here arises specifically from the residue-class decomposition of the interpolation series associated with $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. Its numerator and denominator parameters occur in constrained arithmetic progressions determined by d, $\alpha$, and $\beta$. This structure yields an explicit connection between the higher-order numbers, their interpolation function, and a hypergeometric Hurwitz–Lerch-type series, together with the differential and integral identities derived in Section 3.

Preliminary

We begin with the Stirling numbers of the second kind. For $k\in {\mathbb{N}}_0$, they are determined by

$${\displaystyle \frac{{(e^t-1)}^k}{k!}}={\displaystyle \sum _{n=0}^{\infty }}{S}_2(n,k){\displaystyle \frac{t^n}{n!}}.$$

(1)

See also [14].

Coefficient extraction in (1) gives

$$S_2(n,k)={\displaystyle \frac{1}{k!}}{\displaystyle \sum _{j=0}^k}{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)j^n.$$

(2)

In particular, $S_2(n,k)=0$ whenever $k>n$. For further properties and related identities, see [11,13,15,16,17,18,19,20,21,22,23,24,25].

The Fubini numbers $w_g\left(n\right)$ are characterized by

$${\displaystyle \frac{1}{2-e^t}}={\displaystyle \sum _{n=0}^{\infty }}{w}_g\left(n\right){\displaystyle \frac{t^n}{n!}}.$$

(3)

Combining this expansion with (1) yields the familiar representation

$$w_g\left(n\right)={\displaystyle \sum _{j=0}^n}j!S_2(n,j).$$

(4)

Related definitions and identities can be found in [4,11,17,21,26].

We shall also use the Apostol–Bernoulli polynomials and the higher-order Apostol–Euler polynomials. Their exponential generating functions are given, respectively, by

$${\displaystyle \frac{t{e}^{xt}}{\lambda e^t-1}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{B}}_n(x;\lambda ){\displaystyle \frac{t^n}{n!}},$$

(5)

where $\left|t\right|<2\pi$, when $\lambda =1$; $\left|t\right|<\left|\ln \lambda \right|$, when $\lambda \ne 1$ and

$${\left({\displaystyle \frac{2}{\lambda e^t+1}}\right)}^{\alpha }{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{E}}_n^{\left(\alpha \right)}(x;\lambda ){\displaystyle \frac{t^n}{n!}},$$

(6)

where $\left|t\right|<\left|\ln (-\lambda )\right|;1^{\alpha }:=1$—see, for details, [2,3,5,13,16,27]. The corresponding numbers are defined by

$${\mathcal{B}}_n\left(\lambda \right):={\mathcal{B}}_n(0;\lambda ),{\mathcal{E}}_n^{\left(\alpha \right)}\left(\lambda \right):={\mathcal{E}}_n^{\left(\alpha \right)}(0;\lambda ).$$

When $\alpha =1$, (6) reduces to the usual Apostol–Euler polynomials

$${\displaystyle \frac{2e^{xt}}{\lambda e^t+1}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{E}}_n(x;\lambda ){\displaystyle \frac{t^n}{n!}},$$

where $\left|t\right|<\left|\ln \right(-\lambda \left)\right|$—see, for details, [13].

Simsek [23] constructed a generating function for the polynomials $Y_n(x,u;a)$:

$${\displaystyle \frac{a^{xt}}{a^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{Y}_n(x,u;a){\displaystyle \frac{t^n}{n!}},$$

(7)

where $a\ge 1$; $\left|t\ln \right(a\left)\right|<2\pi$, when $u=1$; $\left|t\ln \left(a\right)\right|<|\ln {\displaystyle \frac{1}{u}}|$, when ${\displaystyle \frac{1}{u}}\ne 1$. In particular, setting $x=0$ gives

$${\displaystyle \frac{1}{a^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{Y}_n(u;a){\displaystyle \frac{t^n}{n!}},$$

which implies that

$$Y_n(u;a):=Y_n(0,u;a)$$

See, for details, [11,23].

For $a,b,c\in {\mathbb{R}}^{+}$, $a\ne b$, $x\in \mathbb{R}$, and $u,\lambda \in \mathbb{C}$ with $u\ne \lambda$, consider

$${\displaystyle \frac{(a^t-u)c^{xt}}{\lambda b^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{H}}_n(x;u;a,b,c;\lambda ){\displaystyle \frac{t^n}{n!}},$$

(8)

where $|t\ln (b)|<2\pi$, when ${\displaystyle \frac{\lambda }{u}}=1$; $|t\ln \left(b\right)|<|\ln {\displaystyle \frac{\lambda }{u}}|$, when ${\displaystyle \frac{\lambda }{u}}\ne 1$ and $1^{\alpha }:=1$ see for details [23].

This expansion defines the generalized Frobenius–Euler-type polynomials in a neighborhood of $t=0$ for which $\lambda b^t-u\ne 0$. Their numerical form is

$${\mathcal{H}}_n(u;a,b,c;\lambda ):={\mathcal{H}}_n(0;u;a,b,c;\lambda ),$$

and hence

$${\displaystyle \frac{a^t-u}{\lambda b^t-u}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{H}}_n(u;a,b,c;\lambda ){\displaystyle \frac{t^n}{n!}}.$$

Relevant extensions and special cases are given in [22,23,24].

The higher-order Frobenius–Euler polynomials are defined by

$${\left({\displaystyle \frac{1-u}{e^t-u}}\right)}^{\alpha }{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{H}_n^{\left(\alpha \right)}(x;u){\displaystyle \frac{t^n}{n!}},$$

where $u\ne 1$ and t lies in a sufficiently small neighborhood of 0 such that $e^t\ne u$; see [11] for details. The corresponding higher-order Frobenius–Euler numbers are given by

$$H_n^{\left(\alpha \right)}\left(u\right):=H_n^{\left(\alpha \right)}(0;u).$$

When $\alpha =1$, the usual Frobenius–Euler polynomials are recovered:

$${\displaystyle \frac{1-u}{e^t-u}}{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{H}_n(x;u){\displaystyle \frac{t^n}{n!}}.$$

This family also follows from (8) by taking $\lambda =a=1$ and $b=c=e$. Further background on these polynomials is available in [1,11,12,13,16,24,25].

