A note on infinite series whose terms involve truncated degenerate exponentials

Abstract

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

Keywords:

• Truncated degenerate exponentials
• degenerate Stirling numbers of the second kind
• generalized falling factorials

Mathematics Subject Classifications:

• 11B83
• 11B65
• 11B73

1. Introduction

The degenerate exponentials play an important role in recent investigations on degenerate versions of many special numbers of polynomials (see [1]). Many of them are introduced by replacing the ordinary exponentials by the degenerate exponentials in their generating functions. These include the degenerate Stirling numbers of the second, the degenerate Bernoulli polynomials, the degenerate Euler polynomials, the partially degenerate Bell polynomials, and the degenerate central factorial numbers, and so on. Not only that, the degenerate gamma function is introduced by replacing the ordinary exponential by the degenerate exponential in the integral representation of the usual gamma function (see [2]). Furthermore, as a degenerate version of the ‘classical’ umbral calculus, the λ-umbral calculus (also called degenerate umbral calculus) is developed again by making the same replacement in the generating function of the Sheffer sequences. As it turns out, the degenerate umbral calculus (see [3]) is more convenient than the umbral calculus when dealing with degenerate special numbers and polynomials. In the same vein, the λ-q-umbral calculus (also called degenerate q-umbral calculus) is recently introduced by replacing the q-exponential by the λ-q-exponential (see [4]). In conclusion, we may say that the study of degenerate versions has been very fruitful (see [2–15]).

The aim of this note is to consider several infinite series whose terms involve the truncated degenerate exponentials, $e_{\lambda }(y)-1-\frac{(1{)}_{1,\lambda }}{1}y-\cdots -\frac{(1{)}_{n,\lambda }}{n!}{y}^n,(n\ge 0)$, and to find either their values or some other expressions of them as finite sums. Some of these infinite series also involve other special numbers, namely binomial coefficients, the generalized falling factorials (see (2(2) $(x{)}_{0,\lambda }=1,(x{)}_{n,\lambda }=x(x-\lambda )(x-2\lambda )\cdots (x-(n-1)\lambda ),(n\ge 1),(\mathrm{see}[8,12]).$(2) )) and the degenerate Stirling numbers of the second kind (see (6(6) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0).$(6) ), (7(7) $\frac{1}{k!}{(e_{\lambda }(t)-1)}^k=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!},(k\ge 0),(\mathrm{see}[8,13,14]).$(7) )).

For any $\lambda \in \mathbb{R}$, the degenerate exponentials are defined by (1) $e_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(x{)}_{n,\lambda }}{n!}{t}^n,\mathrm{and}{e}_{\lambda }(t)=e_{\lambda }^1(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{t}^n,$(1) where the generalized falling factorials are given by (2) $(x{)}_{0,\lambda }=1,(x{)}_{n,\lambda }=x(x-\lambda )(x-2\lambda )\cdots (x-(n-1)\lambda ),(n\ge 1),(\mathrm{see}[8,12]).$(2) From (1(1) $e_{\lambda }^x(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(x{)}_{n,\lambda }}{n!}{t}^n,\mathrm{and}{e}_{\lambda }(t)=e_{\lambda }^1(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{t}^n,$(1) ), we note that $\underset{\lambda \to 0}{lim}{e}_{\lambda }^x(t)={\mathrm{e}}^{xt}$. To simplify expressions, we adopt the following notation which is introduced in [16]: (3) $e_{n,\lambda }(t)=\sum _{k=0}^n(1{)}_{k,\lambda }\frac{t^k}{k!},(n\ge 0).$(3) Then the truncated degenerate exponentials can be expressed as follows: (4) $e_{\lambda }(y)-1-\frac{(1{)}_{1,\lambda }}{1}y-\cdots -\frac{(1{)}_{n,\lambda }}{n!}{y}^n=e_{\lambda }(y)-e_{n,\lambda }(y),(n\ge 0).$(4) The Stirling numbers of the second kind are given by (5) $x^n=\sum _{k=0}^n{S}_2(n,k)(x{)}_k,(n\ge 0),(\mathrm{see}[8]),$(5) where $(x{)}_k=x(x-1)\cdots (x-k+1),(k\ge 1),(x{)}_0=1$.

In [8], the degenerate Stirling numbers of the second kind are defined by (6) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0).$(6) From (6(6) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0).$(6) ), we note that $\underset{\lambda \to 0}{lim}{S}_{2,\lambda }(n,k)=S_2(n,k)$.

