Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid polynomials

Abstract

In this paper, by considering Barnes-type Narumi polynomials of the second kind and Barnes-type Peters polynomials of the second kind, we define and investigate the hybrid polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

MSC:05A15, 05A40, 11B68, 11B75, 11B83.

1 Introduction

In this paper, we consider the polynomials

$${\widehat{\mathrm{NS}}}_n(x)={\widehat{\mathrm{NS}}}_n(x|a;\lambda ;\mu )={\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$$

called Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid polynomials, whose generating function is given by

$$\begin{array}{c}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^x\hfill \\ =\sum _{n=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^n}{n!},\hfill \end{array}$$

(1)

where $a_1,\dots ,a_r,{\lambda }_1,\dots ,{\lambda }_s,{\mu }_1,\dots ,{\mu }_s\in \mathbb{C}$ with $a_1,\dots ,a_r,{\lambda }_1,\dots ,{\lambda }_s\ne 0$. When $x=0$, ${\widehat{\mathrm{NS}}}_n={\widehat{\mathrm{NS}}}_n(0)={\widehat{\mathrm{NS}}}_n(0|a;\lambda ;\mu )={\widehat{\mathrm{NS}}}_n(0|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$ are called Barnes-type Narumi of the second kind and Barnes-type Peters of the second kind hybrid numbers.

Recall that the Barnes-type Narumi polynomials of the second kind, denoted by ${\hat{\mathrm{N}}}_n(x|a_1,\dots ,a_r)$, are given by the generating function as

$$\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right){(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{\hat{\mathrm{N}}}_n(x|a_1,\dots ,a_r)\frac{t^n}{n!}.$$

If $x=0$, we write ${\hat{\mathrm{N}}}_n(a_1,\dots ,a_r)={\hat{\mathrm{N}}}_n(0|a_1,\dots ,a_r)$. Narumi polynomials were mentioned in [[1], p.127]. In addition, the Barnes-type Peters polynomials of the second kind, denoted by ${\hat{s}}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$, are given by the generating function as

$$\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{\hat{s}}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^n}{n!}.$$

If $s=1$, then ${\hat{s}}_n(x|\lambda ;\mu )$ are the Peters polynomials of the second kind. Peters polynomials were mentioned in [[1], p.128] and have been investigated in e.g. [2].

In this paper, by considering Barnes-type Narumi polynomials of the second kind and Barnes-type Peters polynomials of the second kind, we define and investigate the hybrid polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.

2 Umbral calculus

Let ℂ be the complex number field and let ℱ be the set of all formal power series in the variable t:

$$\mathcal{F}=\{f(t)=\sum _{k=0}^{\mathrm{\infty }}\frac{a_k}{k!}{t}^k|a_k\in \mathbb{C}\}.$$

(2)

Let $\mathbb{P}=\mathbb{C}[x]$ and let ${\mathbb{P}}^{\ast }$ be the vector space of all linear functionals on ℙ. $〈L|p(x)〉$ is the action of the linear functional L on the polynomial $p(x)$, and we recall that the vector space operations on ${\mathbb{P}}^{\ast }$ are defined by $〈L+M|p(x)〉=〈L|p(x)〉+〈M|p(x)〉$, $〈cL|p(x)〉=c〈L|p(x)〉$, where c is a complex constant in ℂ. For $f(t)\in \mathcal{F}$, let us define the linear functional on ℙ by setting

$$〈f(t)|x^n〉=a_n(n\ge 0).$$

(3)

In particular,

$$〈t^k|x^n〉=n!{\delta }_{n,k}(n,k\ge 0),$$

(4)

where ${\delta }_{n,k}$ is Kronecker’s symbol.

For $f_L(t)={\sum }_{k=0}^{\mathrm{\infty }}\frac{〈L|x^k〉}{k!}{t}^k$, we have $〈f_L(t)|x^n〉=〈L|x^n〉$. That is, $L=f_L(t)$. The map $L\mapsto f_L(t)$ is a vector space isomorphism from ${\mathbb{P}}^{\ast }$ onto ℱ. Henceforth, ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on ℙ, and so an element $f(t)$ of ℱ will be thought of as both a formal power series and a linear functional. We call ℱ the umbral algebra and the umbral calculus is the study of umbral algebra. The order $O(f(t))$ of a power series $f(t)$ (≠0) is the smallest integer k for which the coefficient of $t^k$ does not vanish. If $O(f(t))=1$, then $f(t)$ is called a delta series; if $O(f(t))=0$, then $f(t)$ is called an invertible series. For $f(t),g(t)\in \mathcal{F}$ with $O(f(t))=1$ and $O(g(t))=0$, there exists a unique sequence $s_n(x)$ ($deg{s}_n(x)=n$) such that $〈g(t)f{(t)}^k|s_n(x)〉=n!{\delta }_{n,k}$, for $n,k\ge 0$. Such a sequence $s_n(x)$ is called the Sheffer sequence for $(g(t),f(t))$ which is denoted by $s_n(x)\sim (g(t),f(t))$.

For $f(t),g(t)\in \mathcal{F}$, and $p(x)\in \mathbb{P}$, we have

$$〈f(t)g(t)|p(x)〉=〈f(t)|g(t)p(x)〉=〈g(t)|f(t)p(x)〉$$

(5)

and

$$\begin{array}{r}f(t)=\sum _{k=0}^{\mathrm{\infty }}〈f(t)|x^k〉\frac{t^k}{k!},\\ p(x)=\sum _{k=0}^{\mathrm{\infty }}〈t^k|p(x)〉\frac{x^k}{k!}\end{array}$$

(6)

[[1], Theorem 2.2.5]. Thus, by (6), we get

$$t^{k}p(x)=p^{(k)}(x)=\frac{d^{k}p(x)}{d{x}^k}\text{and}{e}^{yt}p(x)=p(x+y).$$

(7)

Sheffer sequences are characterized by the generating function of [[1], Theorem 2.3.4].

Lemma 1 The sequence $s_n(x)$ is Sheffer for $(g(t),f(t))$ if and only if

$$\frac{1}{g(\overline{f}(t))}{e}^{y\overline{f}(t)}=\sum _{k=0}^{\mathrm{\infty }}\frac{s_k(y)}{k!}{t}^k(y\in \mathbb{C}),$$

where $\overline{f}(t)$ is the compositional inverse of $f(t)$.

For $s_n(x)\sim (g(t),f(t))$, we have the following equations [[1], Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]:

$$f(t)s_n(x)=n{s}_{n-1}(x)(n\ge 0),$$

(8)

$$s_n(x)=\sum _{j=0}^n\frac{1}{j!}〈g{(\overline{f}(t))}^{-1}\overline{f}{(t)}^j|x^n〉x^j,$$

(9)

$$s_n(x+y)=\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)s_j(x)p_{n-j}(y),$$

(10)

where $p_n(x)=g(t)s_n(x)$.