Simsek [11] introduced the numbers $y_{9,n}(\lambda ;a)$ by

$$Y_1(t;a,\lambda ):={\displaystyle \frac{2}{a^t+\lambda }}={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}(\lambda ;a){\displaystyle \frac{t^n}{n!}},$$

(9)

where $|t\ln \left(a\right)|<|\ln ({\displaystyle \frac{-1}{\lambda }})|$. Their polynomial extension is generated by

$$Y_2(t,x;a,\lambda ):={(1+t)}^x{Y}_1(t;a,\lambda )={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}(x,\lambda ;a){\displaystyle \frac{t^n}{n!}}.$$

(10)

Using

$${(1+t)}^x={\displaystyle \sum _{m=0}^{\infty }}{\left(x\right)}_m{\displaystyle \frac{t^m}{m!}},$$

where

$${\left(x\right)}_m=x(x-1)\cdots (x-m+1),{\left(x\right)}_0=1,$$

and applying the Cauchy product to (10), one obtains

$$y_{9,n}(x,\lambda ;a)={\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\left(x\right)}_{n-j}{y}_{9,j}(\lambda ;a).$$

(11)

Thus, the polynomial coefficients are expressed as a binomial convolution of the numbers $y_{9,j}(\lambda ;a)$ with falling factorials. By using (9), we gave some special values of the numbers as follows:

$$y_{9,0}(\lambda ;1)={\displaystyle \frac{2}{1+\lambda }}.$$

The Fubini numbers are defined by

$${\displaystyle \frac{1}{2-e^t}}={\displaystyle \sum _{n=0}^{\infty }}{w}_g\left(n\right){\displaystyle \frac{t^n}{n!}}$$

See, for details, [4,17,21]. Setting $a=e$ and $\lambda =-2$ in (9), and comparison with (3), consequently yields

$$y_{9,n}(-2;e)=-2w_g\left(n\right).$$

These relations demonstrate that the $y_{9,n}$ family provides a common generating function setting for several known sequences of special numbers. The following sections develop its higher-order extension, derive its principal algebraic identities, and construct the associated interpolation and generalized Hurwitz–Lerch-type zeta functions.

2. Generating Functions for the Higher-Order Numbers ${\mathit{y}}_{\mathbf{9},\mathit{n}}^{\left(\mathit{\alpha }\right)}(\mathit{\lambda };\mathit{a})$ and Polynomials ${\mathit{y}}_{\mathbf{9},\mathit{n}}^{\left(\mathit{\alpha }\right)}(\mathit{x},\mathit{\lambda };\mathit{a})$

In this section, we introduce generating functions for the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ and polynomials $y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)$. Their basic properties and their connections with familiar families of special numbers are derived from the corresponding generating functions.

Let $a\ge 1$, $x\in \mathbb{R}$. We define

$$f(t;\lambda ,\alpha ,a)={\left({\displaystyle \frac{2}{a^t+\lambda }}\right)}^{\alpha }={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(\lambda ;a\right){\displaystyle \frac{t^n}{n!}}$$

(12)

and

$$g(t,x;\lambda ,\alpha ,a)=f(t;\lambda ,\alpha ,a){(1+t)}^x={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right){\displaystyle \frac{t^n}{n!}},$$

(13)

where $|t\ln \left(a\right)|<|\ln (-{\displaystyle \frac{1}{\lambda }})|$; $1^{\alpha }:=1.$

We next derive several special cases and relations associated with the generating functions (12) and (13).

Explicit Formulas and Relations to Other Polynomials

Some special cases of (12) and (13) are as follows:

$$\begin{array}{cc}\hfill y_{9,n}^{\left(\alpha \right)}(\lambda ;a)=y_{9,n}^{\left(\alpha \right)}(0,\lambda ;a), & y_{9,n}(x,\lambda ;a)=y_{9,n}^{\left(1\right)}(x,\lambda ;a),\hfill \\ y_{9,n}(\lambda ;a)=y_{9,n}^{\left(1\right)}(\lambda ;a),\hfill & \hfill y_{9,0}^{\left(\alpha \right)}(\lambda ;1)={\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }.\end{array}$$

For $n\ge 1$,

$$y_{9,n}^{\left(\alpha \right)}(\lambda ;1)=0.$$

When $\alpha =0$, (13) gives

$$y_{9,n}^{\left(\alpha \right)}\left(x,\lambda ;a\right)={\displaystyle \frac{x(x-1)\dots (x-n+1)}{n!}}.$$

Combining this identity with the generating function of the higher-order Apostol–Euler numbers, we obtain

$${\lambda }^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{E}}_n^{\left(\alpha \right)}\left({\displaystyle \frac{1}{\lambda }}\right){\left(\ln a\right)}^n{\displaystyle \frac{t^n}{n!}}.$$

By comparing coefficients ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we have

$$y_{9,n}^{\left(\alpha \right)}(\lambda ;a)={\displaystyle \frac{1}{{\lambda }^{\alpha }}}{\mathcal{E}}_n^{\left(\alpha \right)}\left({\displaystyle \frac{1}{\lambda }}\right){\left(\ln a\right)}^n.$$

By using (5) with $x=0$ and (12) with $\alpha =1$, we also have

$${\displaystyle \frac{1}{2\lambda }}{\displaystyle \sum _{n=1}^{\infty }}n{y}_{9,n-1}\left(-{\displaystyle \frac{1}{\lambda }};e\right){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{\mathcal{B}}_n\left(\lambda \right){\displaystyle \frac{t^n}{n!}}.$$

Thus we obtain

$${\mathcal{B}}_n\left(\lambda \right)={\displaystyle \frac{n}{2\lambda }}{y}_{9,n-1}\left(-{\displaystyle \frac{1}{\lambda }};e\right).$$

Using the generating function of the higher-order Frobenius–Euler numbers together with (12), we obtain

$${\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\left({\displaystyle \frac{1+\lambda }{e^{t\ln a}+\lambda }}\right)}^{\alpha }={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}.$$

Therefore,

$${\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{\left(\ln a\right)}^n{H}_n^{\left(\alpha \right)}(-\lambda ){\displaystyle \frac{t^n}{n!}}.$$

By comparing coefficients ${\displaystyle \frac{t^n}{n!}}$ on both sides of the above equation, we have the following formula:

$$y_{9,n}^{\left(\alpha \right)}(\lambda ;a)={\left({\displaystyle \frac{2}{1+\lambda }}\right)}^{\alpha }{\left(\ln a\right)}^n{H}_n^{\left(\alpha \right)}(-\lambda ).$$

We now give the formula for the polynomials $y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)$. Combining (12) and (13) gives

$${(1+t)}^x{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(x,\lambda ;a){\displaystyle \frac{t^n}{n!}}.$$

For $\left|t\right|<1$, by applying the binomial theorem to the above equation after some calculations, we get the following formula:

Proposition 1.