By (6(6) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0).$(6) ), we easily get (7) $\frac{1}{k!}{(e_{\lambda }(t)-1)}^k=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!},(k\ge 0),(\mathrm{see}[8,13,14]).$(7) We would like to mention the recent work [17] on the r-truncated Stirling numbers of the second kind $S_{2,\lambda }^{[r]}(n,kr)$, defined by $\frac{1}{k!}{(e_{\lambda }(t)-e_{r-1,\lambda }(t))}^k=\sum _{n=kr}^{\mathrm{\infty }}{S}_{2,\lambda }^{[r]}(n,kr)\frac{t^n}{n!}.$Note here that the r-truncated Stirling numbers of the second kind reduce to the degenerate Stirling numbers of the second for r = 1. Explicit expressions, some properties and related identities of those numbers are investigated in connection with several other degenerate special numbers and polynomials in [17].

The backward difference operator ▽ is defined as (8) $▽f(x)=f(x)-f(x-1),(\mathrm{see}[14]).$(8) From (8(8) $▽f(x)=f(x)-f(x-1),(\mathrm{see}[14]).$(8) ), we note that (9) $(\genfrac{}{}{0ex}{}{x-1}{n-1})=▽(\genfrac{}{}{0ex}{}{x}{n})=(\genfrac{}{}{0ex}{}{x}{n})-(\genfrac{}{}{0ex}{}{x-1}{n}),(n\ge 1).$(9) Thus, by (9(9) $(\genfrac{}{}{0ex}{}{x-1}{n-1})=▽(\genfrac{}{}{0ex}{}{x}{n})=(\genfrac{}{}{0ex}{}{x}{n})-(\genfrac{}{}{0ex}{}{x-1}{n}),(n\ge 1).$(9) ), we get (10) $(\genfrac{}{}{0ex}{}{x}{n})=(\genfrac{}{}{0ex}{}{x+1}{n})-(\genfrac{}{}{0ex}{}{x}{n-1}),(n\ge 0),(\mathrm{see}[6,7,18-20]).$(10) In addition, the degenerate Bell polynomials are defined by ${\varphi }_{n,\lambda }(x)={\mathrm{e}}^{-x}\sum _{k=0}^{\mathrm{\infty }}\frac{(k{)}_{n,\lambda }}{k!}{x}^k=\sum _{k=0}^n{S}_{2,\lambda }(n,k)x^k,(n\ge 0),(\mathrm{see}[8,13]),$where the middle one is the Dobinski-like formula for the degenerate Bell polynomials.

2. Infinite series whose terms involve truncated degenerate exponentials

In the section, we will consider infinite series whose terms involve truncated degenerate exponentials. We first observe that (11) $\begin{array}{rl}& \frac{1}{x-1}(e_{\lambda }(xy)-e_{\lambda }(y))=\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\left(\frac{x^k-1}{x-1}\right)\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(1{)}_{k+1,\lambda }}{(k+1)!}{y}^{k+1}\sum _{n=0}^k{x}^n=\sum _{n=0}^{\mathrm{\infty }}{x}^n\sum _{k=n+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\\ & =\sum _{n=0}^{\mathrm{\infty }}{x}^n(e_{\lambda }(y)-e_{n,\lambda }(y)).\end{array}$(11) Taking the limit as $x\to 1$ in (11(11) $\begin{array}{rl}& \frac{1}{x-1}(e_{\lambda }(xy)-e_{\lambda }(y))=\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\left(\frac{x^k-1}{x-1}\right)\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(1{)}_{k+1,\lambda }}{(k+1)!}{y}^{k+1}\sum _{n=0}^k{x}^n=\sum _{n=0}^{\mathrm{\infty }}{x}^n\sum _{k=n+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\\ & =\sum _{n=0}^{\mathrm{\infty }}{x}^n(e_{\lambda }(y)-e_{n,\lambda }(y)).\end{array}$(11) ), we have (12) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\underset{x\to 1}{lim}\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\left(\frac{x^k-1}{x-1}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^{k}k\\ & =y\sum _{k=0}^{\mathrm{\infty }}\frac{(1-\lambda {)}_{k,\lambda }}{k!}{y}^k=y{e}_{\lambda }^{1-\lambda }(y)=\frac{y}{1+\lambda y}{e}_{\lambda }(y).\end{array}$(12) Therefore, by (11(11) $\begin{array}{rl}& \frac{1}{x-1}(e_{\lambda }(xy)-e_{\lambda }(y))=\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\left(\frac{x^k-1}{x-1}\right)\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(1{)}_{k+1,\lambda }}{(k+1)!}{y}^{k+1}\sum _{n=0}^k{x}^n=\sum _{n=0}^{\mathrm{\infty }}{x}^n\sum _{k=n+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\\ & =\sum _{n=0}^{\mathrm{\infty }}{x}^n(e_{\lambda }(y)-e_{n,\lambda }(y)).\end{array}$(11) ) and (12(12) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\underset{x\to 1}{lim}\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\left(\frac{x^k-1}{x-1}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^{k}k\\ & =y\sum _{k=0}^{\mathrm{\infty }}\frac{(1-\lambda {)}_{k,\lambda }}{k!}{y}^k=y{e}_{\lambda }^{1-\lambda }(y)=\frac{y}{1+\lambda y}{e}_{\lambda }(y).\end{array}$(12) ), we obtain the following theorem.