Assume that $p_n(x)\sim (1,f(t))$ and $q_n(x)\sim (1,g(t))$. Then the transfer formula [[1], Corollary 3.8.2] is given by

$$q_n(x)=x{\left(\frac{f(t)}{g(t)}\right)}^n{x}^{-1}{p}_n(x)(n\ge 1).$$

For $s_n(x)\sim (g(t),f(t))$, and $r_n(x)\sim (h(t),l(t))$, assume that

$$s_n(x)=\sum _{m=0}^n{C}_{n,m}{r}_m(x)(n\ge 0).$$

Then we have [[1], p.132]

$$C_{n,m}=\frac{1}{m!}〈\frac{h(\overline{f}(t))}{g(\overline{f}(t))}l{(\overline{f}(t))}^m|x^n〉.$$

(11)

3 Main results

For convenience, we introduce an Appell sequence of polynomials $F_n(a_1,\dots ,a_r)$ ($a_1,\dots ,a_r\ne 0$), defined by

$$\prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t}\right)e^{xt}=\sum _{n=0}^{\mathrm{\infty }}{F}_n(x|a_1,\dots ,a_r)\frac{t^n}{n!}.$$

If $x=0$, we write $F_n(a_1,\dots ,a_r)=F_n(0|a_1,\dots ,a_r)$. By [[1], Theorem 2.5.8] with $y=0$, we have

$$F_n(x|a_1,\dots ,a_r)=\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)F_{n-m}(a_1,\dots ,a_r)x^m.$$

More precisely, one can show that

$$F_n(x|a_1,\dots ,a_r)=\sum _{i=0}^n\sum _{l_1+\cdots +l_r=i}\frac{i!}{(i+r)!}\left(\genfrac{}{}{0ex}{}{n}{i}\right)\left(\genfrac{}{}{0ex}{}{i+r}{l_1+1,\dots ,l_r+1}\right)a_1^{l_1+1}\cdots a_r^{l_r+1}{x}^{n-i},$$

where

$$\left(\genfrac{}{}{0ex}{}{i+r}{l_1+1,\dots ,l_r+1}\right)=\frac{(i+r)!}{(l_1+1)!\cdots (l_r+1)!}.$$

Remember that the generalized Barnes-type Euler polynomials $E_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$ are defined by the generating function

$$\prod _{j=1}^s{\left(\frac{2}{1+e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^n}{n!}.$$

If ${\mu }_1=\cdots ={\mu }_s=1$, then $E_n(x|{\lambda }_1,\dots ,{\lambda }_s)=E_n(x|{\lambda }_1,\dots ,{\lambda }_s;1,\dots ,1)$ are called the Barnes-type Euler polynomials. If further ${\lambda }_1=\cdots ={\lambda }_s=1$, then $E_n^{(r)}(x)=E_n(x|1,\dots ,1;1,\dots ,1)$ are called the Euler polynomials of order s. If $x=0$, we write $E_n({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)=E_n(0|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$. By [[1], Theorem 2.5.8] with $y=0$, we have

$$E_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)=\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right)E_{n-m}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)x^m.$$

We note that

$$\begin{array}{c}t{F}_n(x|a_1,\dots ,a_r)=\frac{d}{dx}{F}_n(x|a_1,\dots ,a_r)=n{F}_{n-1}(x|a_1,\dots ,a_r),\hfill \\ t{E}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)=\frac{d}{dx}{E}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \phantom{t{E}_n(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)}=n{E}_{n-1}(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

From the definition (1), ${\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)$ is the Sheffer sequence for the pair

$$g(t)=\prod _{i=1}^r\left(\frac{t{e}^{a_{i}t}}{e^{a_{i}t}-1}\right)\prod _{j=1}^s{\left(\frac{1+e^{{\lambda }_{j}t}}{e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}\text{and}f(t)=e^t-1.$$

So,

$${\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\sim (\prod _{i=1}^r\left(\frac{t{e}^{a_{i}t}}{e^{a_{i}t}-1}\right)\prod _{j=1}^s{\left(\frac{1+e^{{\lambda }_{j}t}}{e^{{\lambda }_{j}t}}\right)}^{{\mu }_j},e^t-1).$$

(12)

3.1 Explicit expressions

Let ${(n)}_j=n(n-1)\cdots (n-j+1)$ ($j\ge 1$) with ${(n)}_0=1$. The (signed) Stirling numbers of the first kind $S_1(n,m)$ are defined by

$${(x)}_n=\sum _{m=0}^n{S}_1(n,m)x^m.$$

Theorem 1

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times F_l(-x|a_1,\dots ,a_r)\hfill \end{array}$$

(13)

$$\begin{array}{c}=2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times F_l(x+\sum _{j=1}^s{\lambda }_j{\mu }_j-\sum _{i=1}^r{a}_i|a_1,\dots ,a_r)\hfill \end{array}$$

(14)

$$\begin{array}{c}=2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r)\hfill \\ \times E_l(-x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \end{array}$$

(15)

$$=\sum _{j=0}^n\sum _{l=j}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,j){\widehat{\mathrm{NS}}}_{n-l}{x}^j$$

(16)

$$=\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{s}}_{n-l}({\lambda }_1,\dots ,{\lambda }_r;{\mu }_1,\dots ,{\mu }_r){\hat{N}}_l(x|a_1,\dots ,a_r),$$

(17)

$$=\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{N}}_{n-l}(a_1,\dots ,a_r){\hat{s}}_l(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).$$

(18)

Proof Since

$$\prod _{i=1}^r\left(\frac{t{e}^{a_{i}t}}{e^{a_{i}t}-1}\right)\prod _{j=1}^s{\left(\frac{1+e^{{\lambda }_{j}t}}{e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\sim (1,e^t-1)$$

(19)

and

$${(x)}_n\sim (1,e^t-1),$$

(20)

we have

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t{e}^{a_{i}t}}\right)\prod _{j=1}^s{\left(\frac{e^{{\lambda }_{j}t}}{1+e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{(x)}_n\hfill \\ =\sum _{m=0}^n{S}_1(n,m)\prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t{e}^{a_{i}t}}\right)\prod _{j=1}^s{\left(\frac{e^{{\lambda }_{j}t}}{1+e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^m\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^m\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right){(-1)}^m{E}_m(-x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right){(-1)}^m\hfill \\ \times \sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s){(-x)}^l\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\hfill \\ \times \sum _{l=0}^m{(-1)}^l\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)x^l\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\hfill \\ \times \sum _{l=0}^m{(-1)}^l\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s){(-1)}^l{F}_l(-x|a_1,\dots ,a_r).\hfill \end{array}$$

So, we get (13).

