Some properties on degenerate Fubini polynomials

ABSTRACT

The nth Fubini number enumerates the number of ordered partitions of a set with n elements and is the number of possible ways to write the Fubini formula for a summation of integration of order n. Further, Fubini polynomials are natural extensions of the Fubini numbers. There are many variants of Fubini numbers and polynomials. Recently, the degenerate Fubini polynomials were introduced by Kim-Kim-Jang as a degenerate version of the Fubini polynomials. The aim of this article is by using generating functions and certain differential operators to further study some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials.

KEYWORDS:

• Degenerate Fubini polynomials
• higher-order degenerate Fubini polynomials
• degenerate Stirling numbers of the first kind
• degenerate Stirling numbers of the second kind

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 11B73
• 11B83
• 47B39

1. Introduction

Explorations for degenerate versions of some special numbers and polynomials, which were initiated by Carlitz in the pioneering work of [1], have drawn the attention of many mathematicians in recent years (see [2–7] and the references therein). Their interests were not only in combinatorial and arithmetical properties but also in applications to differential equations, identities of symmetry and probability theory. These degenerate versions include the degenerate Stirling numbers of the first and second kinds, degenerate Bernoulli numbers of the second kind and degenerate Bell numbers and polynomials. We remark here that the degenerate Stirling numbers of the first kind and of the second kind appear very frequently when we study degenerate versions of some special numbers and polynomials (see (3(3) $(x{)}_n=\sum _{k=0}^n{S}_{1,\lambda }(n,k\left)\right(x{)}_{k,\lambda },(n\ge 0),$(3) ), (4(4) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k\left)\right(x{)}_k,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[4\left]\right).$(4) )).

Fubini number $F_n=\sum _{k=0}^{n}k!S_2(n,k)$ enumerates the ordered partitions of the set $\left[n\right]=\{1,2,\dots ,n\}$. It is the number of possible ways to write the Fubini formula for a summation of integration of order n. A natural extension of $F_n$ is the Fubini polynomial $F_n\left(x\right)=\sum _{k=0}^{n}k!S_2(n,k)x^k$. There are many variants of Fubini numbers and polynomials. One of them is the degenerate Fubini polynomials which were introduced in [6] as a degenerate version of the Fubini polynomials (see also [7–14]). The motivation or ?>impetus for the present research is the regained recent interests in degenerate versions of some special numbers and polynomials by many mathematicians.

The aim of this article is by using generating functions and certain differential operators to further study some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials (see (12(12) $\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[6\left]\right).$(12) ), (15(15) ${\left(\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}\right)}^r=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(15) )). In more detail, the series $\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+r}{r})(k{)}_{n,\lambda }{x}^k$ is expressed in terms of the higher-order degenerate Fubini polynomials and certain differential operator in connection with the generalized falling factorial. Explicit expressions and recurrence relations are derived for the higher-order degenerate Fubini polynomials. Obtained is the convolution expression of the degenerate Fubini polynomial and the higher-order degenerate Fubini polynomial. Deduced are an identity involving the higher-order degenerate Fubini polynomials and an integral representation for those polynomials in terms of the degenerate Bell polynomials.

The outline of this article is as follows. In Section 1, we recall degenerate exponentials, degenerate logarithms, degenerate Stirling numbers of the first kind and degenerate Stirling numbers of the second kind which are, respectively, degenerate versions of exponentials, logarithms, Stirling numbers of the first kind and Stirling numbers of the second kind. Also, we remind the reader of degenerate Bernoulli numbers, degenerate Bell polynomials, degenerate Fubini polynomials and higher-order degenerate Fubini polynomials which are, respectively, degenerate versions of Bernoulli numbers, Bell polynomials, Fubini polynomials and higher-order Fubini polynomials. In Section 2, we state our main results. In Theorem 2.1, we express the series $\sum _{k=0}^{\mathrm{\infty }}(k{)}_{n,\lambda }{x}^k$ in terms of the degenerate Fubini polynomial and of some differential operator made out of the generalized falling factorial. In Theorem 2.2, the result in Theorem 2.1 is generalized to the case of the series $\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+r}{r})(k{)}_{n,\lambda }{x}^k$, which is expressed in terms of the higher-order degenerate Fubini polynomial and of the same differential operator. In Corollary 2.3, explicit expressions for the higher-order degenerate Fubini polynomials are derived. In Theorem 2.5, the higher-order degenerate Fubini polynomials are represented as an integral over $(0,\mathrm{\infty })$ of suitable functions involving the degenerate Bell polynomials. In Theorem 2.6, we derive an expression for the convolution of the degenerate Fubini polynomial and the higher-order degenerate Fubini polynomial. In Theorem 2.7, we get an identity involving the higher-order degenerate Fubini polynomials. In Theorems 2.8, 2.10 and 2.11, we obtain recurrence relations for the higher-order degenerate Fubini polynomials. In Theorem 2.9, we derive a recurrence relation for the degenerate Bell polynomials. In the rest of this section, we state what are needed throughout this paper.

It is well known that the degenerate exponentials are defined by (1) $e_{\lambda }^x\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{(x{)}_{k,\lambda }}{k!}{t}^n\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[4,14,15\left]\right),$(1) where, for any $\lambda \in \mathbb{R}$, the generalized falling factorials are given by (2) $(x{)}_{0,\lambda }=1,(x{)}_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1\left)\lambda \right),(n\ge 1).$(2) For x = 1, we use the simpler notation $e_{\lambda }\left(t\right)=e_{\lambda }^1\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{t}^n\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[2,3,5,6,10-13\left]\right).$In [4], the degenerate Stirling numbers of the first kind are defined by (3) $(x{)}_n=\sum _{k=0}^n{S}_{1,\lambda }(n,k\left)\right(x{)}_{k,\lambda },(n\ge 0),$(3) where $(x{)}_0=1,(x{)}_n=x(x-1)\cdots (x-n+1),(n\ge 1)$.