Let $a\in {\mathbb{R}}^{+}$, $x\in \mathbb{R}$, $\lambda \in \mathbb{C}$ satisfying the assumptions of (12), $\alpha \in \mathbb{C}$, and $n\in {\mathbb{N}}_0$. Then

$$y_{9,n}^{\left(\alpha \right)}(x,\lambda ;a)={\displaystyle \sum _{j=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right){\left(x\right)}_{n-j}{y}_{9,j}^{\left(\alpha \right)}(\lambda ;a).$$

(14)

By using convergence conditions for binomial theorem in (12), we get

$$2^{\alpha }={(a^t+\lambda )}^{\alpha }{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}.$$

The binomial theorem gives

$${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{a^{kt}}{{\lambda }^k}}{\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}.$$

Since

$$a^{kt}=e^{kt\ln a}={\displaystyle \sum _{j=0}^{\infty }}{\left(k\ln a\right)}^j{\displaystyle \frac{t^j}{j!}},$$

the Cauchy product yields

$${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }={\displaystyle \sum _{m=0}^{\infty }}\left[{\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{\displaystyle \sum _{n=0}^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\left(k\ln a\right)}^{m-n}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a)\right]{\displaystyle \frac{t^m}{m!}}.$$

Thus, we obtain the following identity:

Proposition 2.

Let $a\in {\mathbb{R}}^{+}$, $\alpha \in \mathbb{C}$, $\left|\lambda \right|>1$ and $m\in {\mathbb{N}}_0$. Then

$${\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right){\displaystyle \frac{1}{{\lambda }^k}}{\displaystyle \sum _{n=0}^m}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{n}}\right){\left(k\ln a\right)}^{m-n}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a)=0.$$

Applying the binomial theorem to (12), we have

$${\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(\alpha \right)}(\lambda ;a){\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=0}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\left({\displaystyle \frac{a^t}{\lambda }}\right)}^k.$$

After some calculations, we get the following result:

Proposition 3.

Let $a\in {\mathbb{R}}^{+}$, $\alpha \in \mathbb{C}$, $\left|\lambda \right|>1$ and $n\in {\mathbb{N}}_0$. Then

$$y_{9,n}^{\left(\alpha \right)}(\lambda ;a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{{\left(k\ln a\right)}^n}{{\lambda }^k}}.$$

(15)

3. Interpolation Function for the Numbers ${\mathit{y}}_{\mathbf{9},\mathit{n}}^{\left(\mathit{\alpha }\right)}(\mathit{\lambda };\mathit{a})$

In this section, we investigate interpolation function for the numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. We give some properties of this function. We also give the relation between this function and the multiple Hurwitz–Lerch zeta function [6,10].

By using (15), we set

$$Z_y(s,\lambda ;\alpha ,a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \frac{1}{{\left(\ln a\right)}^s}}{\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{1}{{\lambda }^k{k}^s}},$$

(16)

where $\alpha \in \mathbb{C},$ $a>1$ and $\ln a\ne 1$; $s\in \mathbb{C}$ when $|\lambda |>1$; $\Re \left(\mathrm{s}\right)>\alpha$ when $|{\displaystyle \frac{1}{\lambda }}|=1$.

When $\alpha$ is a positive integer, the function $Z_y(s,\lambda ;\alpha ,a)$ is related to the multiple Hurwitz–Lerch zeta function ${\mathsf{\Phi }}_{\alpha }(\lambda ,s,v)$. That is, for $z=-1/\lambda$,

$$Z_y(s,\lambda ;\alpha ,a)=-{\displaystyle \frac{\alpha }{\lambda }}{\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\left(\ln a\right)}^{-s}{\mathsf{\Phi }}_{\alpha +1}\left(-{\displaystyle \frac{1}{\lambda }},s+1,1\right),$$

where

$${\mathsf{\Phi }}_{\alpha }(\lambda ,s,v)={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{{\lambda }^k}{{(k+v)}^s}}.$$

For the function ${\mathsf{\Phi }}_{\alpha }(\lambda ,s,v)$ see [13] (p. 205). We also note that (16) reduces to Definition 5.9 given in [11] for $\alpha =1$.

Theorem 1.

Let $n\in \mathbb{N}$. Then

$$Z_y(-n,\lambda ;\alpha ,a)=y_{9,n}^{\left(\alpha \right)}(\lambda ;a).$$

(17)

Proof. 

Setting $s=-n$ in (16) gives

$$Z_y(-n,\lambda ;\alpha ,a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\displaystyle \frac{{\left(k\ln a\right)}^n}{{\lambda }^k}}.$$

Combining this with (15) yields (17). □

Remark 1.

Theorem 1 reduces to Theorem 5.10 given in [11] for $\alpha =1$.

3.1. Multiplication Formula for the Function $Z_y(s,\lambda ;\alpha ,a)$

Let $d,k\in \mathbb{N}$. Setting $k=dn+r$, $n\in {\mathbb{N}}_0$, $1\le r\le d$, $d>0$ in (16), we get

$$Z_y(s,\lambda ;\alpha ,a)={\displaystyle \frac{1}{{\left(d\ln a\right)}^s}}{\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}{\displaystyle \sum _{n=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right){\displaystyle \frac{{\left(\frac{{(-1)}^d}{{\lambda }^d}\right)}^n}{{\left(n+\frac{r}{d}\right)}^s}}.$$

(18)

Combining (18) with the following relations,

$$x^{\overline{m}}=x(x+1)\cdots (x+m-1),x^{\overline{0}}=1,$$

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +dn+r-1}{dn+r}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\displaystyle \frac{{(\alpha +r)}^{\overline{dn}}}{{(r+1)}^{\overline{dn}}}},$$

(19)

$$b^{\overline{dn}}=d^{dn}{\displaystyle \prod _{j=0}^{d-1}}{\left({\displaystyle \frac{b+j}{d}}\right)}^{\overline{n}},$$

see also [28] (Equations (5.2.5) and (5.5.6)) and [29], we get

$${\displaystyle \frac{{(\alpha +r)}^{\overline{dn}}}{{(r+1)}^{\overline{dn}}}}={\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}^{\overline{n}}}{{\left(\frac{r+1+j}{d}\right)}^{\overline{n}}}},$$

(20)

The above relation gives the following residue-class decomposition or multiplication formula for the function $Z_y(s,\lambda ;\alpha ,a)$:

Theorem 2.

