New approach to $ \lambda $-Stirling numbers

Dae San Kim; Hye Kyung Kim; Taekyun Kim; Dae San Kim; Hye Kyung Kim; Taekyun Kim

doi:10.3934/math.20231449

AIMS Mathematics

2023, Volume 8, Issue 12: 28322-28333. doi: 10.3934/math.20231449

Previous Article Next Article

Research article Special Issues

New approach to $\lambda$ -Stirling numbers

1.
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2.
Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea
3.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received: 18 August 2023 Revised: 28 September 2023 Accepted: 28 September 2023 Published: 17 October 2023
MSC : 11B73, 05A18

The aim of this paper is to study the $\lambda$ -Stirling numbers of both kinds, which are $\lambda$ -analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when $\lambda$ are positive integers. If $\lambda = 1$ , then the $\lambda$ -Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $\lambda$ -Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $\lambda$ -Stirling numbers of both kinds are obtained for the $\lambda$ -analogues of $r$ -Stirling numbers of both kinds, which are generalizations of those numbers.

Keywords:

Citation: Dae San Kim, Hye Kyung Kim, Taekyun Kim. New approach to $\lambda$ -Stirling numbers[J]. AIMS Mathematics, 2023, 8(12): 28322-28333. doi: 10.3934/math.20231449

Related Papers:

[1]	Taekyun Kim, Dae San Kim, Jin-Woo Park . Degenerate $r$ -truncated Stirling numbers. AIMS Mathematics, 2023, 8(11): 25957-25965. doi: 10.3934/math.20231322
[2]	Sang Jo Yun, Jin-Woo Park . On a generation of degenerate Daehee polynomials. AIMS Mathematics, 2025, 10(5): 12286-12298. doi: 10.3934/math.2025556
[3]	Taekyun Kim, Dae San Kim, Hye Kyung Kim . Some identities involving degenerate Stirling numbers arising from normal ordering. AIMS Mathematics, 2022, 7(9): 17357-17368. doi: 10.3934/math.2022956
[4]	Dojin Kim, Patcharee Wongsason, Jongkyum Kwon . Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions. AIMS Mathematics, 2022, 7(6): 9716-9730. doi: 10.3934/math.2022541
[5]	Feng Qi, Da-Wei Niu, Bai-Ni Guo . Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind. AIMS Mathematics, 2019, 4(2): 170-175. doi: 10.3934/math.2019.2.170
[6]	Aimin Xu . Some identities connecting Stirling numbers, central factorial numbers and higher-order Bernoulli polynomials. AIMS Mathematics, 2025, 10(2): 3197-3206. doi: 10.3934/math.2025148
[7]	Hye Kyung Kim . Note on $r$ -central Lah numbers and $r$ -central Lah-Bell numbers. AIMS Mathematics, 2022, 7(2): 2929-2939. doi: 10.3934/math.2022161
[8]	Waseem Ahmad Khan, Kottakkaran Sooppy Nisar, Dumitru Baleanu . A note on (p, q)-analogue type of Fubini numbers and polynomials. AIMS Mathematics, 2020, 5(3): 2743-2757. doi: 10.3934/math.2020177
[9]	Takao Komatsu, Wenpeng Zhang . Several expressions of truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers. AIMS Mathematics, 2020, 5(6): 5939-5954. doi: 10.3934/math.2020380
[10]	Yulu Feng, Min Qiu . Some results on p-adic valuations of Stirling numbers of the second kind. AIMS Mathematics, 2020, 5(5): 4168-4196. doi: 10.3934/math.2020267

Abstract

1. Introduction

It is remarkable that explorations for degenerate versions of quite a few special numbers and polynomials, which began with the study of degenerate Bernoulli and degenerate Euler polynomials by Carlitz, have led to the discoveries of many interesting results (see ^{[10,12,13,14,15]} and references therein). Specially, the degenerate Stirling numbers of both kinds are degenerate versions of the usual Stirling numbers of both kinds and occur frequently when we study degenerate versions of many special numbers and polynomials. In addition, $\lambda$ -analogues, which are different from degenerate versions, of some special numbers and polynomials, have also been studied (see ^[5,6,9,11]). Here, we consider the $\lambda$ -analogues of Stirling numbers of both kinds.