As the inversion formula of (3(3) $(x{)}_n=\sum _{k=0}^n{S}_{1,\lambda }(n,k\left)\right(x{)}_{k,\lambda },(n\ge 0),$(3) ), the degenerate Stirling numbers of the second kind are defined by (4) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k\left)\right(x{)}_k,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[4\left]\right).$(4) Let ${\log }_{\lambda }t$ be the compositional inverse of $e_{\lambda }\left(t\right)$. Then, ${\log }_{\lambda }\left(t\right)$ is given by (5) ${\log }_{\lambda }(1+t)=\sum _{n=1}^{\mathrm{\infty }}{\lambda }^{n-1}\left(1{)}_{n,1/\lambda }\frac{t^n}{n!}\right(\mathrm{s}\mathrm{e}\mathrm{e}\left[4\right]).$(5) Note that $\underset{\lambda \to 0}{lim}{\log }_{\lambda }(1+t)=\log (1+t)$.

In [1], Carlitz introduced the degenerate Bernoulli numbers given by (6) $\frac{t}{e_{\lambda }\left(t\right)-1}=\sum _{n=0}^{\mathrm{\infty }}{\text{β}}_{n,\lambda }\frac{t^n}{n!}.$(6) Note that $\underset{\lambda \to 0}{lim}{\text{β}}_{n,\lambda }=B_n$ are the ordinary Bernoulli numbers given by $\frac{t}{{\mathrm{e}}^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[1-24\left]\right).$For $s\in \mathbb{C}$, the gamma function is defined by $\mathrm{\Gamma }\left(s\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{s-1}\mathrm{d}t,\mathrm{R}\mathrm{e}\left(s\right)>0\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[21,22\left]\right).$Note that $\mathrm{\Gamma }(s+1)=s\mathrm{\Gamma }\left(s\right),\mathrm{\Gamma }(n+1)=n!,(n\in \mathbb{N})$.

Recently, the degenerate Bell polynomials (also called degenerate exponential polynomials) are defined by (7) ${\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[5\left]\right).$(7) Note that $\underset{\lambda \to 0}{lim}{\varphi }_{n,\lambda }\left(x\right)={\varphi }_n\left(x\right),(n\ge 0),$ where ${\varphi }_n\left(x\right)$ are the ordinary Bell polynomials (also called exponential polynomials) given by (8) ${\varphi }_n\left(x\right)=\sum _{k=0}^n{S}_2(n,k)x^k,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[21\left]\right).$(8) Here, $S_2(n,k)$ are the Stirling numbers of the second kind given by $x^n=\sum _{k=0}^n{S}_2(n,k)(x{)}_k,(n\ge 0\left)\right(\mathrm{s}\mathrm{e}\mathrm{e}[1,8,9,11,16-22]).$It is well known that the Fubini polynomials are defined by (9) $\frac{1}{1-y({\mathrm{e}}^t-1)}=\sum _{n=0}^{\mathrm{\infty }}{F}_n\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[1,6,8-12,18-22\left]\right).$(9) Thus, by (9(9) $\frac{1}{1-y({\mathrm{e}}^t-1)}=\sum _{n=0}^{\mathrm{\infty }}{F}_n\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[1,6,8-12,18-22\left]\right).$(9) ), we get (10) $F_n\left(y\right)=\sum _{k=0}^n{S}_2(n,k)k!y^k,$(10) and (11) ${\left(y\frac{\mathrm{d}}{\mathrm{d}y}\right)}^m\left(\frac{1}{1-y}\right)=\sum _{k=0}^{\mathrm{\infty }}{k}^m{y}^k=\frac{1}{1-y}{F}_m\left(\frac{y}{1-y}\right)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[8,9,11\left]\right).$(11) Recently, the degenerate Fubini polynomials are defined by (12) $\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[6\left]\right).$(12) Thus, we note that (13) $F_{n,\lambda }\left(y\right)=\sum _{k=0}^n{y}^{k}k!S_{2,\lambda }(n,k),$(13) and (14) $F_{n,\lambda }\left(y\right)=(-1{)}^n\frac{y}{y+1}{F}_{n,-\lambda }(-1-y),(n>0\left)\right(\mathrm{s}\mathrm{e}\mathrm{e}\left[6\right]).$(14) For any $r\in \mathbb{N}$, the higher-order degenerate Fubini polynomials are defined by (15) ${\left(\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}\right)}^r=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(15) Thus, by (15(15) ${\left(\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}\right)}^r=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(15) ), we get (16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) We recommend the reader to refer to [3, 6, 10] for further details on degenerate Fubini polynomials.