Let $d\in \mathbb{N}$ and $a>1$. Then

$$Z_y(s,\lambda ;\alpha ,a)={\displaystyle \frac{1}{{\left(d\ln a\right)}^s}}{\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}^{\overline{n}}}{{\left(\frac{r+1+j}{d}\right)}^{\overline{n}}}}\right]{\displaystyle \frac{{\left(\frac{{(-1)}^d}{{\lambda }^d}\right)}^n}{{\left(n+\frac{r}{d}\right)}^s}}.$$

(21)

3.2. A Generalized Hypergeometric Hurwitz–Lerch-Type Zeta Function

Motivated by the residue-class decomposition obtained above, we introduce the following zeta-type function.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s,z\in \mathbb{C}$, and $x\in \mathbb{C}\setminus (-\infty ,0]$, with $\left|z\right|<1$. Suppose that

$${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-},j=0,1,\dots ,d-1.$$

We define

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

(22)

For $d=1$, Equation (24) reduces to

$$y_{9,m}^{\left(\alpha \right)}(\lambda ;a)=-{\displaystyle \frac{\alpha }{\lambda }}{\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\left(\ln a\right)}^m{\mathcal{Z}}_1^{(\alpha +1,2)}\left(-m,1;-{\displaystyle \frac{1}{\lambda }}\right).$$

Remark 2.

For $d=1$, Equation (22) reduces to

$${\mathcal{Z}}_1^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\alpha }^{\overline{n}}}{{\beta }^{\overline{n}}}}{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

If $\beta =1$, then $1^{\overline{n}}=n!$, and hence

$${\mathcal{Z}}_1^{(\alpha ,1)}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\alpha }^{\overline{n}}}{n!}}{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

The connection between (22) and the interpolation function $Z_y$ is given below.

Corollary 1.

Let $d\in \mathbb{N}$, $a>0$ with $a\ne 1$, and $\lambda ,\alpha ,s\in \mathbb{C}$ with $\left|\lambda \right|>1$. Then

$$Z_y(s,\lambda ;\alpha ,a)={\displaystyle \frac{1}{{\left(d\ln a\right)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(s,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).$$

(23)

Proof. 

By (21),

$$Z_y(s,\lambda ;\alpha ,a)={\displaystyle \frac{1}{{\left(d\ln a\right)}^s}}{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +r+j}{d}\right)}^{\overline{n}}}{{\left(\frac{r+1+j}{d}\right)}^{\overline{n}}}}\right]{\displaystyle \frac{{\left(\frac{{(-1)}^d}{{\lambda }^d}\right)}^n}{{\left(n+\frac{r}{d}\right)}^s}}.$$

The inner series is

$${\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(s,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right)$$

by (22). Substitution gives (23). □

Combining (17) with (23) gives the following representation.

Corollary 2.

Let $d,m\in \mathbb{N}$, $a>0$ with $a\ne 1$, $\alpha \in \mathbb{C}$, and $\lambda \in \mathbb{C}\setminus (-\infty ,0]$ with $\left|\lambda \right|>1$ under the branch conditions of Proposition 3. Then

$$y_{9,m}^{\left(\alpha \right)}(\lambda ;a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\left(d\ln a\right)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).$$

(24)

Proof. 

Setting $s=-m$ in (23), we obtain

$$Z_y(-m,\lambda ;\alpha ,a)={\left(d\ln a\right)}^m{\displaystyle \sum _{r=1}^d}{\displaystyle \frac{{(-1)}^r}{{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +r-1}{r}}\right){\mathcal{Z}}_d^{(\alpha +r,r+1)}\left(-m,{\displaystyle \frac{r}{d}};{\displaystyle \frac{{(-1)}^d}{{\lambda }^d}}\right).$$

Using (17) and multiplying both sides by ${\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }$ gives (24). □

Remark 3.

If $\alpha =\beta$, the rising-factorial terms cancel, and

$${\mathcal{Z}}_d^{(\alpha ,\alpha )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}=\mathsf{\Phi }(z,s,x),$$

where

$$\mathsf{\Phi }(z,s,x)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}$$

denotes the classical Lerch transcendent [13,30].

Remark 4.

For $m\in {\mathbb{N}}_0$, setting $s=-m$ in (22) gives

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(-m,x;z)={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]z^n{(n+x)}^m.$$

Finally, setting $\alpha =1$ and $a=e$ in (16) yields

$$Z_y(s,\lambda ;1,e)={\displaystyle \frac{2}{\lambda }}{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(-1)}^k}{{\lambda }^k{k}^s}}=-{\displaystyle \frac{2}{{\lambda }^2}}\mathsf{\Phi }\left(-{\displaystyle \frac{1}{\lambda }},s,1\right).$$

Theorem 3.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s,z\in \mathbb{C}$, $x\in \mathbb{C}\setminus (-\infty ,0]$, and $\left|z\right|<1$. Assume that ${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-}$, $j=0,1,\dots ,d-1$. Then

$${\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)=-s{\mathcal{Z}}_d^{(\alpha ,\beta )}(s+1,x;z).$$

(25)

Proof. 

Since the series in (22) converges locally uniformly with respect to x on compact subsets of $\mathbb{C}\setminus (-\infty ,0]$, it may be differentiated term by term.

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x}}{\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& ={\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]z^n{\displaystyle \frac{\partial }{\partial x}}{(n+x)}^{-s}\hfill \\ & =-s{\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]{\displaystyle \frac{z^n}{{(n+x)}^{s+1}}}.\hfill \end{array}$$

The final series is ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s+1,x;z)$, which proves (25). □

3.3. A Functional-Differential Relation

Let

$$\vartheta =z{\displaystyle \frac{\partial }{\partial z}}$$

(26)

denote the Euler differential operator.

Theorem 4.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,s\in \mathbb{C}$, $\Re \left(x\right)>0$, and $\left|z\right|<1$. Assume that ${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-}$, $j=0,1,\dots ,d-1$. Then

$$\begin{array}{cc}\hfill {\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right){\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& =\left[{\displaystyle \prod _{j=0}^{d-1}}\left({\displaystyle \frac{\beta +j}{d}}-1\right)\right]{\displaystyle \frac{1}{x^s}}\hfill \\ \hfill & +z{\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\alpha +j}{d}}\right){\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x+1;z).\hfill \end{array}$$

(27)

Proof. 