The Stirling number of the second kind ${{n} \brace {k}}$ enumerates the number of partitions of the set $[n] = \left\{1, 2, \dots, n \right\}$ into $k$ nonempty disjoint sets, while the unsigned Stirling number of the first kind ${{n} \brack {k}} = (-1)^{n-k}S_{1}(n, k)$ counts the number of permutations of $n$ elements with $k$ disjoint cycles. Here, we consider the $\lambda$ -analogues of the Stirling numbers of both kinds. The $\lambda$ -analogues of ${{n} \brace {k}}$ are called the $\lambda$ -Stirling numbers of the second kind and denoted by ${{n} \brace {k}}_{\lambda}$ , while those of ${{n} \brack {k}}$ are called the $\lambda$ -Stirling numbers of the first kind and denoted by ${{n} \brack {k}}_{\lambda}$ . Here, $\lambda$ is any real number and we note that ${{n} \brack {k}}_{1} = {{n} \brack {k}} \;{\rm{and}}\; {{n} \brace {k}}_{1} = {{n} \brace {k}}$ . Now, let $\lambda$ be any positive integer, then it is possible to give combinatorial interpretations of ${{n} \brace {k}}_{\lambda}$ and ${{n} \brack {k}}_{\lambda}$ . Indeed, ${{n} \brace {k}}_{\lambda}$ , called the translated Whitney numbers of the second kind ^[2,16], enumerates the number of partitions of the set $[n]$ into $k$ subsets such that each element of each subset can mutate in $\lambda$ ways, except the dominant one. Meanwhile, ${{n} \brack {k}}_{\lambda}$ , called the translated Whitney numbers of the first kind ^[2,16], counts the number of permutations of $n$ elements with $k$ cycles, such that the element of each cycle can mutate in $\lambda$ ways except the dominant one. We also consider generalizations of the $\lambda$ -Stirling numbers of both kinds, namely the $\lambda$ -analogues of $r$ -Stirling numbers of the second kind ${{n+r} \brace {k+r}}_{r, \lambda}$ and those of (unsigned) $r$ -Stirling numbers of the first kind ${{n+r} \brack {k+r}}_{r, \lambda}$ . Here, we note that ${{n} \brace {k}}_{0, \lambda} = {{n} \brace {k}}_{\lambda}$ and ${{n} \brack {k}}_{0, \lambda} = {{n} \brack {k}}_{\lambda}$ .

The aim of this paper is to derive new types of generating functions of the $\lambda$ -Stirling numbers of both kinds (see Theorem 3, Corollary 4, Remark 5). We note that the new type of generating function of the $\lambda$ -Stirling numbers of the second kind is equivalent to the ordinary generating function of those numbers (see Theorem 3, Remark 5). The generating function of the $\lambda$ -Stirling numbers of the first kind follows from that of the second kind by inversion (see Corollary 4). In Theorem 6, we find expressions of the integral over $(a, \infty)$ of the reciprocals of the generalized rising factorials as finite sums by using an identity in (2.5), and the fact that the limit as $b$ tends to $\infty$ of a certain finite sum is zero (see (2.13)). Also, we derive an expression of $\frac{1}{ka^{k}}$ as an infinite sum by integrating the generating function of the $\lambda$ -Stirling numbers of the second kind in Corollary 4. In section three, we derive the corresponding results to those in section two for the $\lambda$ -analogues of $r$ -Stirling numbers of both kinds by using similar methods. We are indebted to ^[3] for many ideas in this paper.

The outline of this paper is as follows. We first recall the generalized falling factorial sequence, the Stirling numbers of the first kind, the Stirling numbers of the second kind and the unsigned Stirling numbers of the first kind. We remind the reader of the $\lambda$ -Stirling numbers of the first kind, the $\lambda$ -Stirling numbers of the second kind and the unsigned $\lambda$ -Stirling numbers of the first kind. We recall the $\lambda$ -analogues of (unsigned) $r$ -Stirling numbers of the first kind and the $\lambda$ -analogues of $r$ -Stirling numbers of the second kind as well as an explicit expression of the $\lambda$ -analogues of $r$ -Stirling numbers of the second kind and the $\lambda$ -analogues of binomial coefficients. Then we remind the reader of orthogonality and inverse relations for the $\lambda$ -analogues of $r$ -Stirling numbers. Section two is the main result of this paper. In Theorem 3, we obtain an expression of the reciprocal of the generalized rising factorial as an infinite series involving the $\lambda$ -Stirling numbers of the second kind, which is equivalent to the ordinary generating function of those numbers. It is noted in Corollary 4 that we get, by inversion, a generating function of the unsigned $\lambda$ -Stirling numbers of the first kind in terms of the reciprocals of the generalized rising factorials. By applying a Frullani's integral in (2.10), we show that the limit $b$ tending to $\infty$ of a certain finite sum is equal to zero (see (2.13)). By using this fact and an expression of the reciprocals of the generalized rising factorials in (2.5), we get explicit expressions as finite sums of the integral of those reciprocals over $(a, \infty)$ in Theorem 6 and those as infinite sums of $\frac{1}{ka^{k}}$ in Corollary 7. In section three, all the results in section two are extended to the $\lambda$ -analogues of $r$ -Stirling numbers and similar results to those in section two are obtained. In the rest of this section, we recall the necessary facts that will be used throughout this paper.

For any $\lambda \in \mathbb{R}$ , the generalized falling factorial sequence is defined by

$\begin{equation} \begin{split} (x)_{0,\lambda} = 1, \ (x)_{n,\lambda} = x(x-\lambda)(x-2\lambda) \ \cdots \ (x-(n-1)\lambda), \quad (n\geq1). \end{split} \end{equation}$

(1.1)

For $n\geq0$ , it is known that the Stirling numbers of the first kind are defined by

$\begin{equation} \begin{split} (x)_n = \sum\limits_{k = 0}^n S_1(n,l)x^l, \end{split} \end{equation}$

(1.2)

see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18]}.

As the inversion formula of (1.2), the Stirling numbers of the second kind are defined by

$\begin{equation} \begin{split} x^n = \sum\limits_{k = 0}^n {n \brace k}(x)_k, \quad (n\geq0), \end{split} \end{equation}$

(1.3)

see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,18,19,20]}.