We can obtain an alternative expression (14) for ${\widehat{\mathrm{NS}}}_n(x)$ as follows:

$$\begin{array}{rcl}{\widehat{\mathrm{NS}}}_n(x)& =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)e^{({\sum }_{j=1}^s{\lambda }_j{\mu }_j-{\sum }_{i=1}^r{a}_i)t}\\ \times \prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t}\right)\prod _{j=1}^s{\left(\frac{2}{1+e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^m\\ =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)e^{({\sum }_{j=1}^s{\lambda }_j{\mu }_j-{\sum }_{i=1}^r{a}_i)t}\\ \times \prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t}\right)E_m(x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\\ =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)e^{({\sum }_{j=1}^s{\lambda }_j{\mu }_j-{\sum }_{i=1}^r{a}_i)t}\\ \times \prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t}\right)\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)x^l\\ =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\\ \times e^{({\sum }_{j=1}^s{\lambda }_j{\mu }_j-{\sum }_{i=1}^r{a}_i)t}\prod _{i=1}^r\left(\frac{e^{a_{i}t}-1}{t}\right)x^l\\ =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\\ \times F_l(x+\sum _{j=1}^s{\lambda }_j{\mu }_j-\sum _{i=1}^r{a}_i|a_1,\dots ,a_r).\end{array}$$

From the proof of (13),

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)x^m\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{S}_1(n,m)\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}{(-1)}^m{F}_m(-x|a_1,\dots ,a_r)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)F_{m-l}(a_1,\dots ,a_r){(-x)}^l\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)F_{m-l}(a_1,\dots ,a_r){(-1)}^l\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^l\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)F_{m-l}(a_1,\dots ,a_r)E_l(-x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r)E_l(-x|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s),\hfill \end{array}$$

which is the identity (15).

By (12), we have

$$\begin{array}{c}〈g{(\overline{f}(t))}^{-1}\overline{f}{(t)}^j|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^j|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|j!\sum _{l=j}^{\mathrm{\infty }}{S}_1(l,j)\frac{t^l}{l!}{x}^n〉\hfill \\ =j!\sum _{l=j}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,j)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|x^{n-l}〉\hfill \\ =j!\sum _{l=j}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,j)〈\sum _{i=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_i\frac{t^i}{i!}|x^{n-l}〉=j!\sum _{l=j}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,j){\widehat{\mathrm{NS}}}_{n-l}.\hfill \end{array}$$

By (9), we get the identity (16).

Next,

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =〈\sum _{i=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_i(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^i}{i!}|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y|x^n〉\hfill \\ =〈\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right){(1+t)}^y{x}^n〉\hfill \\ =〈\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|\sum _{l=0}^{\mathrm{\infty }}{\hat{N}}_l(y|a_1,\dots ,a_r)\frac{t^l}{l!}{x}^n〉\hfill \\ =\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{N}}_l(y|a_1,\dots ,a_r)〈\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|x^{n-l}〉\hfill \\ =\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{N}}_l(y|a_1,\dots ,a_r)〈\sum _{i=0}^{\mathrm{\infty }}{\hat{s}}_i({\lambda }_1,\dots ,{\lambda }_r;{\mu }_1,\dots ,{\mu }_r)\frac{t^i}{i!}|x^{n-l}〉\hfill \\ =\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{N}}_l(y|a_1,\dots ,a_r){\hat{s}}_{n-l}({\lambda }_1,\dots ,{\lambda }_r;{\mu }_1,\dots ,{\mu }_r).\hfill \end{array}$$

Thus, we obtain (17).

Finally, we obtain

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =〈\sum _{i=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_i(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^i}{i!}|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)|\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y{x}^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)|\sum _{l=0}^{\mathrm{\infty }}{\hat{s}}_l(y|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^l}{l!}{x}^n〉\hfill \\ =\sum _{l=0}^n{\hat{s}}_l(y|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\left(\genfrac{}{}{0ex}{}{n}{l}\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)|x^{n-l}〉\hfill \\ =\sum _{l=0}^n{\hat{s}}_l(y|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\left(\genfrac{}{}{0ex}{}{n}{l}\right)〈\sum _{i=0}^{\mathrm{\infty }}{\hat{N}}_i(a_1,\dots ,a_r)\frac{t^i}{i!}|x^{n-l}〉\hfill \\ =\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\hat{s}}_l(y|{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s){\hat{N}}_{n-l}(a_1,\dots ,a_r).\hfill \end{array}$$

Thus, we get the identity (18). □

3.2 Sheffer identity

Theorem 2

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x+y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){\widehat{\mathrm{NS}}}_l(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s){(y)}_{n-l}.\hfill \end{array}$$

(21)

Proof By (12) with

$$\begin{array}{rl}{p}_n(x)& =\prod _{i=1}^r\left(\frac{t{e}^{a_{i}t}}{e^{a_{i}t}-1}\right)\prod _{j=1}^s{\left(\frac{1+e^{{\lambda }_{j}t}}{e^{{\lambda }_{j}t}}\right)}^{{\mu }_j}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\\ ={(x)}_n\sim (1,e^t-1),\end{array}$$

using (10), we have (21). □

3.3 Difference relations

Theorem 3

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x+1|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)-{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =n{\widehat{\mathrm{NS}}}_{n-1}(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

(22)

Proof By (8) with (12), we get

$$\begin{array}{c}(e^t-1){\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =n{\widehat{\mathrm{NS}}}_{n-1}(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

By (7), we have (22). □

3.4 Recurrence

Theorem 4

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_{n+1}(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =(x+\sum _{j=1}^s{\mu }_j{\lambda }_j-\sum _{i=1}^r{a}_i){\widehat{\mathrm{NS}}}_n(x-1|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ +2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\frac{{(-1)}^{m+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)}{l+1}{S}_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times (\sum _{i=1}^r{a}_i{F}_{l+1}(1-x|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r)-r{F}_{l+1}(1-x|a_1,\dots ,a_r))\hfill \\ +2^{-1-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^{m+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r)\hfill \\ \times \sum _{i=1}^s{\mu }_i{\lambda }_i{E}_l(1-x|\lambda ;\mu +e_i).\hfill \end{array}$$

(23)

Proof By applying

$$s_{n+1}(x)=(x-\frac{g^{\prime }(t)}{g(t)})\frac{1}{f^{\prime }(t)}{s}_n(x)$$

(24)

[[1], Corollary 3.7.2] with (12), we get

$${\widehat{\mathrm{NS}}}_{n+1}(x)=x{\widehat{\mathrm{NS}}}_n(x-1)-e^{-t}\frac{g^{\prime }(t)}{g(t)}{\widehat{\mathrm{NS}}}_n(x).$$