Set

$$A_n={\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}.$$

Then

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

Since $\vartheta z^n=n{z}^n$, we have

$${\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right){\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\beta +j}{d}}-1\right){\displaystyle \frac{z^n}{{(n+x)}^s}}.$$

For $n\ge 1$,

$$A_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\beta +j}{d}}-1\right)=A_{n-1}{\displaystyle \prod _{j=0}^{d-1}}\left(n-1+{\displaystyle \frac{\alpha +j}{d}}\right).$$

(28)

Separating the term $n=0$, applying (28), and shifting the index gives

$$\begin{array}{cc}\hfill {\displaystyle \prod _{j=0}^{d-1}}\left(\vartheta +{\displaystyle \frac{\beta +j}{d}}-1\right){\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& =\left[{\displaystyle \prod _{j=0}^{d-1}}\left({\displaystyle \frac{\beta +j}{d}}-1\right)\right]{\displaystyle \frac{1}{x^s}}\hfill \\ & +z{\displaystyle \sum _{n=0}^{\infty }}{A}_n{\displaystyle \prod _{j=0}^{d-1}}\left(n+{\displaystyle \frac{\alpha +j}{d}}\right){\displaystyle \frac{z^n}{{(n+x+1)}^s}}.\hfill \end{array}$$

Recognizing the last series through the Euler operator (26) proves (27). □

Corollary 3.

Under the assumptions of Remark 3, we have

$$\mathsf{\Phi }(z,s,x)={\displaystyle \frac{1}{x^s}}+z\mathsf{\Phi }(z,s,x+1).$$

(29)

Proof. 

When $\alpha =\beta$, the rising-factorial terms in (22) cancel. Hence

$${\mathcal{Z}}_d^{(\alpha ,\alpha )}(s,x;z)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{{(n+x)}^s}}=\mathsf{\Phi }(z,s,x).$$

Separating the term $n=0$ and shifting the index yields (29). □

3.4. An Integral Representation

Recall that the generalized hypergeometric function is defined by

$${}_{p}F_q\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array};z\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(a_1\right)}^{\overline{n}}\cdots {\left(a_p\right)}^{\overline{n}}}{{\left(b_1\right)}^{\overline{n}}\cdots {\left(b_q\right)}^{\overline{n}}}}{\displaystyle \frac{z^n}{n!}},$$

where $b_j\notin {\mathbb{Z}}_0^{-}$ for $j=1,\dots ,q$; see, for example, [7,8,13].

Theorem 5.

Let $d\in \mathbb{N}$, $\alpha ,\beta ,z\in \mathbb{C}$, $\Re \left(s\right)>0$, $\Re \left(x\right)>0$, and $\left|z\right|<1$. Assume that ${\displaystyle \frac{\beta +j}{d}}\notin {\mathbb{Z}}_0^{-}$, $j=0,1,\dots ,d-1$. Then

$$\begin{array}{cc}\hfill {\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-xt}\hfill \\ \hfill & \times {}_{d+1}F_d\left(\begin{array}{c}1,{\displaystyle \frac{\alpha }{d}},{\displaystyle \frac{\alpha +1}{d}},\dots ,{\displaystyle \frac{\alpha +d-1}{d}}\\ {\displaystyle \frac{\beta }{d}},{\displaystyle \frac{\beta +1}{d}},\dots ,{\displaystyle \frac{\beta +d-1}{d}}\end{array};z{e}^{-t}\right)dt.\hfill \end{array}$$

(30)

Proof. 

Using the following known integral formula

$${\displaystyle \frac{1}{{(n+x)}^s}}={\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-(n+x)t}dt,$$

valid for $\Re \left(s\right)>0$ and $\Re \left(x\right)>0$, in (22), we obtain

$${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}{\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]z^n{\int }_0^{\infty }{t}^{s-1}{e}^{-(n+x)t}dt.$$

The quotient of the Pochhammer factors grows at most polynomially in n. Hence, for some constants $C,M>0$, the absolute value of the integrand series is bounded by

$$C{t}^{\Re \left(s\right)-1}{e}^{-\Re \left(x\right)t}{\displaystyle \sum _{n=0}^{\infty }}{(1+n)}^M{\left|z\right|}^n.$$

Since $\left|z\right|<1$, $\Re \left(s\right)>0$, and $\Re \left(x\right)>0$, this majorant is integrable on $(0,\infty )$. Therefore, the interchange of summation and integration follows from the dominated convergence theorem. Hence,

$$\begin{array}{cc}\hfill {\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}{\int }_0^{\infty }{t}^{s-1}{e}^{-xt}\hfill \\ & \times {\displaystyle \sum _{n=0}^{\infty }}\left[{\displaystyle \prod _{j=0}^{d-1}}{\displaystyle \frac{{\left(\frac{\alpha +j}{d}\right)}^{\overline{n}}}{{\left(\frac{\beta +j}{d}\right)}^{\overline{n}}}}\right]{\left(z{e}^{-t}\right)}^{n}dt.\hfill \end{array}$$

Because

$${\displaystyle \frac{1^{\overline{n}}}{n!}}=1,$$

the inner series is

$${}_{d+1}F_d\left(\begin{array}{c}1,{\displaystyle \frac{\alpha }{d}},{\displaystyle \frac{\alpha +1}{d}},\dots ,{\displaystyle \frac{\alpha +d-1}{d}}\\ {\displaystyle \frac{\beta }{d}},{\displaystyle \frac{\beta +1}{d}},\dots ,{\displaystyle \frac{\beta +d-1}{d}}\end{array};z{e}^{-t}\right).$$

This proves (30). □

3.5. Numerical Consistency Checks

The examples below are not intended as applications; they serve as independent checks of the coefficient and interpolation formulas. A genuine application is developed in Section 4.

Example 1.