The unsigned Stirling numbers of the first kind are defined by ${n \brack k} = (-1)^{n-k}S_1(n, k), \quad (n, k\geq0)$ . In ^[9,11], the $\lambda$ -Stirling numbers of the first kind are given by

$\begin{equation} \begin{split} (x)_{n,\lambda} = \sum\limits_{k = 0}^n S_{1,\lambda}(n,k)x^k, \ \ (n\geq0). \end{split} \end{equation}$

(1.4)

The unsigned $\lambda$ -Stirling numbers of the first kind are defined by

$\begin{equation} \begin{split} < x > _{n,\lambda} = (-1)^n (-x)_{n,\lambda} = \sum\limits_{k = 0}^n {n \brack k}_\lambda x^k, \end{split} \end{equation}$

(1.5)

where the generalized rising factorial sequence is given by

$\begin{equation} \begin{split} < x > _{0,\lambda} = 1, \ < x > _{n,\lambda} = x(x+\lambda) \ \cdots \ (x+(n-1)\lambda), \quad (n\geq1). \end{split} \end{equation}$

(1.6)

From (1.4) and (1.5), we note that ${n \brack k}_{\lambda} = (-1)^{n-k}S_{1, \lambda}(n, k)$ .

The $\lambda$ -Stirling numbers of the second kind are defined by

$\begin{equation} \begin{split} x^n = \sum\limits_{k = 0}^n{n \brace k}_{\lambda}(x)_{k,\lambda}, \quad (n\geq0), \ \ [9,11]. \end{split} \end{equation}$

(1.7)

For $r\in\mathbb{N}\cup\{0\}$ , the $\lambda$ -analogues of (unsigned) $r$ -Stirling numbers of the first kind are defined by

$\begin{equation} \begin{split} < x+r > _{n,\lambda} = \sum\limits_{k = 0}^n {{n+r} \brack {k+r}}_{r,\lambda} x^k, \ \ [9,11], \end{split} \end{equation}$

(1.8)

where we note that ${{n} \brack {k}}_{0, \lambda} = {{n} \brack {k}}_{\lambda}$ , for $r = 0$ , (see (1.5), (1.8)).

Thus, by (1.8), we get

$\begin{equation} \begin{split} (x)_{n,\lambda} = \sum\limits_{k = 0}^n(-1)^{n-k}{{n+r} \brack {k+r}}_{r,\lambda}(x+r)^k, \quad (n\geq0). \end{split} \end{equation}$

(1.9)

The $\lambda$ -analogues of $r$ -Stirling numbers of the second kind are defined by

$\begin{equation} \begin{split} (x+r)^n = \sum\limits_{k = 0}^n {{n+r} \brace {k+r}}_{r,\lambda}(x)_{k,\lambda}, \quad (n\geq0), \ \ [9,11]. \end{split} \end{equation}$

(1.10)

Note that

$\begin{equation} \begin{split} {{n+r} \brace {k+r}}_{r,\lambda} = \frac{1}{\lambda^k}\frac{1}{k!}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^{k-l}(l\lambda+r)^n. \end{split} \end{equation}$

(1.11)

This follows easily from Theorem 1 ^[9]. As a special case of (1.11), we observe that

$\begin{equation} {{n} \brace {k}}_{\lambda} = {{n} \brace {k}}_{0,\lambda} = \frac{1}{k!}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^{k-l}\lambda^{n-k}l^{n},\; {\text{see}}\,\, (1.7)\; {\text{and}}\; (1.10). \end{equation}$

(1.12)

For $n\geq0$ , the $\lambda$ -analogues of binomial coefficients are defined by

$\begin{equation} \begin{split} {\binom{x}{n}}_{\lambda} = \frac{(x)_{n,\lambda}}{n!} = \frac{x(x-\lambda) \ \cdots \ (x-(n-1)\lambda))}{n!}, \ [15]. \end{split} \end{equation}$

(1.13)

We are going to use the next theorem and its corollary several times in the sequel. Both (a) and (b) follow from (1.9) and (1.10), while (c)–(e) can be derived from (a) and (b).

Theorem 1. The $\lambda$ -analogues of $r$ -Stirling numbers enjoy the following orthogonality and inverse relations:

$\begin{align*} &(a)\,\, \sum\limits_{k = l}^n (-1)^{n-k}{{n+r} \brack {k+r}}_{r,\lambda}{{k+r} \brace {l+r}}_{r,\lambda} = \delta_{n,l}, \\ &(b)\,\,\sum\limits_{k = l}^n (-1)^{k-l}{{k+r} \brack {l+r}}_{r,\lambda}{{n+r} \brace {k+r}}_{r,\lambda} = \delta_{n,l}, \\ &(c)\,\, a_{n} = \sum\limits_{k = 0}^{n}{{n+r} \brace {k+r}}_{r,\lambda} c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = 0}^{n}(-1)^{n-k}{{n+r}\brack {k+r}}_{r,\lambda}a_{k},\\ &(d)\,\, a_{n} = \sum\limits_{k = n}^{m}{{k+r} \brace {n+r}}_{r,\lambda}c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = n}^{m}(-1)^{k-n}{{k+r} \brack {n+r}}_{r,\lambda}a_{k}, \\ &(e)\,\, a_{n} = \sum\limits_{k = n}^{\infty}{{k+r} \brace {n+r}}_{r,\lambda}c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = n}^{\infty}(-1)^{k-n}{{k+r} \brack {n+r}}_{r,\lambda}a_{k}. \\ \end{align*}$

In the special case of $r = 0$ , we obtain the next orthogonality and inverse relations for the $\lambda$ -Stirling numbers.