Theorem 2.1

The following identities hold true. $\begin{array}{rl}& \frac{1}{x-1}(e_{\lambda }(xy)-e_{\lambda }(y))=\sum _{n=0}^{\mathrm{\infty }}(e_{\lambda }(y)-e_{n,\lambda }(y))x^n,\\ & \frac{y}{1+\lambda y}{e}_{\lambda }(y)=\sum _{n=0}^{\mathrm{\infty }}(e_{\lambda }(y)-e_{n,\lambda }(y)).\end{array}$

The degenerate hyperbolic cosine function is defined by ${\cosh }_{\lambda }(x)=\frac{e_{\lambda }(-x)+e_{\lambda }(x)}{2}.$Note that $\underset{\lambda \to 0}{lim}{\cosh }_{\lambda }(x)=\cosh (x)$. The next corollary is immediate from Theorem 2.1.

Corollary 2.2

The following identities hold true. $\begin{array}{rl}& \sum _{n=1}^{\mathrm{\infty }}(e_{\lambda }(1)-e_{n,\lambda }(1))x^n=\frac{e_{\lambda }(x)-x{e}_{\lambda }(1)}{x-1}+1,\\ & \sum _{n=1}^{\mathrm{\infty }}(e_{\lambda }(1)-e_{n,\lambda }(1))=1-\frac{\lambda }{1+\lambda }{e}_{\lambda }(1),\end{array}$and $\sum _{n=1}^{\mathrm{\infty }}(e_{\lambda }(1)-e_{n,\lambda }(1))(-1{)}^n=1-{\cosh }_{\lambda }(1).$

From (10(10) $(\genfrac{}{}{0ex}{}{x}{n})=(\genfrac{}{}{0ex}{}{x+1}{n})-(\genfrac{}{}{0ex}{}{x}{n-1}),(n\ge 0),(\mathrm{see}[6,7,18-20]).$(10) ), we note that (13) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{n=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})\sum _{k=n+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k=\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\sum _{n=p}^{k-1}(\genfrac{}{}{0ex}{}{n}{p})\\ & =\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\sum _{n=p}^{k-1}((\genfrac{}{}{0ex}{}{n+1}{p+1})-(\genfrac{}{}{0ex}{}{n}{p+1}))=\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k(\genfrac{}{}{0ex}{}{k}{p+1})\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(1{)}_{k+p+1,\lambda }}{(k+p+1)!}{y}^{k+p+1}(\genfrac{}{}{0ex}{}{k+p+1}{p+1})=\frac{y^{p+1}(1{)}_{p+1,\lambda }}{(p+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{(1-(p+1)\lambda {)}_{k,\lambda }}{k!}{y}^k\\ & =\frac{y^{p+1}}{(p+1)!}(1{)}_{p+1,\lambda }{e}_{\lambda }^{1-(p+1)\lambda }(y)=\frac{y^{p+1}}{(p+1)!}(1{)}_{p+1,\lambda }(1+\lambda y{)}^{-(p+1)}{e}_{\lambda }(y).\end{array}$(13) Therefore, by (13(13) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{n=p}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})\sum _{k=n+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k=\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\sum _{n=p}^{k-1}(\genfrac{}{}{0ex}{}{n}{p})\\ & =\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k\sum _{n=p}^{k-1}((\genfrac{}{}{0ex}{}{n+1}{p+1})-(\genfrac{}{}{0ex}{}{n}{p+1}))=\sum _{k=p+1}^{\mathrm{\infty }}\frac{(1{)}_{k,\lambda }}{k!}{y}^k(\genfrac{}{}{0ex}{}{k}{p+1})\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(1{)}_{k+p+1,\lambda }}{(k+p+1)!}{y}^{k+p+1}(\genfrac{}{}{0ex}{}{k+p+1}{p+1})=\frac{y^{p+1}(1{)}_{p+1,\lambda }}{(p+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{(1-(p+1)\lambda {)}_{k,\lambda }}{k!}{y}^k\\ & =\frac{y^{p+1}}{(p+1)!}(1{)}_{p+1,\lambda }{e}_{\lambda }^{1-(p+1)\lambda }(y)=\frac{y^{p+1}}{(p+1)!}(1{)}_{p+1,\lambda }(1+\lambda y{)}^{-(p+1)}{e}_{\lambda }(y).\end{array}$(13) ), we obtain the following theorem.