2. Degenerate Fubini polynomials

In view of (11(11) ${\left(y\frac{\mathrm{d}}{\mathrm{d}y}\right)}^m\left(\frac{1}{1-y}\right)=\sum _{k=0}^{\mathrm{\infty }}{k}^m{y}^k=\frac{1}{1-y}{F}_m\left(\frac{y}{1-y}\right)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[8,9,11\left]\right).$(11) ), we consider (17) $\begin{array}{rl}\frac{1}{1-x}{F}_{n,\lambda }\left(\frac{x}{1-x}\right)& =\frac{1}{1-x}\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!{\left(\frac{x}{1-x}\right)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!x^k{\left(\frac{1}{1-x}\right)}^{k+1}\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!x^k\sum _{l=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+l}{l})x^l\\ & =\sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!(\genfrac{}{}{0ex}{}{k+l}{k})x^{k+l}\\ & =\sum _{l=0}^{\mathrm{\infty }}{x}^l\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!(\genfrac{}{}{0ex}{}{l}{k})=\sum _{l=0}^{\mathrm{\infty }}{x}^l(l{)}_{n,\lambda }.\end{array}$(17) By (2(2) $(x{)}_{0,\lambda }=1,(x{)}_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1\left)\lambda \right),(n\ge 1).$(2) ), we easily get (18) ${\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }\frac{1}{1-x}=\sum _{l=0}^{\mathrm{\infty }}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{x}^l=\sum _{l=0}^{\mathrm{\infty }}(l{)}_{n,\lambda }{x}^l.$(18) Therefore, by (17(17) $\begin{array}{rl}\frac{1}{1-x}{F}_{n,\lambda }\left(\frac{x}{1-x}\right)& =\frac{1}{1-x}\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!{\left(\frac{x}{1-x}\right)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!x^k{\left(\frac{1}{1-x}\right)}^{k+1}\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!x^k\sum _{l=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+l}{l})x^l\\ & =\sum _{l=0}^{\mathrm{\infty }}\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!(\genfrac{}{}{0ex}{}{k+l}{k})x^{k+l}\\ & =\sum _{l=0}^{\mathrm{\infty }}{x}^l\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!(\genfrac{}{}{0ex}{}{l}{k})=\sum _{l=0}^{\mathrm{\infty }}{x}^l(l{)}_{n,\lambda }.\end{array}$(17) ) and (18(18) ${\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }\frac{1}{1-x}=\sum _{l=0}^{\mathrm{\infty }}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{x}^l=\sum _{l=0}^{\mathrm{\infty }}(l{)}_{n,\lambda }{x}^l.$(18) ), we obtain the following theorem.

Theorem 2.1

For $n\ge 0$, we have $\frac{1}{1-x}{F}_{n,\lambda }\left(\frac{x}{1-x}\right)={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }\frac{1}{1-x}=\sum _{k=0}^{\mathrm{\infty }}(k{)}_{n,\lambda }{x}^k.$

Let $f\left(x\right)=\sum _{k=0}^n{a}_k{x}^k\in \mathbb{R}\left[x\right]$. Then, we have (19) $\begin{array}{rl}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{m,\lambda }f\left(x\right)& =\sum _{j=0}^m{S}_{2,\lambda }(m,j){\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{j}f\left(x\right)\\ & =\sum _{j=0}^m{S}_{2,\lambda }(m,j)x^j{f}^{\left(j\right)}\left(x\right),\end{array}$(19) where $f^{\left(j\right)}\left(x\right)=\left(\frac{\mathrm{d}}{\mathrm{d}x}{)}^{j}f\right(x)$.

For $n\ge 0$ and $r\in \mathbb{N}$, by (16(16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) ), we get (20) $\begin{array}{rl}{\left(\frac{1}{1-x}\right)}^{r+1}{F}_{n,\lambda }^{(r+1)}\left(\frac{x}{1-x}\right)& ={\left(\frac{1}{1-x}\right)}^{r+1}\sum _{l=0}^n{S}_{2,\lambda }(n,l)(\genfrac{}{}{0ex}{}{l+r}{l})l!{\left(\frac{x}{1-x}\right)}^l\\ & =\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)l!(\genfrac{}{}{0ex}{}{l+r}{l})(\genfrac{}{}{0ex}{}{k+l+r}{k})x^{k+l}\\ & =\sum _{k=l}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)l!(\genfrac{}{}{0ex}{}{l+r}{l})(\genfrac{}{}{0ex}{}{k+r}{k-l})x^k\\ & =\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)(k{)}_l(\genfrac{}{}{0ex}{}{k+r}{k})x^k\\ & =\sum _{k=0}^{\mathrm{\infty }}(k{)}_{n,\lambda }(\genfrac{}{}{0ex}{}{k+r}{k})x^k={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{\left(\frac{1}{1-x}\right)}^{r+1}.\end{array}$(20) Therefore, by (20(20) $\begin{array}{rl}{\left(\frac{1}{1-x}\right)}^{r+1}{F}_{n,\lambda }^{(r+1)}\left(\frac{x}{1-x}\right)& ={\left(\frac{1}{1-x}\right)}^{r+1}\sum _{l=0}^n{S}_{2,\lambda }(n,l)(\genfrac{}{}{0ex}{}{l+r}{l})l!{\left(\frac{x}{1-x}\right)}^l\\ & =\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)l!(\genfrac{}{}{0ex}{}{l+r}{l})(\genfrac{}{}{0ex}{}{k+l+r}{k})x^{k+l}\\ & =\sum _{k=l}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)l!(\genfrac{}{}{0ex}{}{l+r}{l})(\genfrac{}{}{0ex}{}{k+r}{k-l})x^k\\ & =\sum _{k=0}^{\mathrm{\infty }}\sum _{l=0}^n{S}_{2,\lambda }(n,l)(k{)}_l(\genfrac{}{}{0ex}{}{k+r}{k})x^k\\ & =\sum _{k=0}^{\mathrm{\infty }}(k{)}_{n,\lambda }(\genfrac{}{}{0ex}{}{k+r}{k})x^k={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{\left(\frac{1}{1-x}\right)}^{r+1}.\end{array}$(20) ), we obtain the following theorem.

Theorem 2.2

For $n,r\ge 0$, we have ${\left(\frac{1}{1-x}\right)}^{r+1}{F}_{n,\lambda }^{(r+1)}\left(\frac{x}{1-x}\right)={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{\left(\frac{1}{1-x}\right)}^{r+1}=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+r}{r})(k{)}_{n,\lambda }{x}^k.$

By (16(16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) ), we have (21) $\begin{array}{rl}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)& =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(\genfrac{}{}{0ex}{}{k+r-1}{k})k!(-1{)}^k(-x{)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(-r{)}_k(-x{)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(-x{)}^k\sum _{j=0}^k{S}_{1,\lambda }(k,j\left)\right(-r{)}_{j,\lambda }.\end{array}$(21) From (21(21) $\begin{array}{rl}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)& =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(\genfrac{}{}{0ex}{}{k+r-1}{k})k!(-1{)}^k(-x{)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(-r{)}_k(-x{)}^k\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)(-x{)}^k\sum _{j=0}^k{S}_{1,\lambda }(k,j\left)\right(-r{)}_{j,\lambda }.\end{array}$(21) ), we obtain the following corollary.