Observe that

$$\begin{array}{rl}\frac{g^{\prime }(t)}{g(t)}& ={(lng(t))}^{\prime }\\ ={(rlnt+\left(\sum _{i=1}^r{a}_i\right)t-\sum _{i=1}^{r}ln(e^{a_{i}t}-1)+\sum _{j=1}^s{\mu }_{j}ln(1+e^{{\lambda }_{j}t})-\left(\sum _{j=1}^s{\mu }_j{\lambda }_j\right)t)}^{\prime }\\ =\frac{r}{t}+\sum _{i=1}^r{a}_i-\sum _{i=1}^r\frac{a_i{e}^{a_{i}t}}{e^{a_{i}t}-1}+\sum _{j=1}^s\frac{{\mu }_j{\lambda }_j{e}^{{\lambda }_{j}t}}{1+e^{{\lambda }_{j}t}}-\sum _{j=1}^s{\mu }_j{\lambda }_j\\ =\frac{r-{\sum }_{i=1}^r\frac{a_{i}t{e}^{a_{i}t}}{e^{a_{i}t}-1}}{t}+\sum _{i=1}^r{a}_i+\sum _{j=1}^s\frac{{\mu }_j{\lambda }_j{e}^{{\lambda }_{j}t}}{1+e^{{\lambda }_{j}t}}-\sum _{j=1}^s{\mu }_j{\lambda }_j,\end{array}$$

where

$$r-\sum _{i=1}^r\frac{a_{i}t{e}^{a_{i}t}}{e^{a_{i}t}-1}=-\frac{1}{2}\left(\sum _{i=1}^r{a}_i\right)t+\cdots$$

has the order at least one. Since from the proofs of (13) and (15)

$$\begin{array}{rcl}{\widehat{\mathrm{NS}}}_n(x)& =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\\ \times \sum _{l=0}^m{(-1)}^l\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)x^l\\ =& 2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\\ \times \sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)F_{m-l}(a_1,\dots ,a_r){(-1)}^l\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^l,\end{array}$$

we have

$$\begin{array}{c}\frac{g^{\prime }(t)}{g(t)}{\widehat{\mathrm{NS}}}_n(x)\hfill \\ =(\sum _{i=1}^r{a}_i-\sum _{j=1}^s{\mu }_j{\lambda }_j){\widehat{\mathrm{NS}}}_n(x)\hfill \\ +2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m{(-1)}^l\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times \prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)\frac{r-{\sum }_{i=1}^r\frac{a_{i}t}{1-e^{-a_{i}t}}}{t}{x}^l\hfill \\ +2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)F_{m-l}(a_1,\dots ,a_r){(-1)}^l\hfill \\ \times \sum _{i=1}^s\frac{{\mu }_i{\lambda }_i}{1+e^{-{\lambda }_{i}t}}\prod _{j=1}^s{\left(\frac{2}{1+e^{-{\lambda }_{j}t}}\right)}^{{\mu }_j}{x}^l.\hfill \end{array}$$

The third term is equal to

$$\begin{array}{c}{2}^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r){(-1)}^l\hfill \\ \times \sum _{i=1}^s\frac{{\mu }_i{\lambda }_i}{2}{(-1)}^l{E}_l(-x|\lambda ;\mu +e_i)\hfill \\ =2^{-1-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^m\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r)\hfill \\ \times \sum _{i=1}^s{\mu }_i{\lambda }_i{E}_l(-x|\lambda ;\mu +e_i),\hfill \end{array}$$

where $\lambda =({\lambda }_1,\dots ,{\lambda }_s)$, $\mu =({\mu }_1,\dots ,{\mu }_s)$, and . The second term is

$$\begin{array}{c}{2}^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\frac{{(-1)}^l}{l+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times \prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)(r-\sum _{i=1}^r\frac{a_{i}t}{1-e^{-a_{i}t}})x^{l+1}\hfill \\ =r{2}^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\frac{{(-1)}^l}{l+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times \prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)x^{l+1}\hfill \\ -2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\frac{{(-1)}^l}{l+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times \sum _{i=1}^r{a}_i\frac{-t}{e^{-a_{i}t}-1}\prod _{i=1}^r\left(\frac{e^{-a_{i}t}-1}{-t}\right)x^{l+1}\hfill \\ =r{2}^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\frac{{(-1)}^l}{l+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times {(-1)}^{l+1}{F}_{l+1}(-x|a_1,\dots ,a_r)\hfill \\ -2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n{(-1)}^m{S}_1(n,m)\sum _{l=0}^m\frac{{(-1)}^l}{l+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times \sum _{i=1}^r{a}_i{(-1)}^{l+1}{F}_{l+1}(-x|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r)\hfill \\ =2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\frac{{(-1)}^{m+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)}{l+1}{S}_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times (r{F}_{l+1}(-x|a_1,\dots ,a_r)-\sum _{i=1}^r{a}_i{F}_{l+1}(-x|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r)).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_{n+1}(x)\hfill \\ =(x+\sum _{j=1}^s{\mu }_j{\lambda }_j-\sum _{i=1}^r{a}_i){\widehat{\mathrm{NS}}}_n(x-1)\hfill \\ +2^{-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m\frac{{(-1)}^{m+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)}{l+1}{S}_1(n,m)E_{m-l}({\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ \times (\sum _{i=1}^r{a}_i{F}_{l+1}(1-x|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r)-r{F}_{l+1}(1-x|a_1,\dots ,a_r))\hfill \\ +2^{-1-{\sum }_{j=1}^s{\mu }_j}\sum _{m=0}^n\sum _{l=0}^m{(-1)}^{m+1}\left(\genfrac{}{}{0ex}{}{m}{l}\right)S_1(n,m)F_{m-l}(a_1,\dots ,a_r)\hfill \\ \times \sum _{i=1}^s{\mu }_i{\lambda }_i{E}_l(1-x|\lambda ;\mu +e_i),\hfill \end{array}$$

which is the identity (23). □

3.5 Differentiation

Theorem 5

$$\begin{array}{c}\frac{d}{dx}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =n!\sum _{l=0}^{n-1}\frac{{(-1)}^{n-l-1}}{l!(n-l)}{\widehat{\mathrm{NS}}}_l(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

(25)

Proof We shall use

$$\frac{d}{dx}{s}_n(x)=\sum _{l=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{l}\right)〈\overline{f}(t)|x^{n-l}〉s_l(x)$$