Corollary 2. The $\lambda$ -Stirling numbers enjoy the following orthogonality and inverse relations:

$\begin{align*} &(a)\,\, \sum\limits_{k = l}^n (-1)^{n-k}{{n} \brack {k}}_{\lambda}{{k} \brace {l}}_{\lambda} = \delta_{n,l}, \\ &(b)\,\,\sum\limits_{k = l}^n (-1)^{k-l}{{k} \brack {l}}_{\lambda}{{n} \brace {k}}_{\lambda} = \delta_{n,l}, \\ &(c)\,\, a_{n} = \sum\limits_{k = 0}^{n}{{n} \brace {k}}_{\lambda} c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = 0}^{n}(-1)^{n-k}{{n}\brack {k}}_{\lambda}a_{k},\\ &(d)\,\, a_{n} = \sum\limits_{k = n}^{m}{{k} \brace {n}}_{\lambda}c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = n}^{m}(-1)^{k-n}{{k} \brack {n}}_{\lambda}a_{k}, \\ &(e)\,\, a_{n} = \sum\limits_{k = n}^{\infty}{{k} \brace {n}}_{\lambda}c_{k} \Longleftrightarrow c_{n} = \sum\limits_{k = n}^{\infty}(-1)^{k-n}{{k} \brack {n}}_{\lambda}a_{k}. \\ \end{align*}$

2. New approach to $\lambda$ -Stirling numbers

Let

$\begin{equation} \begin{split} \frac{1}{x(x+\lambda)(x+2\lambda) \ \cdots \ (x+k\lambda)} = \sum\limits_{l = 0}^k \frac{A_{l,\lambda}}{x+l \lambda}. \end{split} \end{equation}$

(2.1)

Then, for $0\leq m\leq k$ , we have

$\begin{equation} \begin{split} \lim\limits_{x \rightarrow -m\lambda}(x+m\lambda)\frac{1}{x(x+\lambda)(x+2\lambda) \ \cdots \ (x+k\lambda)} = \lim\limits_{x \rightarrow -m\lambda}(x+m\lambda)\sum\limits_{l = 0}^k \frac{A_{l,\lambda}}{x+l\lambda} = A_{m,\lambda}. \end{split} \end{equation}$

(2.2)

On the other hand, by (2.2), we get

$\begin{equation} \begin{split} \lim\limits_{x \rightarrow -m\lambda}&\frac{x+m\lambda}{x(x+\lambda) \ \cdots \ (x+(m-1)\lambda)(x+m\lambda)(x+(m+1)\lambda) \ \cdots \ (x+k\lambda)}\\ & \quad = \frac{1}{(-m\lambda)\cdot(-m+1)\lambda \ \cdots \ (-\lambda) \ \cdot \ \lambda \ \cdot \ 2\lambda \ \cdots \ (k-m)\lambda}\\ & \quad = \frac{(-1)^m}{{\lambda}^m m! \lambda^{k-m}(k-m)!} = \frac{(-1)^m}{\lambda^k}\frac{1}{k!}\binom{k}{m}. \end{split} \end{equation}$

(2.3)

From (2.2) and (2.3), we note that

$\begin{equation} \begin{split} A_{m,\lambda} = \frac{1}{\lambda^k}\frac{(-1)^m}{k!}\binom{k}{m}. \end{split} \end{equation}$

(2.4)

By (2.1) and (2.4), we get

$\begin{equation} \begin{split} \frac{1}{x(x+\lambda) \ \cdots \ (x+k\lambda)}& = \sum\limits_{l = 0}^k \frac{1}{\lambda^k}\frac{(-1)^l}{k!}\binom{k}{l}\frac{1}{x+l\lambda}\\ & = \frac{\lambda^{-k}}{k!}\sum\limits_{l = 0}^k (-1)^l \binom{k}{l}\sum\limits_{n = 0}^\infty (-1)^n l^n \lambda^n \bigg( \frac{1}{x}\bigg)^{n+1}\\ & = \sum\limits_{n = 0}^\infty \bigg(\frac{(-1)^n}{k!}\sum\limits_{l = 0}^k (-1)^l \binom{k}{l}\lambda^{n-k}l^n\bigg) \bigg(\frac{1}{x}\bigg)^{n+1}\\ & = \sum\limits_{n = 0}^\infty \frac{(-1)^{n-k}}{k!}\bigg(\sum\limits_{l = 0}^k (-1)^{k-l}\binom{k}{l}\lambda^{n-k} l^n\bigg)\bigg(\frac{1}{x}\bigg)^{n+1}\\ & = \sum\limits_{n = k}^\infty (-1)^{n-k} {{n} \brace {k}}_\lambda \bigg(\frac{1}{x}\bigg)^{n+1}, \end{split} \end{equation}$

(2.5)

where we used (1.12).

Therefore, by (2.5), we obtain the following theorem.

Theorem 3. For $k\geq0$ , we have

$\begin{equation} \begin{split} \frac{1}{x(x+\lambda) \ \cdots \ (x+k\lambda)} = \sum\limits_{n = k}^\infty (-1)^{n-k} {{n} \brace {k}}_\lambda \bigg(\frac{1}{x}\bigg)^{n+1}. \end{split} \end{equation}$

(2.6)

Then, from Corollary 2 (e), we obtain the following result, which is the generating function of the unsigned $\lambda$ -Stirling numbers of the first kind in terms of the inverses of the generalized rising factorials.