Corollary 2.3

For $r\in \mathbb{N}$ and $n\ge 0$, we have $F_{n,\lambda }^{\left(r\right)}(-1)=(-r{)}_{n,\lambda },F_{n,\lambda }^{\left(r\right)}(x)=\sum _{k=0}^n\sum _{j=0}^k{S}_{2,\lambda }(n,k\left)S_{1,\lambda }\right(k,j\left)\right(-x{)}^k(-r{)}_{j,\lambda }.$

By (16(16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) ), we get $\begin{array}{rl}\sum _{r=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)y^r& =\sum _{r=0}^{\mathrm{\infty }}\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k(\genfrac{}{}{0ex}{}{k+r-1}{k})k!y^r\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^{k}k!\sum _{r=1}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+r-1}{k})y^r\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^{k}k!\sum _{r=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+r}{k})y^{r+1}\\ & =y\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^{k}k!{\left(\frac{1}{1-y}\right)}^{k+1}=\frac{y}{1-y}\sum _{k=0}^n{S}_{2,\lambda }(n,k)k!{\left(\frac{x}{1-y}\right)}^k\\ & =\frac{y}{1-y}{F}_{n,\lambda }\left(\frac{x}{1-y}\right).\end{array}$From (7(7) ${\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[5\left]\right).$(7) ), we note that (22) $\begin{array}{rl}& {\int }_0^x{\mathrm{e}}^{y\left(e_{\lambda }\right(t)-1)}\mathrm{d}y=\frac{1}{e_{\lambda }\left(t\right)-1}({\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}-1)\\ & =\frac{1}{t}\frac{t}{e_{\lambda }\left(t\right)-1}({\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}-1)=\frac{1}{t}\sum _{l=0}^{\mathrm{\infty }}{\text{β}}_{l,\lambda }\frac{t^l}{l!}\sum _{k=1}^{\mathrm{\infty }}{\varphi }_{k,\lambda }\left(x\right)\frac{t^k}{k!}\\ & =\frac{1}{t}\sum _{n=1}^{\mathrm{\infty }}\sum _{k=1}^n(\genfrac{}{}{0ex}{}{n}{k}){\text{β}}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(x\right)\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left(\frac{1}{n+1}\sum _{k=1}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{k}){\text{β}}_{n+1-k,\lambda }{\varphi }_{k,\lambda }\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(22) By (22(22) $\begin{array}{rl}& {\int }_0^x{\mathrm{e}}^{y\left(e_{\lambda }\right(t)-1)}\mathrm{d}y=\frac{1}{e_{\lambda }\left(t\right)-1}({\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}-1)\\ & =\frac{1}{t}\frac{t}{e_{\lambda }\left(t\right)-1}({\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}-1)=\frac{1}{t}\sum _{l=0}^{\mathrm{\infty }}{\text{β}}_{l,\lambda }\frac{t^l}{l!}\sum _{k=1}^{\mathrm{\infty }}{\varphi }_{k,\lambda }\left(x\right)\frac{t^k}{k!}\\ & =\frac{1}{t}\sum _{n=1}^{\mathrm{\infty }}\sum _{k=1}^n(\genfrac{}{}{0ex}{}{n}{k}){\text{β}}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(x\right)\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left(\frac{1}{n+1}\sum _{k=1}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{k}){\text{β}}_{n+1-k,\lambda }{\varphi }_{k,\lambda }\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(22) ), we get the next lemma.

Lemma 2.4

For $n\ge 0$, we have ${\int }_0^x{\varphi }_{n,\lambda }\left(y\right)\mathrm{d}y=\frac{1}{n+1}\sum _{k=1}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{k}){\text{β}}_{n+1-k,\lambda }{\varphi }_{k,\lambda }\left(x\right).$

From (7(7) ${\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[5\left]\right).$(7) ), we see that (23) ${\varphi }_{n,\lambda }\left(x\right)=\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k,(n\ge 0).$(23) By making use of (16(16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) ) and (23(23) ${\varphi }_{n,\lambda }\left(x\right)=\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k,(n\ge 0).$(23) ), we have (24) $\begin{array}{rl}{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\varphi }_{n,\lambda }\left(xy\right){\mathrm{e}}^{-y}\mathrm{d}y& =\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k{\int }_0^{\mathrm{\infty }}{y}^{r+k-1}{\mathrm{e}}^{-y}\mathrm{d}y\\ & =\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k(r+k-1)!\\ & =\mathrm{\Gamma }\left(r\right)\sum _{k=0}^n{S}_{2,\lambda }(n,k)(\genfrac{}{}{0ex}{}{r+k-1}{k})k!x^k=\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$(24)

Theorem 2.5

For $n\ge 0$ and r>0, we have $F_{n,\lambda }^{\left(r\right)}\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\varphi }_{n,\lambda }\left(xy\right){\mathrm{e}}^{-y}\mathrm{d}y.$