(cf. [[1], Theorem 2.3.12]). Since

$$\begin{array}{rl}〈\overline{f}(t)|x^{n-l}〉& =〈ln(1+t)|x^{n-l}〉\\ =〈\sum _{m=1}^{\mathrm{\infty }}\frac{{(-1)}^{m-1}{t}^m}{m}|x^{n-l}〉\\ =\sum _{m=1}^{n-l}\frac{{(-1)}^{m-1}}{m}〈t^m|x^{n-l}〉\\ =\sum _{m=1}^{n-l}\frac{{(-1)}^{m-1}}{m}(n-l)!{\delta }_{m,n-l}\\ ={(-1)}^{n-l-1}(n-l-1)!\end{array}$$

with (12), we have

$$\begin{array}{c}\frac{d}{dx}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\sum _{l=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{l}\right){(-1)}^{n-l-1}(n-l-1)!{\widehat{\mathrm{NS}}}_l(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =n!\sum _{l=0}^{n-1}\frac{{(-1)}^{n-l-1}}{l!(n-l)}{\widehat{\mathrm{NS}}}_l(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s),\hfill \end{array}$$

which is the identity (25). □

3.6 One more relation

The classical Cauchy numbers of the first kind $c_n$ are defined by

$$\frac{t}{ln(1+t)}=\sum _{n=0}^{\mathrm{\infty }}{c}_n\frac{t^n}{n!}$$

(see e.g. [3, 4]).

Theorem 6

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a;\lambda ;\mu )\hfill \\ =(x-\sum _{i=1}^r{a}_i){\widehat{\mathrm{NS}}}_{n-1}(x-1|a;\lambda ;\mu )+\sum _{j=1}^s{\lambda }_j{\mu }_j{\widehat{\mathrm{NS}}}_{n-1}(x-{\lambda }_j-1|a;\lambda ;\mu +e_j)\hfill \\ +\frac{1}{n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)c_l(\sum _{i=1}^r{a}_i{\widehat{\mathrm{NS}}}_{n-l}(x-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )\hfill \\ -r{\widehat{\mathrm{NS}}}_{n-l}(x-1|a;\lambda ;\mu )).\hfill \end{array}$$

(26)

Proof For $n\ge 1$, we have

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =〈\sum _{l=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_l(y|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^l}{l!}|x^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y|x^n〉\hfill \\ =〈{\partial }_t\left(\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y\right)|x^{n-1}〉\hfill \\ =〈\left({\partial }_t\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^y|x^{n-1}〉\hfill \\ +〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left({\partial }_t\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\right){(1+t)}^y|x^{n-1}〉\hfill \\ +〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\left({\partial }_t{(1+t)}^y\right)|x^{n-1}〉.\hfill \end{array}$$

The third term is

$$\begin{array}{c}y〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-1}|x^{n-1}〉\hfill \\ =y{\widehat{\mathrm{NS}}}_{n-1}(y-1|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

Since

$$\begin{array}{c}{\partial }_t\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\hfill \\ =\frac{1}{1+t}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\frac{{\sum }_{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})}{t}\hfill \\ -\frac{1}{1+t}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\sum _{l=1}^r{a}_l\right),\hfill \end{array}$$

with

$$\sum _{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})$$

having order ≥1, the first term is

$$\begin{array}{c}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-1}|\frac{{\sum }_{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})}{t}{x}^{n-1}〉\hfill \\ -\left(\sum _{l=1}^r{a}_l\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-1}|x^{n-1}〉\hfill \\ =-\left(\sum _{l=1}^r{a}_l\right){\widehat{\mathrm{NS}}}_{n-1}(y-1)+\frac{1}{n}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{y-1}|\sum _{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})x^n〉\hfill \\ =-\left(\sum _{l=1}^r{a}_l\right){\widehat{\mathrm{NS}}}_{n-1}(y-1)\hfill \\ +\frac{1}{n}(\sum _{\nu =1}^r{a}_{\nu }〈\frac{{(1+t)}^{a_{\nu }}ln(1+t)}{{(1+t)}^{a_{\nu }}-1}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{y-1}|\frac{t}{ln(1+t)}{x}^n〉\hfill \\ -r〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-1}|\frac{t}{ln(1+t)}{x}^n〉)\hfill \\ =-\left(\sum _{l=1}^r{a}_l\right){\widehat{\mathrm{NS}}}_{n-1}(y-1)\hfill \\ +\frac{1}{n}(\sum _{\nu =1}^r{a}_{\nu }〈\frac{{(1+t)}^{a_{\nu }}ln(1+t)}{{(1+t)}^{a_{\nu }}-1}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{y-1}|\sum _{l=0}^{\mathrm{\infty }}{c}_l\frac{t^l}{l!}{x}^n〉\hfill \\ -r〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-1}|\sum _{l=0}^{\mathrm{\infty }}{c}_l\frac{t^l}{l!}{x}^n〉)\hfill \\ =-\left(\sum _{l=1}^r{a}_l\right){\widehat{\mathrm{NS}}}_{n-1}(y-1)\hfill \\ +\frac{1}{n}(\sum _{\nu =1}^r\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)a_{\nu }{c}_l{\widehat{\mathrm{NS}}}_{n-l}(y-1|a_1,\dots ,a_{\nu -1},a_{\nu +1},\dots ,a_r;\lambda ;\mu )\hfill \\ -r\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)c_l{\widehat{\mathrm{NS}}}_{n-l}(y-1|a;\lambda ;\mu ))\hfill \\ =-\left(\sum _{\nu =1}^r{a}_{\nu }\right){\widehat{\mathrm{NS}}}_{n-1}(y-1)\hfill \\ +\frac{1}{n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)c_l(\sum _{\nu =1}^r{a}_{\nu }{\widehat{\mathrm{NS}}}_{n-l}(y-1|a_1,\dots ,a_{\nu -1},a_{\nu +1},\dots ,a_r;\lambda ;\mu )\hfill \\ -r{\widehat{\mathrm{NS}}}_{n-l}(y-1|a;\lambda ;\mu )).\hfill \end{array}$$

Since

$${\partial }_t\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}=\sum _{l=1}^s{\lambda }_l{\mu }_l{(1+t)}^{-{\lambda }_l-1}\left(\frac{{(1+t)}^{{\lambda }_l}}{1+{(1+t)}^{{\lambda }_l}}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j},$$

the second term is

$$\begin{array}{c}\sum _{l=1}^s{\lambda }_l{\mu }_l〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\frac{{(1+t)}^{{\lambda }_l}}{1+{(1+t)}^{{\lambda }_l}}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{y-{\lambda }_l-1}|x^{n-1}〉\hfill \\ =\sum _{l=1}^s{\lambda }_l{\mu }_l{\widehat{\mathrm{NS}}}_{n-1}(y-{\lambda }_l-1|a;\lambda ;\mu +e_l).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a;\lambda ;\mu )\hfill \\ =(x-\sum _{i=1}^r{a}_i){\widehat{\mathrm{NS}}}_{n-1}(x-1|a;\lambda ;\mu )+\sum _{j=1}^s{\lambda }_j{\mu }_j{\widehat{\mathrm{NS}}}_{n-1}(x-{\lambda }_j-1|a;\lambda ;\mu +e_j)\hfill \\ +\frac{1}{n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)c_l(\sum _{i=1}^r{a}_i{\widehat{\mathrm{NS}}}_{n-l}(x-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )\hfill \\ -r{\widehat{\mathrm{NS}}}_{n-l}(x-1|a;\lambda ;\mu )),\hfill \end{array}$$