Let

$$a=e,\lambda =2,\alpha =2.$$

Then (12) becomes

$${\left({\displaystyle \frac{2}{e^t+2}}\right)}^2={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(2\right)}(2;e){\displaystyle \frac{t^n}{n!}}.$$

(31)

Expanding the left-hand side of (31) at $t=0$ gives

$${\left({\displaystyle \frac{2}{e^t+2}}\right)}^2={\displaystyle \frac{4}{9}}-{\displaystyle \frac{8}{27}}t+{\displaystyle \frac{8}{243}}{t}^3+{\displaystyle \frac{2}{729}}{t}^4-{\displaystyle \frac{14}{3645}}{t}^5+O\left(t^6\right).$$

Since the coefficient of $t^n$ is $y_{9,n}^{\left(2\right)}(2;e)/n!$, this expansion gives, for example,

$$y_{9,3}^{\left(2\right)}(2;e)=3!\left({\displaystyle \frac{8}{243}}\right)={\displaystyle \frac{16}{81}}.$$

All entries in Table 1 were computed using exact rational arithmetic and independently verified using Proposition 3; therefore, no numerical rounding was involved.

Table 1. The first values of $y_{9,n}^{\left(2\right)}(2;e)$.

n	${\mathit{y}}_{9,\mathit{n}}^{\left(2\right)}(2;\mathit{e})$
0	${\displaystyle \frac{4}{9}}$
1	$-{\displaystyle \frac{8}{27}}$
2	0
3	${\displaystyle \frac{16}{81}}$
4	${\displaystyle \frac{16}{243}}$
5	$-{\displaystyle \frac{112}{243}}$

These values illustrate how the generating function and the explicit formula in Proposition 3 can be used for the effective computation of the higher-order numbers.

Example 2.

As a non-first-order numerical test, let

$$\alpha =2,\lambda =3,a=e.$$

Then

$$Z_y(-n,3;2,e)={\left({\displaystyle \frac{2}{3}}\right)}^2{\displaystyle \sum _{k=1}^{\infty }}{(-1)}^k(k+1){\displaystyle \frac{k^n}{3^k}}.$$

The coefficients $y_{9,n}^{\left(2\right)}(3;e)$ are computed independently from

$${\left({\displaystyle \frac{2}{e^t+3}}\right)}^2={\displaystyle \sum _{n=0}^{\infty }}{y}_{9,n}^{\left(2\right)}(3;e){\displaystyle \frac{t^n}{n!}}.$$

To evaluate the interpolation series independently, let $D=z{\displaystyle \frac{d}{dz}}$. For $n\ge 1$,

$$Z_y(-n,3;2,e)={\left({\displaystyle \frac{2}{3}}\right)}^2{\left.D^n\left({\displaystyle \frac{1}{{(1-z)}^2}}\right)\right|}_{z=-1/3}.$$

The resulting exact values are compared with $y_{9,n}^{\left(2\right)}(3;e)$ in Table 2.

Table 2. For $\alpha =2$, $\lambda =3$, and $a=e$, some values of the numbers $y_{9,n}^{\left(2\right)}(3;e)$ and the function $Z_y(-n,3;2,e)$.

n	${\mathit{y}}_{9,\mathit{n}}^{\left(2\right)}(3;\mathit{e})$	${\left(\frac{3}{2}\right)}^2{\mathit{y}}_{9,\mathit{n}}^{\left(2\right)}(3;\mathit{e})$	${\mathit{Z}}_{\mathit{y}}(-\mathit{n},3;2,\mathit{e})$
1	$-{\displaystyle \frac{1}{8}}$	$-{\displaystyle \frac{9}{32}}$	$-{\displaystyle \frac{1}{8}}$
2	$-{\displaystyle \frac{1}{32}}$	$-{\displaystyle \frac{9}{128}}$	$-{\displaystyle \frac{1}{32}}$
3	${\displaystyle \frac{1}{16}}$	${\displaystyle \frac{9}{64}}$	${\displaystyle \frac{1}{16}}$
4	${\displaystyle \frac{11}{128}}$	${\displaystyle \frac{99}{512}}$	${\displaystyle \frac{11}{128}}$
5	$-{\displaystyle \frac{17}{256}}$	$-{\displaystyle \frac{153}{1024}}$	$-{\displaystyle \frac{17}{256}}$

We now give some values of the numbers $y_{9,n}^{\left(2\right)}(3;e)$ and the function $Z_y(-n,3;2,e)$ in the following table.

4. An Application: A Probability Law Associated with the Interpolation Function

In this section we show that the interpolation function $Z_y$ admits a natural probabilistic interpretation. For $\lambda <-1$ the weights appearing in the interpolation series (16) are all positive, and after normalization they define a probability distribution on $\{1,2,3,\dots \}$ whose moments are given by the values of $Z_y$ at nonpositive integers. This provides a concrete application of the interpolation function and connects the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ to the classical theory of discrete probability distributions [31].

4.1. Construction of the Probability Distribution

Throughout this section we fix $a>1$, $\lambda <-1$, and $\alpha >0$. Set

$$\mu =-\lambda >1,q={\displaystyle \frac{1}{\mu }}\in (0,1).$$

(32)

Consider the weights

$$w_k=\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right)q^k,k\ge 1.$$

(33)

Since $q\in (0,1)$ and $\alpha >0$, we have $w_k>0$ for every $k\ge 1$. Moreover, by the binomial series

$${(1-q)}^{-\alpha }={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right)q^k,$$

the total mass is

$$S:={\displaystyle \sum _{k=1}^{\infty }}{w}_k={(1-q)}^{-\alpha }-1={\left({\displaystyle \frac{\mu }{\mu -1}}\right)}^{\alpha }-1\in (0,\infty ).$$

(34)

Therefore,

$$p_k:={\displaystyle \frac{w_k}{S}},k\ge 1,$$

(35)

defines a probability distribution on $\{1,2,3,\dots \}$, which is the zero-truncated negative binomial distribution with parameters $(\alpha ,q)$ [31].

Remark 5.

The weights (33) are exactly the absolute values of the coefficients appearing in the interpolation series (16) when $\lambda <0$. Indeed, for $\lambda =-\mu$,

$${(-1)}^k{\lambda }^{-k}={(-1)}^k{(-\mu )}^{-k}={\mu }^{-k}=q^k.$$

Thus, the passage from the analytic object $Z_y$ to the probabilistic object X is canonical.

4.2. Moment Generating Function and Moments

Let X be a random variable with distribution (35). Its moment generating function is

$$M_X\left(t\right)=\mathbb{E}\left[e^{tX}\right]={\displaystyle \frac{1}{S}}{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\left(q{e}^t\right)}^k={\displaystyle \frac{{(1-q{e}^t)}^{-\alpha }-1}{{(1-q)}^{-\alpha }-1}},$$

(36)

for $t<\ln \mu =-\ln q$. Substituting $q=1/\mu$ gives the equivalent form

$$M_X\left(t\right)={\displaystyle \frac{{(1-e^t/\mu )}^{-\alpha }-1}{{(1-1/\mu )}^{-\alpha }-1}},t<\ln \mu .$$

(37)

The following theorem is the main result of this section: it identifies the moments of X with the values of the interpolation function $Z_y$ at nonpositive integers.