Note that $\begin{array}{rl}{\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r& =\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}=\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{y}^{r-1}\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(xy\right)\frac{t^n}{n!}{\mathrm{e}}^{-y}\mathrm{d}y\\ & =\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\mathrm{e}}^{xy\left(e_{\lambda }\right(t)-1)}{\mathrm{e}}^{-y}\mathrm{d}y\\ & =\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\mathrm{e}}^{y\left(x{e}_{\lambda }\right(t)-x-1)}\mathrm{d}y.\end{array}$Differentiating both sides of (15(15) ${\left(\frac{1}{1-y\left(e_{\lambda }\right(t)-1)}\right)}^r=\sum _{n=0}^{\mathrm{\infty }}{F}_{n,\lambda }^{\left(r\right)}\left(y\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(15) ) with respect to t, on the one hand, we have (25) $\frac{rx{e}_{\lambda }^{1-\lambda }\left(t\right)}{{(1-x(e_{\lambda }\left(t\right)-1\left)\right)}^{r+1}}=\sum _{n=0}^{\mathrm{\infty }}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.$(25) On the other hand, we also have (26) $\begin{array}{rl}\frac{rx{e}_{\lambda }^{1-\lambda }\left(t\right)}{{(1-x(e_{\lambda }\left(t\right)-1\left)\right)}^{r+1}}& =\frac{r}{1+\lambda t}(\frac{1+x}{1-x\left(e_{\lambda }\right(t)-1)}-1){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\frac{r(1+x)}{1+\lambda t}\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & -\frac{r}{1+\lambda t}{\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r.\end{array}$(26) By (25(25) $\frac{rx{e}_{\lambda }^{1-\lambda }\left(t\right)}{{(1-x(e_{\lambda }\left(t\right)-1\left)\right)}^{r+1}}=\sum _{n=0}^{\mathrm{\infty }}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}.$(25) ) and (26(26) $\begin{array}{rl}\frac{rx{e}_{\lambda }^{1-\lambda }\left(t\right)}{{(1-x(e_{\lambda }\left(t\right)-1\left)\right)}^{r+1}}& =\frac{r}{1+\lambda t}(\frac{1+x}{1-x\left(e_{\lambda }\right(t)-1)}-1){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\frac{r(1+x)}{1+\lambda t}\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & -\frac{r}{1+\lambda t}{\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r.\end{array}$(26) ), we get (27) $\begin{array}{rl}& r(1+x)\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\sum _{n=0}^{\mathrm{\infty }}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}(1+\lambda t)+r{\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(F_{n+1,\lambda }^{\left(r\right)}\right(x)+(n\lambda +r\left)F_{n,\lambda }^{\left(r\right)}\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(27) It is immediate to show that (28) $\begin{array}{rl}\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r& =\sum _{l=0}^{\mathrm{\infty }}{F}_{l,\lambda }\left(x\right)\frac{t^l}{l!}\cdot \sum _{k=0}^{\mathrm{\infty }}{F}_{k,\lambda }^{\left(r\right)}\left(x\right)\frac{t^k}{k!}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})F_{k,\lambda }^{\left(r\right)}\right(x\left)F_{n-k,\lambda }\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(28) Therefore, by (27(27) $\begin{array}{rl}& r(1+x)\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\sum _{n=0}^{\mathrm{\infty }}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)\frac{t^n}{n!}(1+\lambda t)+r{\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(F_{n+1,\lambda }^{\left(r\right)}\right(x)+(n\lambda +r\left)F_{n,\lambda }^{\left(r\right)}\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(27) ) and (28(28) $\begin{array}{rl}\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right){\left(\frac{1}{1-x\left(e_{\lambda }\right(t)-1)}\right)}^r& =\sum _{l=0}^{\mathrm{\infty }}{F}_{l,\lambda }\left(x\right)\frac{t^l}{l!}\cdot \sum _{k=0}^{\mathrm{\infty }}{F}_{k,\lambda }^{\left(r\right)}\left(x\right)\frac{t^k}{k!}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})F_{k,\lambda }^{\left(r\right)}\right(x\left)F_{n-k,\lambda }\right(x\left)\right)\frac{t^n}{n!}.\end{array}$(28) ), we obtain the following theorem.

Theorem 2.6

Convolution

For $n\ge 0$ and r>0, we have $\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})F_{k,\lambda }^{\left(r\right)}\left(x\right)F_{n-k,\lambda }\left(x\right)=\frac{1}{r(1+x)}\left(F_{n+1,\lambda }^{\left(r\right)}\right(x)+(n\lambda +r\left)F_{n,\lambda }^{\left(r\right)}\right(x\left)\right).$