Corollary 4. For $k\geq0$ , we have

$\begin{equation} \begin{split} \bigg(\frac{1}{x}\bigg)^{k+1}& = \sum\limits_{n = k}^\infty (-1)^{n-k}S_{1,\lambda}(n,k)\frac{1}{x(x+\lambda) \ \cdots \ (x+n\lambda)}\\ & = \sum\limits_{n = k}^\infty {n \brack k}_{\lambda}\frac{1}{x(x+\lambda) \ \cdots \ (x+n\lambda)}. \end{split} \end{equation}$

(2.7)

Remark 5. It is immediate to see that (2.6) is equivalent to the ordinary generating function of the $\lambda$ -Stirling numbers of the second kind given by

$\begin{equation*} \frac{x^k}{(1-\lambda x)(1-2 \lambda x) \cdots (1-k \lambda x)} = \sum\limits_{n = k}^{\infty}{{n}\brace{k}}_{\lambda}x^{n}. \end{equation*}$

On the other hand, the exponential generating function of the $\lambda$ -Stirling numbers of the second kind is shown in Eq (11) of ^[9] to be equal to

$\begin{equation*} \frac{1}{\lambda^{k}}\frac{1}{k!}\big(e^{\lambda t}-1\big)^{k} = \sum\limits_{n = k}^{\infty}{n \brace k}_{\lambda}\frac{t^{n}}{n!}. \end{equation*}$

Let $f$ be a nonnegative real valued function that has value at $\infty$ , which is denoted by $f(\infty)$ .

Then, for $a, b > 0$ , we have

$\begin{equation} \begin{split} \int_{0}^\infty \frac{f(ax)-f(bx)}{x}dx& = \int_{0}^\infty \int_{b}^a \frac{\partial}{\partial t}f(xt)dtdx\\ & = \int_{b}^a \int_{0}^\infty \frac{\partial}{\partial t}f(xt)dxdt = \int_{b}^a \frac{f(\infty)-f(0)}{t}dt\\ & = (f(\infty)-f(0))\log\frac{a}{b}. \end{split} \end{equation}$

(2.8)

The integral in (2.8) is the Frullani's integral, and its evaluation in (2.9) was first published by Cauchy in 1823. The Eq (2.8) is valid for 'nice' functions $f$ ^[1,4,17].

Thus, by (2.8), we get

$\begin{equation} \begin{split} \int_{0}^\infty \frac{f(ax)-f(bx)}{x}dx = (f(\infty)-f(0))\log\frac{a}{b}, \ \ \ [4]. \end{split} \end{equation}$

(2.9)

Let us take $f(x) = e^{-x}$ and $a = 1$ , then we have

$\begin{equation} \begin{split} \int_{0}^\infty \frac{e^{-x}-e^{-bx}}{x}dx = -\log\frac{1}{b} = \log b,\quad (b > 0). \end{split} \end{equation}$

(2.10)

Let $k$ be any positive integer, and let $b, \lambda > 0$ . Then, from (2.10), we note that

$\begin{equation} \begin{split} \sum\limits_{l = 0}^k &\binom{k}{l}(-1)^l \log(b+l\lambda)\\ & = \sum\limits_{l = 0}^k \binom{k}{l}(-1)^l \int_{0}^\infty \frac{e^{-x}-e^{-(b+l\lambda)x}}{x}dx\\ & = \int_{0}^\infty \left\{\sum\limits_{l = 0}^k \binom{k}{l}(-1)^l \frac{e^{-x}}{x} - \sum\limits_{l = 0}^k\binom{k}{l}(-1)^l \frac{e^{-(b+l\lambda)x}}{x}\right\} dx\\ & = \int_{0}^\infty \left\{(1-1)^{k}\frac{e^{-x}}{x} - \sum\limits_{l = 0}^k\binom{k}{l}(-1)^l \frac{e^{-(b+l\lambda)x}}{x}\right\} dx\\ & = -\int_0^\infty \frac{e^{-bx}}{x}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^l e^{-l\lambda x}dx = -\int_{0}^\infty \frac{e^{-bx}}{x}(1-e^{-\lambda x})^k dx. \end{split} \end{equation}$

(2.11)

As $\frac{(1-e^{-\lambda x})^{k}}{x}$ is bounded on $[0, \infty)$ , say by $M$ , we have

$\begin{equation} \left| \int_{0}^\infty \frac{e^{-bx}}{x}(1-e^{-\lambda x})^k dx \right| \le M \int_{0}^{\infty} e^{-bx} dx = \frac{M}{b}, \end{equation}$

(2.12)

and hence the integral $\int_{0}^\infty \frac{e^{-bx}}{x}(1-e^{-\lambda x})^k dx$ is convergent for any $b, \lambda > 0$ and any positive integer $k$ .