Theorem 6.

Let $a>1$, $\lambda <-1$, $\mu =-\lambda >1$, and $\alpha >0$. Let X be the random variable with distribution (35). Then, for every $n\in {\mathbb{N}}_0$,

$$\mathbb{E}\left[X^n\right]={\displaystyle \frac{Z_y(-n,\lambda ;\alpha ,a)}{{\left(\ln a\right)}^n{Z}_y(0,\lambda ;\alpha ,a)}}.$$

(38)

Proof. 

By (37), the n-th derivative of $M_X$ at $t=0$ is

$$M_X^{\left(n\right)}\left(0\right)={\displaystyle \frac{1}{S}}{\displaystyle \frac{d^n}{d{t}^n}}{\left[{(1-e^t/\mu )}^{-\alpha }\right]}_{t=0}.$$

Expanding the binomial series,

$${(1-e^t/\mu )}^{-\alpha }={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\mu }^{-k}{e}^{kt},$$

so that

$${\displaystyle \frac{d^n}{d{t}^n}}{\left[{(1-e^t/\mu )}^{-\alpha }\right]}_{t=0}={\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\mu }^{-k}{k}^n,$$

the term $k=0$ is vanishing for $n\ge 1$ and contributing 1 for $n=0$. Hence, for $n\ge 1$,

$$\mathbb{E}\left[X^n\right]={\displaystyle \frac{1}{S}}{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\mu }^{-k}{k}^n.$$

(39)

On the other hand, from (16) with $s=-n$ and $\lambda =-\mu$,

$$Z_y(-n,\lambda ;\alpha ,a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }{\left(\ln a\right)}^n{\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right){\mu }^{-k}{k}^n$$

and, for $n=0$,

$$Z_y(0,\lambda ;\alpha ,a)={\left({\displaystyle \frac{2}{\lambda }}\right)}^{\alpha }S.$$

Dividing the first identity by the second yields (38) for $n\ge 1$; the case $n=0$ is immediate since both sides equal 1. □

Remark 6.

Formula (38) shows that the interpolation function $Z_y$ is the natural moment generating object attached to the family $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. Equivalently, using the notation $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)=Z_y(-n,\lambda ;\alpha ,a)$ for $n\ge 1$ (Theorem 1), one may write

$$\mathbb{E}\left[X^n\right]={\displaystyle \frac{y_{9,n}^{\left(\alpha \right)}(\lambda ;a)}{{\left(\ln a\right)}^n{Z}_y(0,\lambda ;\alpha ,a)}},n\ge 1.$$

4.3. Mean, Variance, and the Geometric Case

From (37) one computes directly

$$\mathbb{E}\left[X\right]={\displaystyle \frac{\alpha {\mu }^{\alpha }}{(\mu -1)\left({\mu }^{\alpha }-{(\mu -1)}^{\alpha }\right)}},$$

(40)

and, more generally, the falling factorial moments

$$\mathbb{E}\left[X(X-1)\cdots (X-n+1)\right]={\displaystyle \frac{{\alpha }^{\overline{n}}{\mu }^{\alpha }}{{(\mu -1)}^n\left({\mu }^{\alpha }-{(\mu -1)}^{\alpha }\right)}}.$$

The variance follows from the second factorial moment.

Example 3.

Take $\alpha =1$, so that

$$w_k=q^k,S={\displaystyle \frac{q}{1-q}},p_k=(1-q)q^{k-1}.$$

Thus, X is a geometric random variable on $\{1,2,\dots \}$ with parameter $1-q$. For $q=1/2$ (i.e., $\lambda =-2$) and $a=e$, we have $Z_y(0,-2;1,e)=-1$ and $Z_y(-1,-2;1,e)=-2$, so

$$\mathbb{E}\left[X\right]={\displaystyle \frac{Z_y(-1,-2;1,e)}{\ln e·Z_y(0,-2;1,e)}}={\displaystyle \frac{-2}{-1}}=2,$$

in agreement with the classical value $1/(1-q)=2$ for the geometric distribution. This provides an independent verification of (38).

4.4. Discussion

The probabilistic interpretation (38) has three consequences.

First, it shows that the sequence ${\left\{y_{9,n}^{\left(\alpha \right)}(\lambda ;a)\right\}}_{n\ge 0}$ is, up to the explicit normalization ${\left(\ln a\right)}^n{Z}_y(0,\lambda ;\alpha ,a)$, the moment sequence of a discrete probability distribution. This connects the higher-order numbers to the classical Hausdorff moment problem on $\{1,2,3,\dots \}$ [32], and complements recent probabilistic approaches to Stirling-type numbers [15].

Second, the moment generating function (37) gives immediate access to all factorial moments of X through differentiation, and hence to combinatorial interpretations of the higher-order numbers in terms of factorial moments of the zero-truncated negative binomial law.

Third, the same construction applies verbatim to the generalized zeta-type function ${\mathcal{Z}}_d^{(\alpha ,\beta )}$ introduced in Section 3: whenever the weights are positive, one obtains a probability distribution whose moments are encoded by ${\mathcal{Z}}_d^{(\alpha ,\beta )}$ at nonpositive integers. This suggests a systematic probabilistic reading of the interpolation formulas proved in this paper.

4.5. Asymptotic Behavior of the Higher-Order Numbers

The singularity structure of $f(t;\lambda ,\alpha ,a)$ yields the asymptotic growth of $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ when $\lambda <-1$. The poles of f solve $a^t=-\lambda$; for $\lambda <-1$ the dominant pole is the real number $t_0=\ln \left|\lambda \right|/\ln a>0$.

Proposition 4.

Let $a>1$, $\lambda <-1$, and $\alpha \in \mathbb{N}$. Then, as $n\to \infty$,

$$y_{9,n}^{\left(\alpha \right)}(\lambda ;a)={(-1)}^{\alpha }{\left({\displaystyle \frac{2}{\left|\lambda \right|}}\right)}^{\alpha }{\displaystyle \frac{{\left(\ln a\right)}^n}{{\left(\ln \right|\lambda \left|\right)}^{n+\alpha }}}{\displaystyle \frac{\mathsf{\Gamma }(n+\alpha )}{\mathsf{\Gamma }\left(\alpha \right)}}\left(1+O\left({\displaystyle \frac{1}{n}}\right)\right).$$

(41)

Proof. 