which is the identity (26). □

3.7 A relation involving the Stirling numbers of the first kind

Theorem 7 For $n-1\ge m\ge 1$, we have

$$\begin{array}{c}\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(n-l,m){\widehat{\mathrm{NS}}}_l(a;\lambda ;\mu )\hfill \\ =-\left(\sum _{i=1}^r{a}_i\right)\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu )\hfill \\ +\frac{1}{n}\sum _{k=0}^{n-m}\sum _{l=k}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{l}{k}\right)S_1(n-l,m)c_{l-k}\hfill \\ \times (\sum _{i=1}^r{a}_i{\widehat{\mathrm{NS}}}_k(-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )-r{\widehat{\mathrm{NS}}}_k(-1|a;\lambda ;\mu ))\hfill \\ +\sum _{j=1}^s{\lambda }_j{\mu }_j\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-{\lambda }_j-1|a;\lambda ;\mu +e_j)\hfill \\ +\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m-1){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu ).\hfill \end{array}$$

(27)

Proof We shall compute

$$〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|x^n〉$$

in two different ways. On the one hand, it is equal to

$$\begin{array}{c}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|{(ln(1+t))}^m{x}^n〉\hfill \\ =〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|m!\sum _{l=m}^{\mathrm{\infty }}{S}_1(l,m)\frac{t^l}{l!}{x}^n〉\hfill \\ =m!\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|x^{n-l}〉\hfill \\ =m!\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m)〈\sum _{i=0}^{\mathrm{\infty }}{\widehat{\mathrm{NS}}}_i(a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\frac{t^i}{i!}|x^{n-l}〉\hfill \\ =m!\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m){\widehat{\mathrm{NS}}}_{n-l}(a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =m!\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(n-l,m){\widehat{\mathrm{NS}}}_l(a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

On the other hand, it is equal to

$$\begin{array}{c}〈{\partial }_t\left(\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m\right)|x^{n-1}〉\hfill \\ =〈\left({\partial }_t\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|x^{n-1}〉\hfill \\ +〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left({\partial }_t\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\right){(ln(1+t))}^m|x^{n-1}〉\hfill \\ +〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\left({\partial }_t{(ln(1+t))}^m\right)|x^{n-1}〉.\hfill \end{array}$$

(28)

The third term of (28) is equal to

$$\begin{array}{c}m〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}|{(ln(1+t))}^{m-1}{x}^{n-1}〉\hfill \\ =m〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{-1}|(m-1)!\sum _{l=m-1}^{\mathrm{\infty }}{S}_1(l,m-1)\frac{t^l}{l!}{x}^{n-1}〉\hfill \\ =m!\sum _{l=m-1}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m-1)\hfill \\ \times 〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}|x^{n-1-l}〉\hfill \\ =m!\sum _{l=m-1}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m-1)\hfill \\ \times {\widehat{\mathrm{NS}}}_{n-1-l}(-1|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =m!\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m-1)\hfill \\ \times {\widehat{\mathrm{NS}}}_l(-1|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s).\hfill \end{array}$$

Since

$$\begin{array}{c}{\partial }_t\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\hfill \\ =\frac{1}{1+t}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\frac{{\sum }_{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})}{t}\hfill \\ -\frac{1}{1+t}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\sum _{\nu =1}^r{a}_{\nu }\right),\hfill \end{array}$$

the first term of (28) is equal to

$$\begin{array}{c}\frac{1}{n}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{-1}\sum _{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})|{(ln(1+t))}^m{x}^n〉\hfill \\ -\left(\sum _{\nu =1}^r{a}_{\nu }\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{-1}|{(ln(1+t))}^m{x}^{n-1}〉\hfill \\ =\frac{1}{n}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}\hfill \\ \times \sum _{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})|m!\sum _{l=m}^{\mathrm{\infty }}{S}_1(l,m)\frac{t^l}{l!}{x}^n〉\hfill \\ -\left(\sum _{\nu =1}^r{a}_{\nu }\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{-1}|m!\sum _{l=m}^{\mathrm{\infty }}{S}_1(l,m)\frac{t^l}{l!}{x}^{n-1}〉\hfill \\ =\frac{m!}{n}\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\hfill \\ \times {(1+t)}^{-1}\sum _{\nu =1}^r(\frac{a_{\nu }t{(1+t)}^{a_{\nu }}}{{(1+t)}^{a_{\nu }}-1}-\frac{t}{ln(1+t)})|x^{n-l}〉\hfill \\ -m!\left(\sum _{\nu =1}^r{a}_{\nu }\right)\sum _{l=m}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m)\hfill \\ \times 〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}|x^{n-l-1}〉\hfill \\ =-m!\left(\sum _{\nu =1}^r{a}_{\nu }\right)\sum _{l=m}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m){\widehat{\mathrm{NS}}}_{n-l-1}(-1)\hfill \\ +\frac{m!}{n}\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m)(\left(\sum _{\nu =1}^r{a}_{\nu }\right)〈\frac{{(1+t)}^{a_{\nu }}ln(1+t)}{{(1+t)}^{a_{\nu }}-1}\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\hfill \\ \times \prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}|\frac{t}{ln(1+t)}{x}^{n-l}〉\hfill \\ -r〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-1}|\frac{t}{ln(1+t)}{x}^{n-l}〉)\hfill \\ =-m!\left(\sum _{\nu =1}^r{a}_{\nu }\right)\sum _{l=m}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m){\widehat{\mathrm{NS}}}_{n-l-1}(-1)\hfill \\ +\frac{m!}{n}\sum _{l=m}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(l,m)\sum _{k=0}^{n-l}\left(\genfrac{}{}{0ex}{}{n-l}{k}\right)c_k\hfill \\ \times (\left(\sum _{\nu =1}^r{a}_{\nu }\right){\widehat{\mathrm{NS}}}_{n-l-k}(-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )-r{\widehat{\mathrm{NS}}}_{n-l-k}(-1|a;\lambda ;\mu ))\hfill \\ =-m!\left(\sum _{\nu =1}^r{a}_{\nu }\right)\sum _{l=0}^{n-1-m}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-1)\hfill \\ +\frac{m!}{n}\sum _{k=0}^{n-m}\sum _{l=k}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{l}{k}\right)S_1(n-l,m)c_{l-k}\hfill \\ \times (\left(\sum _{\nu =1}^r{a}_{\nu }\right){\widehat{\mathrm{NS}}}_k(-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )-r{\widehat{\mathrm{NS}}}_k(-1|a;\lambda ;\mu )).\hfill \end{array}$$