Moreover, for any positive integer $k$ and any $\lambda > 0$ , by (2.12) we get

$\begin{equation} \begin{split} \lim\limits_{b \rightarrow \infty}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^l \log(b+l\lambda) = -\lim\limits_{b \rightarrow \infty}\int_0^\infty \frac{e^{-bx}}{x}(1-e^{-\lambda x})^k dx = 0. \end{split} \end{equation}$

(2.13)

From (2.13) and (2.5), for any $\lambda, a > 0$ and any positive integer $k$ , we note that

$\begin{equation} \begin{split} \int_a^\infty \frac{1}{x(x+\lambda) \ \cdots \ (x+k\lambda)}dx& = \lim\limits_{b \rightarrow \infty}\int_a^b \frac{1}{x(x+\lambda) \ \cdots \ (x+k\lambda)}dx\\ & = \frac{\lambda^{-k}}{k!}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^l \lim\limits_{b \rightarrow \infty}\int_a^b \frac{1}{x+l\lambda}dx\\ & = \frac{\lambda^{-k}}{k!}\lim\limits_{b \rightarrow \infty}\sum\limits_{l = 0}^k\binom{k}{l}(-1)^l (\log(b+l\lambda)-\log(a+l\lambda))\\ & = \frac{\lambda^{-k}}{k!}\sum\limits_{l = 0}^k\binom{k}{l}(-1)^{l+1}\log(a+l\lambda). \end{split} \end{equation}$

(2.14)

Thus, by (2.13), we get the following theorem.

Theorem 6. For any $\lambda, a > 0$ and any positive integer $k$ , the following identity holds.

$\begin{equation} \begin{split} \int_a^\infty \frac{1}{x(x+\lambda) \ \cdots \ (x+k\lambda)}dx = \frac{\lambda^{-k}}{k!}\sum\limits_{l = 0}^k\binom{k}{l}(-1)^{l+1}\log(a+l\lambda). \end{split} \end{equation}$

(2.15)

From Corollary 4 and (2.15), for any positive integer $k$ and any $\lambda, a > 0$ , we have

$\begin{equation} \begin{split} \frac{1}{ka^k} = \int_a^\infty \frac{1}{x^{k+1}}dx& = \sum\limits_{n = k}^\infty {n \brack k}_\lambda \int_a^\infty \frac{1}{x(x+\lambda) \ \cdots \ (x+n\lambda)}dx\\ & = \sum\limits_{n = k}^\infty {n \brack k}_\lambda \frac{1}{n! \lambda^n}\sum\limits_{l = 0}^n \binom{n}{l}(-1)^{l+1}\log(a+l\lambda). \end{split} \end{equation}$

(2.16)

We state this as a corollary.

Corollary 7. For any positive integer $k$ and any $\lambda, a > 0$ , the following holds true.

$\begin{equation*} \frac{1}{ka^k} = \sum\limits_{n = k}^\infty {n \brack k}_\lambda \frac{1}{n! \lambda^n}\sum\limits_{l = 0}^n \binom{n}{l}(-1)^{l+1}\log(a+l\lambda). \end{equation*}$

By Theorem 3, we get

$\begin{equation} \begin{split} \bigg(\frac{1}{x}\bigg)^k = \sum\limits_{n = k}^\infty {n \brack k}_\lambda \frac{1}{(x+\lambda)(x+2\lambda)\ \cdots \ (x+n\lambda)}. \end{split} \end{equation}$

(2.17)

Replacing $x$ by $x-\lambda$ and $n$ by $n+1$ , we get

$\begin{equation} \begin{split} \bigg(\frac{1}{x-\lambda}\bigg)^k = \sum\limits_{n = k-1}^\infty {{n+1} \brack k}_\lambda \frac{1}{x(x+\lambda) \ \cdots \ (x+n\lambda)}. \end{split} \end{equation}$

(2.18)

In particular, for $k = 1$ , we have

$\begin{equation} \begin{split} \frac{1}{x-\lambda}& = \sum\limits_{n = 0}^{\infty}\frac{\lambda^{n}n!}{x(x+\lambda) \ \cdots \ (x+n\lambda)} = \sum\limits_{n = 0}^\infty \frac{\lambda^{n}}{\binom{x+(n-1)\lambda}{n}_\lambda (x+n\lambda)}. \end{split} \end{equation}$

(2.19)

3. $\lambda$ -analogues of $r$ -Stirling numbers

Let

$\begin{equation} \begin{split} \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)} = \sum\limits_{l = 0}^k \frac{K_{l,\lambda}}{x+r+l\lambda}. \end{split} \end{equation}$

(3.1)

For $0\leq m\leq k$ , we have

$\begin{equation} \begin{split} K_{m,\lambda}& = \lim\limits_{x \rightarrow {-r-m\lambda}}\sum\limits_{l = 0}^k \frac{K_{l,\lambda}}{x+r+l\lambda}(x+r+m\lambda)\\ & = \lim\limits_{x \rightarrow {-r-m\lambda}}\frac{x+r+m\lambda}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)}\\ & = \frac{(-1)^m}{\lambda^k}\frac{k!}{m!(k-m)!}\frac{1}{k!} = \frac{(-1)^m}{\lambda^k}\frac{1}{k!}\binom{k}{m}. \end{split} \end{equation}$

(3.2)