From (7(7) ${\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[5\left]\right).$(7) ), we note that (29) $\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}={\mathrm{e}}^{-x}\sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}{e}_{\lambda }^k\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{\mathrm{e}}^{-x}\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k\frac{t^n}{n!}.$(29) Comparing the coefficients on both sides of (32(32) $\begin{array}{rl}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n+m,\lambda }{\mathrm{e}}^x& ={\varphi }_{n+m,\lambda }\left(x\right){\mathrm{e}}^x=x^{(n+m)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n+m}{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{x}^{m\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^m{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\left({\varphi }_{m,\lambda }\right(x\left){\mathrm{e}}^x\right)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}{\varphi }_{m,\lambda }\left(x\right)x^{k\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^k{\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}\sum _{j=0}^m{S}_{2,\lambda }(m,j)x^j{\varphi }_{k,\lambda }\left(x\right){\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\sum _{j=0}^m{S}_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x){\mathrm{e}}^x.\end{array}$(32) ), we have (30) ${\varphi }_{n,\lambda }\left(x\right)={\mathrm{e}}^{-x}\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k,(n\ge 0).$(30) By (30(30) ${\varphi }_{n,\lambda }\left(x\right)={\mathrm{e}}^{-x}\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k,(n\ge 0).$(30) ), we get (31) $x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{\mathrm{e}}^x=\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k={\varphi }_{n,\lambda }\left(x\right){\mathrm{e}}^x={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{\mathrm{e}}^x.$(31) In view of (31(31) $x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{\mathrm{e}}^x=\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k={\varphi }_{n,\lambda }\left(x\right){\mathrm{e}}^x={\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n,\lambda }{\mathrm{e}}^x.$(31) ), we observe that (32) $\begin{array}{rl}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n+m,\lambda }{\mathrm{e}}^x& ={\varphi }_{n+m,\lambda }\left(x\right){\mathrm{e}}^x=x^{(n+m)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n+m}{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{x}^{m\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^m{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\left({\varphi }_{m,\lambda }\right(x\left){\mathrm{e}}^x\right)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}{\varphi }_{m,\lambda }\left(x\right)x^{k\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^k{\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}\sum _{j=0}^m{S}_{2,\lambda }(m,j)x^j{\varphi }_{k,\lambda }\left(x\right){\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\sum _{j=0}^m{S}_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x){\mathrm{e}}^x.\end{array}$(32) From (32(32) $\begin{array}{rl}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n+m,\lambda }{\mathrm{e}}^x& ={\varphi }_{n+m,\lambda }\left(x\right){\mathrm{e}}^x=x^{(n+m)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n+m}{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{x}^{m\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^m{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\left({\varphi }_{m,\lambda }\right(x\left){\mathrm{e}}^x\right)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}{\varphi }_{m,\lambda }\left(x\right)x^{k\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^k{\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}\sum _{j=0}^m{S}_{2,\lambda }(m,j)x^j{\varphi }_{k,\lambda }\left(x\right){\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\sum _{j=0}^m{S}_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x){\mathrm{e}}^x.\end{array}$(32) ), we can derive the following equation: (33) ${\varphi }_{n+m,\lambda }\left(x\right)=\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x).$(33) Thus, by (33(33) ${\varphi }_{n+m,\lambda }\left(x\right)=\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x).$(33) ), we get (34) $\begin{array}{rl}{F}_{n+m,\lambda }^{\left(r\right)}\left(x\right)& =\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n+m,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{k,\lambda }\right(xy)y^{j+r-1}\mathrm{d}y\\ & =\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j\frac{\mathrm{\Gamma }(j+r)}{\mathrm{\Gamma }\left(r\right)}{F}_{k,\lambda }^{(j+r)}\right(x).\end{array}$(34) Therefore, by (34(34) $\begin{array}{rl}{F}_{n+m,\lambda }^{\left(r\right)}\left(x\right)& =\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n+m,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j\frac{1}{\mathrm{\Gamma }\left(r\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{k,\lambda }\right(xy)y^{j+r-1}\mathrm{d}y\\ & =\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j\frac{\mathrm{\Gamma }(j+r)}{\mathrm{\Gamma }\left(r\right)}{F}_{k,\lambda }^{(j+r)}\right(x).\end{array}$(34) ), we obtain the following theorem.

Theorem 2.7

For $n,m\ge 0$, we have $F_{n+m,\lambda }^{\left(r\right)}\left(x\right)=\sum _{k=0}^n\sum _{j=0}^m(\genfrac{}{}{0ex}{}{n}{k})S_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }\right(r{)}_j{x}^j{F}_{k,\lambda }^{(r+j)}\left(x\right).$

Remark

From the proof in (32(32) $\begin{array}{rl}{\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n+m,\lambda }{\mathrm{e}}^x& ={\varphi }_{n+m,\lambda }\left(x\right){\mathrm{e}}^x=x^{(n+m)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n+m}{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{x}^{m\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^m{\mathrm{e}}^x\\ & =x^{n\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n\left({\varphi }_{m,\lambda }\right(x\left){\mathrm{e}}^x\right)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}{\varphi }_{m,\lambda }\left(x\right)x^{k\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^k{\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{(n-k)\lambda }{\left(x^{1-\lambda }\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{n-k}\sum _{j=0}^m{S}_{2,\lambda }(m,j)x^j{\varphi }_{k,\lambda }\left(x\right){\mathrm{e}}^x\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\sum _{j=0}^m{S}_{2,\lambda }(m,j)\left(j{)}_{n-k,\lambda }{x}^j{\varphi }_{k,\lambda }\right(x){\mathrm{e}}^x.\end{array}$(32) ), we note that (35) ${\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n+m,\lambda }{\mathrm{e}}^x=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k}){\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{n-k,\lambda }{\varphi }_{m,\lambda }\left(x\right){\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)}_{k,\lambda }{\mathrm{e}}^x.$(35) By using (16(16) $F_{n,\lambda }^{\left(r\right)}\left(y\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})y^k{S}_{2,\lambda }(n,k)k!,(n\ge 0)\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[3\left]\right).$(16) ), we have (36) $\begin{array}{rl}{F}_{n,\lambda }^{(r+1)}\left(x\right)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r}{k})S_{2,\lambda }(n,k)k!x^k\\ & =\sum _{k=0}^n\{(\genfrac{}{}{0ex}{}{k+r-1}{k-1})+(\genfrac{}{}{0ex}{}{k+r-1}{k})\}{S}_{2,\lambda }(n,k)k!x^k\\ & =\sum _{k=0}^n\frac{k}{r}(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^k+\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^k\\ & =\frac{x}{r}\sum _{k=0}^{n}k(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^{k-1}+F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =\frac{x}{r}\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+F_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$(36) Therefore, by (36(36) $\begin{array}{rl}{F}_{n,\lambda }^{(r+1)}\left(x\right)& =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r}{k})S_{2,\lambda }(n,k)k!x^k\\ & =\sum _{k=0}^n\{(\genfrac{}{}{0ex}{}{k+r-1}{k-1})+(\genfrac{}{}{0ex}{}{k+r-1}{k})\}{S}_{2,\lambda }(n,k)k!x^k\\ & =\sum _{k=0}^n\frac{k}{r}(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^k+\sum _{k=0}^n(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^k\\ & =\frac{x}{r}\sum _{k=0}^{n}k(\genfrac{}{}{0ex}{}{k+r-1}{k})S_{2,\lambda }(n,k)k!x^{k-1}+F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =\frac{x}{r}\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+F_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$(36) ), we obtain the following theorem.