The second term of (28) is equal to

$$\begin{array}{c}\sum _{\nu =1}^s{\lambda }_{\nu }{\mu }_{\nu }〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\frac{{(1+t)}^{{\lambda }_{\nu }}}{1+{(1+t)}^{{\lambda }_{\nu }}}\right)\hfill \\ \times \prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-{\lambda }_{\nu }-1}|{(ln(1+t))}^m{x}^{n-1}〉\hfill \\ =\sum _{\nu =1}^s{\lambda }_{\nu }{\mu }_{\nu }〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\frac{{(1+t)}^{{\lambda }_{\nu }}}{1+{(1+t)}^{{\lambda }_{\nu }}}\right)\hfill \\ \times \prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-{\lambda }_{\nu }-1}|m!\sum _{l=m}^{\mathrm{\infty }}{S}_1(l,m)\frac{t^l}{l!}{x}^{n-1}〉\hfill \\ =m!\sum _{\nu =1}^s{\lambda }_{\nu }{\mu }_{\nu }\sum _{l=m}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\left(\frac{{(1+t)}^{{\lambda }_{\nu }}}{1+{(1+t)}^{{\lambda }_{\nu }}}\right)\hfill \\ \times \prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(1+t)}^{-{\lambda }_{\nu }-1}|x^{n-l-1}〉\hfill \\ =m!\sum _{\nu =1}^s{\lambda }_{\nu }{\mu }_{\nu }\sum _{l=m}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(l,m){\widehat{\mathrm{NS}}}_{n-l-1}(-{\lambda }_{\nu }-1|a;\lambda ;\mu +e_{\nu })\hfill \\ =m!\sum _{\nu =1}^s{\lambda }_{\nu }{\mu }_{\nu }\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-{\lambda }_{\nu }-1|a;\lambda ;\mu +e_{\nu }).\hfill \end{array}$$

Therefore, we get, for $n-1\ge m\ge 1$,

$$\begin{array}{c}m!\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(n-l,m){\widehat{\mathrm{NS}}}_l(a;\lambda ;\mu )\hfill \\ =-m!\left(\sum _{i=1}^r{a}_i\right)\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu )\hfill \\ +\frac{m!}{n}\sum _{k=0}^{n-m}\sum _{l=k}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{l}{k}\right)S_1(n-l,m)c_{l-k}\hfill \\ \times (\sum _{i=1}^r{a}_i{\widehat{\mathrm{NS}}}_k(-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )-r{\widehat{\mathrm{NS}}}_k(-1|a;\lambda ;\mu ))\hfill \\ +m!\sum _{j=1}^s{\lambda }_j{\mu }_j\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-{\lambda }_j-1|a;\lambda ;\mu +e_j)\hfill \\ +m!\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m-1){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu ).\hfill \end{array}$$

Dividing both sides by m!, we obtain, for $n-1\ge m\ge 1$,

$$\begin{array}{c}\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)S_1(n-l,m){\widehat{\mathrm{NS}}}_l(a;\lambda ;\mu )\hfill \\ =-\left(\sum _{i=1}^r{a}_i\right)\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu )\hfill \\ +\frac{1}{n}\sum _{k=0}^{n-m}\sum _{l=k}^{n-m}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{l}{k}\right)S_1(n-l,m)c_{l-k}\hfill \\ \times (\sum _{i=1}^r{a}_i{\widehat{\mathrm{NS}}}_k(-1|a_1,\dots ,a_{i-1},a_{i+1},\dots ,a_r;\lambda ;\mu )-r{\widehat{\mathrm{NS}}}_k(-1|a;\lambda ;\mu ))\hfill \\ +\sum _{j=1}^s{\lambda }_j{\mu }_j\sum _{l=0}^{n-m-1}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m){\widehat{\mathrm{NS}}}_l(-{\lambda }_j-1|a;\lambda ;\mu +e_j)\hfill \\ +\sum _{l=0}^{n-m}\left(\genfrac{}{}{0ex}{}{n-1}{l}\right)S_1(n-l-1,m-1){\widehat{\mathrm{NS}}}_l(-1|a;\lambda ;\mu ).\hfill \end{array}$$

Thus, we get (27). □

3.8 A relation with the falling factorials

Theorem 8

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\sum _{m=0}^n\left(\genfrac{}{}{0ex}{}{n}{m}\right){\widehat{\mathrm{NS}}}_{n-m}(a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s){(x)}_m.\hfill \end{array}$$

(29)

Proof For (12) and (20), assume that

$${\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)=\sum _{m=0}^n{C}_{n,m}{(x)}_m.$$

By (11), we have

$$\begin{array}{rl}{C}_{n,m}& =\frac{1}{m!}〈\frac{1}{{\prod }_{i=1}^r(\frac{ln(1+t)e^{a_{i}ln(1+t)}}{e^{a_{i}ln(1+t)}-1}){\prod }_{j=1}^s{(\frac{1+e^{{\lambda }_{j}ln(1+t)}}{e^{{\lambda }_{j}ln(1+t)}})}^{{\mu }_j}}{t}^m|x^n〉\\ =\frac{1}{m!}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|t^m{x}^n〉\\ =\left(\genfrac{}{}{0ex}{}{n}{m}\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}|x^{n-m}〉\\ =\left(\genfrac{}{}{0ex}{}{n}{m}\right){\widehat{\mathrm{NS}}}_{n-m}.\end{array}$$

Thus, we get the identity (29). □

3.9 A relation with higher-order Frobenius-Euler polynomials

For $\alpha \in \mathbb{C}$ with $\alpha \ne 1$, the Frobenius-Euler polynomials of order r, $H_n^{(r)}(x|\alpha )$ are defined by the generating function

$${\left(\frac{1-\alpha }{e^t-\alpha }\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{H}_n^{(r)}(x|\alpha )\frac{t^n}{n!}$$

(see e.g. [5]).