By (3.1) and (3.2), we get

$\begin{equation} \begin{split} \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)}& = \sum\limits_{l = 0}^k \frac{(-1)^l}{\lambda^k k!}\binom{k}{l}\frac{1}{x+r+l\lambda}\\ & = \frac{1}{\lambda^k k!}\sum\limits_{l = 0}^k(-1)^l\binom{k}{l}\frac{1}{x} \frac{1}{1+\frac{r+l\lambda}{x}}\\ & = \sum\limits_{n = 0}^\infty (-1)^n \frac{1}{\lambda^k k!}\sum\limits_{l = 0}^k(-1)^l\binom{k}{l}(r+l\lambda)^n \bigg(\frac{1}{x}\bigg)^{n+1}\\ & = \sum\limits_{n = 0}^\infty (-1)^{n-k}\bigg(\frac{1}{\lambda^k}\frac{1}{k!}\sum\limits_{l = 0}^k(-1)^{k-l}\binom{k}{l}(r+l\lambda)^n\bigg)\bigg(\frac{1}{x}\bigg)^{n+1}\\ & = \sum\limits_{n = k}^\infty (-1)^{n-k}{{n+r} \brace {k+r}}_{r,\lambda}\bigg(\frac{1}{x}\bigg)^{n+1}, \end{split} \end{equation}$

(3.3)

where we used (1.11).

Therefore, by (3.3), we obtain the following theorem.

Theorem 8. For $k\geq0$ , we have

$\begin{equation} \begin{split} \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)} = \sum\limits_{n = k}^\infty (-1)^{n-k}{{n+r} \brace {k+r}}_{r,\lambda} \bigg(\frac{1}{x}\bigg)^{n+1}. \end{split} \end{equation}$

(3.4)

Now, from Corollary 2 (e) and using the notation in (1.13), we obtain the following result.

Corollary 9. For $k\geq0$ , we have

$\begin{equation*} \begin{split} \bigg(\frac{1}{x}\bigg)^{k+1}& = \sum\limits_{n = k}^\infty {{n+r} \brack {k+r}}_{r,\lambda} \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+n\lambda)}\\ & = \sum\limits_{n = k}^\infty {{n+r} \brack {k+r}}_{r,\lambda}\frac{1}{n!{\binom{x+r+(n-1)\lambda}{n}}_\lambda (x+r+n\lambda)}. \end{split} \end{equation*}$

Remark 10. The Eq (3.4) is equivalent to the ordinary generating function of the $\lambda$ -analogues of $r$ -Stirling numbers of the second kind given by

$\begin{equation*} \frac{x^{k}}{(1-rx)(1-(r+\lambda)x)\cdots(1-(r+k \lambda)x)} = \sum\limits_{n = k}^{\infty}{{n+r} \brace {k+r}}_{r,\lambda}x^{n}. \end{equation*}$

On the other hand, it is shown in Theorem 1 of ^[9] that the exponential generating function of the $\lambda$ -analogues of $r$ -Stirling numbers of the second kind is given by

$\begin{equation*} \frac{1}{\lambda^{k}}\frac{1}{k!}\big(e^{\lambda t}-1\big)^{k}e^{rt} = \sum\limits_{n = k}^{\infty}{n+r \brace k+r}_{r,\lambda}\frac{t^{n}}{n!},\quad (k\ge 0). \end{equation*}$

Let $k$ be any positive integer and let $\lambda, a > 0$ . By using (2.13) and (3.3), we obtain

$\begin{equation} \begin{split} \int_a^\infty \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)}dx = \frac{1}{k!}\frac{1}{\lambda^k}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^{l+1}\log(a+r+l\lambda). \end{split} \end{equation}$

(3.5)

From (3.5) and Corollary 6, we have

$\begin{equation} \begin{split} \frac{1}{ka^k}& = \int_a^\infty \frac{1}{x^{k+1}}dx\\ & = \sum\limits_{n = k}^\infty {{n+r} \brack {k+r}}_{r,\lambda}\frac{1}{n!} \int_a^\infty \frac{1}{n!{\binom{x+r+(n-1)\lambda}{n}}_\lambda (x+r+n\lambda)}dx\\ & = \sum\limits_{n = k}^\infty {{n+r} \brack {k+r}}_{r,\lambda}\frac{1}{n!\lambda^n}\sum\limits_{l = 0}^n \binom{n}{l}(-1)^{l+1}\log(a+r+l\lambda). \end{split} \end{equation}$

(3.6)

Thus we have shown the following theorem.

Theorem 11. For any positive integer $k$ and any $\lambda, a > 0$ , the following hold true.

$\begin{align*} &\int_a^\infty \frac{1}{(x+r)(x+r+\lambda) \ \cdots \ (x+r+k\lambda)}dx = \frac{1}{k!}\frac{1}{\lambda^k}\sum\limits_{l = 0}^k \binom{k}{l}(-1)^{l+1}\log(a+r+l\lambda), \\ &\frac{1}{ka^k} = \sum\limits_{n = k}^\infty {{n+r} \brack {k+r}}_{r,\lambda}\frac{1}{n!\lambda^n}\sum\limits_{l = 0}^n \binom{n}{l}(-1)^{l+1}\log(a+r+l\lambda). \end{align*}$

4. Conclusions

We have witnessed that studying both degenerate versions and $\lambda$ -analogues of some special numbers and polynomials yielded many fascinating results. Here, we note that these two are different in nature. Indeed, the degenerate versions tend to the non-degenerate ones as $\lambda$ tends to zero, while the $\lambda$ -analogues as $\lambda$ tends to one. In this paper, we studied the $\lambda$ -Stirling numbers of both kinds, which are $\lambda$ -analogues of the usual Stirling numbers. We derived new types of generating functions for those numbers as well as for the $\lambda$ -analogues of $r$ -Stirling numbers of both kinds. All of these generating functions are related to reciprocals of the generalized rising factorials. Furthermore, from those new types of generating functions we obtained expressions of the integral over $(a, \infty)$ of the reciprocals of the generalized rising factorials as finite sums and an expression of $\frac{1}{ka^{k}}$ as an infinite sum, involving the $\lambda$ -Stirling numbers of the first kind or the $\lambda$ -analogues of $r$ -Stirling numbers of the first kind.