Theorem 2.8

For $n\ge 1$, we have $F_{n,\lambda }^{(r+1)}\left(x\right)=\frac{x}{r}\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+F_{n,\lambda }^{\left(r\right)}\left(x\right).$

From (7(7) ${\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(x\right)\frac{t^n}{n!}\left(\mathrm{s}\mathrm{e}\mathrm{e}\right[5\left]\right).$(7) ), we note that (37) $\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)\frac{t^n}{n!}& =\left(e_{\lambda }\right(t)-1){\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\right(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(x\right)-{\varphi }_{n,\lambda })\left(x\right)\frac{t^n}{n!},\end{array}$(37) where ${\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}{\varphi }_{n,\lambda }\left(x\right)$.

Comparing the coefficients on both sides of (37(37) $\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)\frac{t^n}{n!}& =\left(e_{\lambda }\right(t)-1){\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}\\ & =\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\right(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(x\right)-{\varphi }_{n,\lambda })\left(x\right)\frac{t^n}{n!},\end{array}$(37) ), we have (38) ${\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\right(x)-{\varphi }_{n,\lambda }(x),(n\ge 0).$(38) It is immediate to see that (39) $\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}=\frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=x{e}_{\lambda }^{1-\lambda }\left(t\right){\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}.$(39) Thus, by (38(38) ${\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\right(x)-{\varphi }_{n,\lambda }(x),(n\ge 0).$(38) ) and (39(39) $\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}=\frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}=x{e}_{\lambda }^{1-\lambda }\left(t\right){\mathrm{e}}^{x\left(e_{\lambda }\right(t)-1)}.$(39) ), we get (40) $\begin{array}{rl}(1+\lambda t)\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}& =x\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\right(x)\frac{t^n}{n!}\\ & =x\sum _{n=0}^{\mathrm{\infty }}\left({\varphi }_{n,\lambda }^{\mathrm{\prime }}\right(x)+{\varphi }_{n,\lambda }(x\left)\right)\frac{t^n}{n!},\end{array}$(40) and (41) $(1+\lambda t)\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left({\varphi }_{n+1,\lambda }\right(x)+\lambda n{\varphi }_{n,\lambda }(x\left)\right)\frac{t^n}{n!}.$(41) Therefore, by (40(40) $\begin{array}{rl}(1+\lambda t)\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}& =x\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\right(x)\frac{t^n}{n!}\\ & =x\sum _{n=0}^{\mathrm{\infty }}\left({\varphi }_{n,\lambda }^{\mathrm{\prime }}\right(x)+{\varphi }_{n,\lambda }(x\left)\right)\frac{t^n}{n!},\end{array}$(40) ) and (41(41) $(1+\lambda t)\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n+1,\lambda }\left(x\right)\frac{t^n}{n!}=\sum _{n=0}^{\mathrm{\infty }}\left({\varphi }_{n+1,\lambda }\right(x)+\lambda n{\varphi }_{n,\lambda }(x\left)\right)\frac{t^n}{n!}.$(41) ), we obtain the following theorem.

Theorem 2.9

For $n\ge 0$, we have ${\varphi }_{n+1,\lambda }\left(x\right)=x\left({\varphi }_{n,\lambda }^{\mathrm{\prime }}\right(x)+{\varphi }_{n,\lambda }(x\left)\right)-\lambda n{\varphi }_{n,\lambda }\left(x\right).$

Invoking Theorem 2.10 and integrating by parts, we note that (42) $\begin{array}{rl}& \mathrm{\Gamma }\left(r\right)F_{n+1,\lambda }^{\left(r\right)}\left(x\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n+1,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^r\mathrm{d}y+x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^r\mathrm{d}y-\lambda n{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)+x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^r\mathrm{d}y-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)-{\int }_0^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(xy\right)(r{y}^{r-1}{\mathrm{e}}^{-y}-{\mathrm{e}}^{-y}{y}^r)\mathrm{d}y\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)-r\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)+\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)\\ & =\mathrm{\Gamma }\left(r\right)\left(r\right(x+1\left)F_{n,\lambda }^{(r+1)}\right(x)-(\lambda n+r\left)F_{n,\lambda }^{\left(r\right)}\right(x\left)\right).\end{array}$(42) Therefore, by comparing the coefficients on both sides of (42(42) $\begin{array}{rl}& \mathrm{\Gamma }\left(r\right)F_{n+1,\lambda }^{\left(r\right)}\left(x\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n+1,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^r\mathrm{d}y+x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^r\mathrm{d}y-\lambda n{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)+x{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^r\mathrm{d}y-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)-{\int }_0^{\mathrm{\infty }}{\varphi }_{n,\lambda }\left(xy\right)(r{y}^{r-1}{\mathrm{e}}^{-y}-{\mathrm{e}}^{-y}{y}^r)\mathrm{d}y\\ & =x\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)-\lambda n{F}_{n,\lambda }^{\left(r\right)}\left(x\right)\mathrm{\Gamma }\left(r\right)-r\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)+\mathrm{\Gamma }(r+1)F_{n,\lambda }^{(r+1)}\left(x\right)\\ & =\mathrm{\Gamma }\left(r\right)\left(r\right(x+1\left)F_{n,\lambda }^{(r+1)}\right(x)-(\lambda n+r\left)F_{n,\lambda }^{\left(r\right)}\right(x\left)\right).\end{array}$(42) ), we obtain the following theorem.