Set $s=t-t_0$. Since $a^{t_0}=\left|\lambda \right|$, we have $a^t+\lambda =\left|\lambda \right|(a^s-1)=\left|\lambda \right|\left(\ln a\right)s(1+O\left(s\right))$ as $s\to 0$; hence,

$$f(t;\lambda ,\alpha ,a)={\left({\displaystyle \frac{2}{\left|\lambda \right|\ln a}}\right)}^{\alpha }{(t-t_0)}^{-\alpha }\left(1+O(t-t_0)\right).$$

The transfer theorem for algebraic singularities [33] (Theorem VI.1) gives (41). □

Remark 7.

For $\alpha =1$, $\lambda =-2$, and $a=e$, Proposition 4 gives $y_{9,n}(-2;e)\sim -n!/{\left(\ln 2\right)}^{n+1}$, which, through the identity $y_{9,n}(-2;e)=-2w_g\left(n\right)$ established in the Introduction, recovers the classical asymptotic formula $w_g\left(n\right)\sim n!/\left(2{\left(\ln 2\right)}^{n+1}\right)$ for the Fubini numbers [17,21]. This provides an independent verification of (41).

5. Computational Directions and Algorithmic Outlook

The identities of Section 2 and Section 3 suggest four computational directions that we outline below.

5.1. Numerical Evaluation of $Z_y$ and ${\mathcal{Z}}_d^{(\alpha ,\beta )}$

Three representations of $Z_y$ are available: the binomial series (16), valid and geometrically convergent for $\left|\lambda \right|>1$; the residue-class decomposition (21), which splits the series into d sub-series indexed by residue classes; and the hypergeometric integral (30) for ${\mathcal{Z}}_d^{(\alpha ,\beta )}$, valid when $\Re \left(s\right)>0$ and $\Re \left(x\right)>0$. The first is efficient for moderate s and $\alpha$, the second for large $\alpha$ and small d, and the third when x is bounded away from zero. A systematic comparison, with error bounds in terms of $\left|\lambda \right|$, s, and d, is a natural next step [13,28].

5.2. Parameter-Dependent Nonlinear Equations

The differential identity (25) and the functional-differential relation (27) express the dependence of ${\mathcal{Z}}_d^{(\alpha ,\beta )}$ on x and z through local operators. Given prescribed values of $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$ and all but one parameter among $\{x,\alpha ,z\}$, these identities reduce the problem to a nonlinear equation in a single unknown. This reduction is relevant in regimes where the parameter dependence is smooth but the solution set may be multimodal—precisely the setting in which niching and evolutionary strategies for nonlinear equations, such as those developed by Wang et al. [34], are most effective. The differential identities supply the symbolic input that such solvers would otherwise approximate by finite differences.

5.3. High-Dimensional Parameter Regimes

When $d\gg 1$, evaluating the product of d Pochhammer ratios in (22) directly becomes expensive, and the residue-class decomposition (21) is no longer directly usable. This is the same regime in which dimensionality-reduction techniques for high-dimensional parameter spaces become essential. The approach of Shao et al. [35], which learns a low-dimensional representation of the parameter space while preserving the objective, suggests a methodological template: one may ask whether ${\mathcal{Z}}_d^{(\alpha ,\beta )}(s,x;z)$ admits an approximate factorization in terms of a small number of effective parameters when $d\gg 1$, and how such a reduction propagates to the interpolation formula (17). We leave this question open.

5.4. Combinatorial Representations and Symbolic Computation

The binomial expansion (15) expresses $y_{9,n}^{\left(\alpha \right)}$ as a signed sum over compositions of n, while the residue-class decomposition (21) splits this sum into d sub-sums indexed by residue classes modulo d. Both structures can be encoded symbolically and manipulated by computer algebra systems. Permutation-based representations of combinatorial objects, such as those recently developed by Li et al. [36] for mass functions, provide a further template for encoding the higher-order numbers and translating the identities of Section 2 into enumerative algorithms.

6. Conclusions

In this paper, we developed a higher-order extension of the numbers $y_{9,n}(\lambda ;a)$ and their associated polynomials through their generating functions. We derived explicit formulas, convolution identities, and relations with the Apostol–Euler, Apostol–Bernoulli, Frobenius–Euler, Fubini, and Stirling families.

A principal contribution of the paper is the construction of an interpolation function for the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. Its values at negative integers recover these numbers up to an explicit normalization factor. By decomposing the interpolation series into residue classes, we obtained a representation in terms of a generalized hypergeometric Hurwitz–Lerch-type function whose numerator and denominator parameters occur in arithmetic progressions.

The zeta-type function considered here is a structured specialization of a generalized hypergeometric Hurwitz–Lerch function rather than an entirely unrestricted new class. Its distinctive role in the present work arises from its natural occurrence in the residue-class decomposition and its explicit connection with the higher-order numbers $y_{9,n}^{\left(\alpha \right)}(\lambda ;a)$. We also established its reduction to the classical Lerch transcendent, a differential identity, a functional-differential relation, and a Mellin-type integral representation.

Section 4 shows that the interpolation function $Z_y$ admits a natural probabilistic interpretation for $\lambda <-1$: the higher-order numbers are, up to an explicit normalization, the moments of a zero-truncated negative binomial law, and the dominant singularity of the generating function yields their asymptotic behavior as $n\to \infty$. This connects the analytic objects of Section 3 to discrete probability theory, to the classical moment problem, and to analytic combinatorics.

The numerical examples provide exact-arithmetic verification of the coefficient and interpolation formulas. They also illustrate how the generating functions and explicit series representations can be used to compute the higher-order numbers.

Several analytic questions remain open, including analytic continuation beyond the disk of convergence, singularity structure, functional equations, special values, asymptotic expansions, and alternative integral representations. Section 5 outlines four computational directions that follow from the results of this paper: high-precision evaluation of $Z_y$ and ${\mathcal{Z}}_d^{(\alpha ,\beta )}$, solution of parameter-dependent nonlinear equations via the differential identities, approximation in high-dimensional parameter regimes, and symbolic combinatorial representations. It would also be useful to investigate summation formulas involving classical zeta functions, polylogarithms, and Apostol-type numbers [9,15,19,26,37].