Theorem 9

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\sum _{m=0}^n(\sum _{j=0}^{n-m}\sum _{l=0}^{n-m-j}\left(\genfrac{}{}{0ex}{}{\sigma }{j}\right)\left(\genfrac{}{}{0ex}{}{n-j}{l}\right){(n)}_j{(1-\alpha )}^{-j}{S}_1(n-j-l,m){\widehat{\mathrm{NS}}}_l)H_m^{(\sigma )}(x|\alpha ).\hfill \end{array}$$

(30)

Proof For (12) and

$$H_n^{(\sigma )}(x|\alpha )\sim ({\left(\frac{e^t-\alpha }{1-\alpha }\right)}^{\sigma },t),$$

(31)

assume that ${\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)={\sum }_{m=0}^n{C}_{n,m}{H}_m^{(\sigma )}(x|\alpha )$. By (11), similarly to the proof of (27), we have

$$\begin{array}{rcl}{C}_{n,m}& =& \frac{1}{m!}〈\frac{{(\frac{e^{ln(1+t)}-\alpha }{1-\alpha })}^{\sigma }}{{\prod }_{i=1}^r(\frac{ln(1+t)e^{a_{i}ln(1+t)}}{e^{a_{i}ln(1+t)}-1}){\prod }_{j=1}^s{(\frac{1+e^{{\lambda }_{j}ln(1+t)}}{e^{{\lambda }_{j}ln(1+t)}})}^{{\mu }_j}}{(ln(1+t))}^m|x^n〉\\ =& \frac{1}{m!{(1-\alpha )}^{\sigma }}\\ \times 〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m{(1-\alpha +t)}^{\sigma }|x^n〉\\ =& \frac{1}{m!{(1-\alpha )}^{\sigma }}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}\\ \times {(ln(1+t))}^m|\sum _{\nu =0}^{min\{\sigma ,n\}}\left(\genfrac{}{}{0ex}{}{\sigma }{\nu }\right){(1-\alpha )}^{\sigma -\nu }{t}^{\nu }{x}^n〉\\ =& \frac{1}{m!{(1-\alpha )}^{\sigma }}\sum _{\nu =0}^{n-m}\left(\genfrac{}{}{0ex}{}{\sigma }{\nu }\right){(1-\alpha )}^{\sigma -\nu }{(n)}_{\nu }\\ \times 〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|x^{n-\nu }〉\\ =& \frac{1}{m!{(1-\alpha )}^{\sigma }}\sum _{\nu =0}^{n-m}\left(\genfrac{}{}{0ex}{}{\sigma }{\nu }\right){(1-\alpha )}^{\sigma -\nu }{(n)}_{\nu }\sum _{l=0}^{n-m-\nu }m!\left(\genfrac{}{}{0ex}{}{n-\nu }{l}\right)S_1(n-\nu -l,m){\widehat{\mathrm{NS}}}_l\\ =& \sum _{\nu =0}^{n-m}\sum _{l=0}^{n-m-\nu }\left(\genfrac{}{}{0ex}{}{\sigma }{\nu }\right)\left(\genfrac{}{}{0ex}{}{n-\nu }{l}\right){(n)}_{\nu }{(1-\alpha )}^{-\nu }{S}_1(n-\nu -l,m){\widehat{\mathrm{NS}}}_l.\end{array}$$

Thus, we get the identity (30). □

3.10 A relation with higher-order Bernoulli polynomials

Bernoulli polynomials ${\mathfrak{B}}_n^{(r)}(x)$ of order r are defined by

$${\left(\frac{t}{e^t-1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{B}}_n^{(r)}(x)}{n!}{t}^n$$

(see e.g. [[1], Section 2.2]). In addition, the Cauchy numbers of the first kind ${\mathfrak{C}}_n^{(r)}$ of order r are defined by

$${\left(\frac{t}{ln(1+t)}\right)}^r=\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathfrak{C}}_n^{(r)}}{n!}{t}^n$$

(see e.g. [[6], equation (2.1)], [[7], equation (6)]).

Theorem 10

$$\begin{array}{c}{\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)\hfill \\ =\sum _{m=0}^n(\sum _{i=0}^{n-m}\sum _{l=0}^{n-m-i}\left(\genfrac{}{}{0ex}{}{n}{i}\right)\left(\genfrac{}{}{0ex}{}{n-i}{l}\right){\mathfrak{C}}_i^{(\sigma )}{S}_1(n-i-l,m){\widehat{\mathrm{NS}}}_l){\mathfrak{B}}_m^{(\sigma )}(x).\hfill \end{array}$$

(32)

Proof For (12) and

$${\mathfrak{B}}_n^{(\sigma )}(x)\sim ({\left(\frac{e^t-1}{t}\right)}^{\sigma },t),$$

(33)

assume that ${\widehat{\mathrm{NS}}}_n(x|a_1,\dots ,a_r;{\lambda }_1,\dots ,{\lambda }_s;{\mu }_1,\dots ,{\mu }_s)={\sum }_{m=0}^n{C}_{n,m}{\mathfrak{B}}_m^{(s)}(x)$. By (11), similarly to the proof of (27), we have

$$\begin{array}{rl}{C}_{n,m}& =\frac{1}{m!}〈\frac{{(\frac{e^{ln(1+t)}-1}{ln(1+t)})}^{\sigma }}{{\prod }_{i=1}^r(\frac{ln(1+t)e^{a_{i}ln(1+t)}}{e^{a_{i}ln(1+t)}-1}){\prod }_{j=1}^s{(\frac{1+e^{{\lambda }_{j}ln(1+t)}}{e^{{\lambda }_{j}ln(1+t)}})}^{{\mu }_j}}{(ln(1+t))}^m|x^n〉\\ =\frac{1}{m!}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|{\left(\frac{t}{ln(1+t)}\right)}^{\sigma }{x}^n〉\\ =\frac{1}{m!}〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|\sum _{i=0}^{\mathrm{\infty }}{\mathfrak{C}}_i^{(\sigma )}\frac{t^i}{i!}{x}^n〉\\ =\frac{1}{m!}\sum _{i=0}^{n-m}{\mathfrak{C}}_i^{(\sigma )}\left(\genfrac{}{}{0ex}{}{n}{i}\right)〈\prod _{i=1}^r\left(\frac{{(1+t)}^{a_i}-1}{{(1+t)}^{a_i}ln(1+t)}\right)\prod _{j=1}^s{\left(\frac{{(1+t)}^{{\lambda }_j}}{1+{(1+t)}^{{\lambda }_j}}\right)}^{{\mu }_j}{(ln(1+t))}^m|x^{n-i}〉\\ =\frac{1}{m!}\sum _{i=0}^{n-m}{\mathfrak{C}}_i^{(\sigma )}\left(\genfrac{}{}{0ex}{}{n}{i}\right)\sum _{l=0}^{n-m-i}m!\left(\genfrac{}{}{0ex}{}{n-i}{l}\right)S_1(n-i-l,m){\widehat{\mathrm{NS}}}_l\\ =\sum _{i=0}^{n-m}\sum _{l=0}^{n-m-i}\left(\genfrac{}{}{0ex}{}{n}{i}\right)\left(\genfrac{}{}{0ex}{}{n-i}{l}\right){\mathfrak{C}}_i^{(\sigma )}{S}_1(n-i-l,m){\widehat{\mathrm{NS}}}_l.\end{array}$$

Thus, we get the identity (32). □