Theorem 2.10

For $r\ge 1$ and $n\ge 0$, we have $F_{n+1,\lambda }^{\left(r\right)}\left(x\right)=r(x+1)F_{n,\lambda }^{(r+1)}\left(x\right)-(\lambda n+r)F_{n,\lambda }^{\left(r\right)}\left(x\right).$

By Theorems 2.9 and 2.11, we get (43) $\begin{array}{rl}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)& =r(x+1)\left(\frac{x}{r}\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\right(x)+F_{n,\lambda }^{\left(r\right)}(x\left)\right)-(\lambda n+r)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =x(x+1)\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+(rx+r-\lambda n-r)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =x(x+1)\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+(rx-\lambda n)F_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$(43) Comparing the coefficients on both sides of (43(43) $\begin{array}{rl}{F}_{n+1,\lambda }^{\left(r\right)}\left(x\right)& =r(x+1)\left(\frac{x}{r}\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\right(x)+F_{n,\lambda }^{\left(r\right)}(x\left)\right)-(\lambda n+r)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =x(x+1)\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+(rx+r-\lambda n-r)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =x(x+1)\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+(rx-\lambda n)F_{n,\lambda }^{\left(r\right)}\left(x\right).\end{array}$(43) ), we have $F_{n+1,\lambda }^{\left(r\right)}\left(x\right)=x(x+1)\frac{\mathrm{d}}{\mathrm{d}x}{F}_{n,\lambda }^{\left(r\right)}\left(x\right)+(rx-\lambda n)F_{n,\lambda }^{\left(r\right)}\left(x\right).$By exploiting (38(38) ${\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(x\right)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\right(x)-{\varphi }_{n,\lambda }(x),(n\ge 0).$(38) ) and integrating by parts, we have (44) $\begin{array}{rl}& \mathrm{\Gamma }\left(r\right)\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{F}_{k,\lambda }^{\left(r\right)}\right(x)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{k,\lambda }\right(xy)y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\right(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(xy\right))y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}\left({\varphi }_{n,\lambda }^{\mathrm{\prime }}\right(xy)+{\varphi }_{n,\lambda }(xy\left)\right)y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^{r-1}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)+{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\varphi }_{m,\lambda }^{\mathrm{\prime }}\left(xy\right){\mathrm{e}}^{-y}\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)-\frac{1}{x}{\int }_0^{\mathrm{\infty }}{\varphi }_{n,\lambda }(x,y)\left(\right(r-1)y^{r-2}{\mathrm{e}}^{-y}-y^{r-1}{\mathrm{e}}^{-y})\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)-\frac{1}{x}\mathrm{\Gamma }(r-1)(r-1)F_{n,\lambda }^{(r-1)}\left(x\right)+\frac{1}{x}\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =\mathrm{\Gamma }\left(r\right)\left(F_{n,\lambda }^{\left(r\right)}\right(x)(1+\frac{1}{x})-\frac{1}{x}{F}_{n,\lambda }^{(r-1)}(x\left)\right).\end{array}$(44) Comparing the coefficients on both sides of (44(44) $\begin{array}{rl}& \mathrm{\Gamma }\left(r\right)\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{F}_{k,\lambda }^{\left(r\right)}\right(x)\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{k,\lambda }\right(xy)y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}\left(\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\right(1{)}_{n-k,\lambda }{\varphi }_{k,\lambda }\left(xy\right))y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}\left({\varphi }_{n,\lambda }^{\mathrm{\prime }}\right(xy)+{\varphi }_{n,\lambda }(xy\left)\right)y^{r-1}\mathrm{d}y\\ & ={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }^{\mathrm{\prime }}\left(xy\right)y^{r-1}\mathrm{d}y+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-y}{\varphi }_{n,\lambda }\left(xy\right)y^{r-1}\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)+{\int }_0^{\mathrm{\infty }}{y}^{r-1}{\varphi }_{m,\lambda }^{\mathrm{\prime }}\left(xy\right){\mathrm{e}}^{-y}\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)-\frac{1}{x}{\int }_0^{\mathrm{\infty }}{\varphi }_{n,\lambda }(x,y)\left(\right(r-1)y^{r-2}{\mathrm{e}}^{-y}-y^{r-1}{\mathrm{e}}^{-y})\mathrm{d}y\\ & =\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)-\frac{1}{x}\mathrm{\Gamma }(r-1)(r-1)F_{n,\lambda }^{(r-1)}\left(x\right)+\frac{1}{x}\mathrm{\Gamma }\left(r\right)F_{n,\lambda }^{\left(r\right)}\left(x\right)\\ & =\mathrm{\Gamma }\left(r\right)\left(F_{n,\lambda }^{\left(r\right)}\right(x)(1+\frac{1}{x})-\frac{1}{x}{F}_{n,\lambda }^{(r-1)}(x\left)\right).\end{array}$(44) ), we obtain the following theorem.

Theorem 2.11

For $n\ge 0$ and r>0, we have $\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{F}_{k,\lambda }^{\left(r\right)}\right(x)=F_{n,\lambda }^{\left(r\right)}(x)(1+\frac{1}{x})-\frac{1}{x}{F}_{n,\lambda }^{(r-1)}(x).$In particular, for $n\in \mathbb{N}$ and r = 1, we get $\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})\left(1{)}_{n-k,\lambda }{F}_{k,\lambda }\right(x)=F_{n,\lambda }(x)(1+\frac{1}{x}).$

3. Conclusion

Carlitz initiated an exploration of degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials. Along the same line as Carlitz's pioneering work, intensive studies have been done for degenerate versions of quite a few special polynomials and numbers. There are various ways of studying special numbers and polynomials, to mention a few, generating functions, p-adic analysis, umbral calculus, operator theory, combinatorial methods, differential equations, special functions, probability theory and analytic number theory. In this article, by using generating functions and certain differential operators, we further studied some identities and properties on the degenerate Fubini polynomials and the higher-order degenerate Fubini polynomials.

It is one of our future projects to continue to explore various degenerate versions of many special polynomials and numbers and their applications to physics, science and engineering by using aforementioned tools.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).