Theorem 2.3

For $p\ge 0$, we have $\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})(e_{\lambda }(y)-e_{n,\lambda }(y))=\frac{y^{p+1}}{(p+1)!}(1{)}_{p+1,\lambda }(1+\lambda y{)}^{-(p+1)}{e}_{\lambda }(y).$Especially, for y = 1, we obtain $\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{p})(e_{\lambda }(1)-e_{n,\lambda }(1))=\frac{(1{)}_{p+1,\lambda }}{(p+1)!}(1+\lambda {)}^{-(p+1)}{e}_{\lambda }(1).$

From Theorem 2.3, we note that (14) $\sum _{n=0}^{\mathrm{\infty }}(n{)}_p(e_{\lambda }(y)-e_{n,\lambda }(y))=\frac{y^{p+1}}{p+1}(1{)}_{p+1,\lambda }(1+\lambda y{)}^{-(p+1)}{e}_{\lambda }(y).$(14) By (6(6) $(x{)}_{n,\lambda }=\sum _{k=0}^n{S}_{2,\lambda }(n,k)(x{)}_k,(n\ge 0).$(6) ) and (14(14) $\sum _{n=0}^{\mathrm{\infty }}(n{)}_p(e_{\lambda }(y)-e_{n,\lambda }(y))=\frac{y^{p+1}}{p+1}(1{)}_{p+1,\lambda }(1+\lambda y{)}^{-(p+1)}{e}_{\lambda }(y).$(14) ), we get (15) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(n{)}_{p,\lambda }(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^p{S}_{2,\lambda }(p,k)(n{)}_k(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\sum _{n=0}^{\mathrm{\infty }}(n{)}_k(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\frac{y^{k+1}}{k+1}(1{)}_{k+1,\lambda }(1+\lambda y{)}^{-(k+1)}{e}_{\lambda }(y).\end{array}$(15) Therefore, by (15(15) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(n{)}_{p,\lambda }(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^p{S}_{2,\lambda }(p,k)(n{)}_k(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\sum _{n=0}^{\mathrm{\infty }}(n{)}_k(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\frac{y^{k+1}}{k+1}(1{)}_{k+1,\lambda }(1+\lambda y{)}^{-(k+1)}{e}_{\lambda }(y).\end{array}$(15) ), we obtain the following theorem.

Theorem 2.4

For $p\ge 0$, we have $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(n{)}_{p,\lambda }(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\frac{y^{k+1}}{k+1}(1{)}_{k+1,\lambda }(1+\lambda y{)}^{-(k+1)}{e}_{\lambda }(y).\end{array}$In particular, for y = 1, we get $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}(n{)}_{p,\lambda }(e_{\lambda }(1)-e_{n,\lambda }(1))\\ & =\sum _{k=0}^p{S}_{2,\lambda }(p,k)\frac{(1{)}_{k+1,\lambda }}{k+1}(1+\lambda {)}^{-(k+1)}{e}_{\lambda }(1).\end{array}$

From (7(7) $\frac{1}{k!}{(e_{\lambda }(t)-1)}^k=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!},(k\ge 0),(\mathrm{see}[8,13,14]).$(7) ), we note that (16) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!}=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!}=\frac{1}{k!}{(e_{\lambda }(t)-1)}^k\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{e}_{\lambda }^j(t)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=0}^{\mathrm{\infty }}\frac{(j{)}_{n,\lambda }}{n!}{t}^n\\ & =\sum _{n=0}^{\mathrm{\infty }}(\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}(j{)}_{n,\lambda })\frac{t^n}{n!}.\end{array}$(16) Comparing the coefficients on both sides of (16(16) $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!}=\sum _{n=k}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)\frac{t^n}{n!}=\frac{1}{k!}{(e_{\lambda }(t)-1)}^k\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{e}_{\lambda }^j(t)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=0}^{\mathrm{\infty }}\frac{(j{)}_{n,\lambda }}{n!}{t}^n\\ & =\sum _{n=0}^{\mathrm{\infty }}(\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}(j{)}_{n,\lambda })\frac{t^n}{n!}.\end{array}$(16) ), we obtain (17) $S_{2,\lambda }(n,k)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}(j{)}_{n,\lambda },(n,k\ge 0).$(17) Taking the limit as $\lambda \to 0$ in (17(17) $S_{2,\lambda }(n,k)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}(j{)}_{n,\lambda },(n,k\ge 0).$(17) ), we have (18) $S_2(n,k)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{j}^n,(n,k\ge 0).$(18) By using (18(18) $S_2(n,k)=\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{j}^n,(n,k\ge 0).$(18) ), we derive the following: (19) $\begin{array}{rl}& \frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\frac{e_{\lambda }(jy)-e_{\lambda }(y)}{j-1}\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\left(\frac{j^n-1}{j-1}\right)\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}{j}^l\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{j}^l\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}{S}_2(l,k)=\sum _{l=0}^{\mathrm{\infty }}{S}_2(l,k)\sum _{n=l+1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n.\end{array}$(19) Therefore, by (19(19) $\begin{array}{rl}& \frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\frac{e_{\lambda }(jy)-e_{\lambda }(y)}{j-1}\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\left(\frac{j^n-1}{j-1}\right)\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}{j}^l\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}{j}^l\\ & =\sum _{n=1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n\sum _{l=0}^{n-1}{S}_2(l,k)=\sum _{l=0}^{\mathrm{\infty }}{S}_2(l,k)\sum _{n=l+1}^{\mathrm{\infty }}\frac{(1{)}_{n,\lambda }}{n!}{y}^n.\end{array}$(19) ), we obtain the following theorem.

Theorem 2.5

For $k\ge 0$, we have $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{S}_2(n,k)(e_{\lambda }(y)-e_{n,\lambda }(y))\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\frac{e_{\lambda }(jy)-e_{\lambda }(y)}{j-1}.\end{array}$In particular, for y = 1, we get $\begin{array}{rl}& \sum _{n=0}^{\mathrm{\infty }}{S}_2(n,k)(e_{\lambda }(1)-e_{n,\lambda }(1))\\ & =\frac{1}{k!}\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j})(-1{)}^{k-j}\frac{e_{\lambda }(j)-e_{\lambda }(1)}{j-1}.\end{array}$

Remark 2.6

We may naturally consider the following problem.

For any $k\ge 0$, find the value of $\sum _{n=0}^{\mathrm{\infty }}{S}_{2,\lambda }(n,k)(e_{\lambda }(y)-e_{n,\lambda }(y)).$

Remark 2.7

Much work has been done as to degenerate and truncated theories. These theories have some applications to mathematics, engineering and physics. Researchers interested in these may refer to [1–24].

3. Conclusion

In this note, we studied infinite series whose terms involve the truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and determined either their values or some other expressions of them as finite sums.

In recent years, we have witnessed that the study of degenerate versions yielded many fascinating and fruitful results. They were found by using various tools such as combinatorial methods, generating functions, p-adic analysis, umbral calculus techniques, differential equations, probability theory, operator theory, special functions and analytic number theory. It is noteworthy that the exploration for degenerate versions is extended to transcendental functions like gamma functions, not just limited to polynomials and numbers. Moreover, it led to the introduction of degenerate umbral calculus and degenerate q-umbral calculus.

We would like to continue to study degenerate versions of many special numbers and polynomials and to find some applications of them to physics, science and engineering as well as to mathematics.

Acknowledgments

The authors would like to thank the referees for the detailed and valuable comments that helped improve the original manuscript in its present form. Also, the authors thank Jangjeon Institute for Mathematical Sciences for the support of this research.

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).Ethics approval and consent to participate The authors declare that there is no ethical problem in the production of this paper. Consent for publication The authors want to publish this paper in this journal.