It is one of our future projects to continue to study various $\lambda$ -analogues of many special numbers and polynomials and to find their applications to physics, science and engineering as well as to mathematics.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to thank the Jangjeon Institute for Mathematical Sciences for supporting this research. This article was supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).

Conflict of interest

The authors declare that there is no conflict of interest.

References

[1]	J. Arias-de Reyna, On the theorem of Frullani, Proc. Amer. Math. Soc., 109 (1990), 165–175.
[2]	H. Belbachir, I. Bousbaa, Translated Whitney and r-Whitney numbers: a combinatorial approach, J. Integer Seq, 16 (2013), Article 13.8.6.
[3]	K. Boyadzhiev, Integrating inverse factorial series and series identities with binomial sums, preprint paper, 2023.
[4]	S. Bravo, I. Gonzalez, K. Kohl, V. H. Moll, Integrals of Frullani type and the method of brackets, Open Math., 15 (2017). https://doi.org/10.1515/math-2017-0001 doi: 10.1515/math-2017-0001
[5]	L. Carlitz, Weighted Stirling numbers of the first and second kind. Ⅱ, Fibonacci Quart., 18 (1980), 242–257.
[6]	L. Carlitz, Weighted Stirling numbers of the first and second kind. Ⅰ, Fibonacci Quart., 18 (1980), 147–162.
[7]	L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Berlin: Springer, 1974.
[8]	S. Araci, U. Duran, M. Acikgoz, On $(q.r, w)$ -Stirling numbers of the second kind, J. Inequal. Spec. Funct. 9 (2018), 9–16.
[9]	D. S. Kim, H. K. Kim, T. Kim, Some identities on $\lambda$ -analogues of $r$ -Stirling numbers of the second kind, Eur. J. Pure Appl. Math., 15 (2022), 1054–1066.
[10]	T. Kim, D. S. Kim, Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys., 30 (2023), 62–75. https://doi.org/10.1134/S1061920823010041 doi: 10.1134/S1061920823010041
[11]	T. Kim, D. S. Kim, Some identities on $\lambda$ -analogues of $r$ -Stirling numbers of the first kind, Filomat, 34 (2020), 451–460. https://doi.org/10.2298/FIL2002451K doi: 10.2298/FIL2002451K
[12]	T. Kim, D. S. Kim, Degenerate $r$ -Whitney numbers and degenerate $r$ -Dowling polynomials via boson operators, Adv. Appl. Math., 140 (2022), 102394. https://doi.org/10.1016/j.aam.2022.102394 doi: 10.1016/j.aam.2022.102394
[13]	T. Kim, D. S. Kim, Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math., 148 (2023), 102535. https://doi.org/10.1016/j.aam.2023.102535 doi: 10.1016/j.aam.2023.102535
[14]	T. Kim, D. S. Kim, Degenerate Whitney numbers of first and second kind of Dowling lattices, Russ. J. Math. Phys., 29 (2022), 358–377. https://doi.org/10.1134/S1061920822030050 doi: 10.1134/S1061920822030050
[15]	T. Kim, D. S. Kim, H. Lee, J. Kwon, Degenerate binomial coefficients and degenerate hypergeometric functions, Adv. Differ. Equ., 2020 (2020), 115. https://doi.org/10.1186/s13662-020-02575-3 doi: 10.1186/s13662-020-02575-3
[16]	M. M. Mangontraum, O. I. Cauntongan, A. M. Dibagulun, A note on the translated Whitney numbers and their $q$ -analogues, Turkish J. Anal. Number Theory, 4 (2016), 74–81. http://dx.doi.org/10.12691/tjant-4-3-4 doi: 10.12691/tjant-4-3-4
[17]	A. M. Ostrowski, On Cauchy-Frullani integrals, Comment. Math. Helvetici, 51 (1976), 57–91.
[18]	L. Ronald, D. E. Knuth, O. Patashnik, Concrete Mathematics, New York: Addison-Wesley, 1994.
[19]	Y. Simsek, Identities and relations related to combinatorial numbers and polynomials, Proc. Jangieon Math. Soc., 20 (2017), 127–135.
[20]	D. S. Kim, T. K. Kim, Normal ordering associated with $\lambda$ -Whitney numbers of the first kind in $\lambda$ -shift algebra, Russ. J. Math. Phys., 30 (2023), 310–319. https://doi.org/10.1134/S1061920823030044 doi: 10.1134/S1061920823030044

This article has been cited by:

Jinyu Wang, Yuankui Ma, Taekyun Kim, Dae San Kim, Probabilistic degenerate Bernstein polynomials, 2025, 33, 2769-0911, 10.1080/27690911.2024.2448191

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1344) PDF downloads(90) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

New approach to $\lambda$ -Stirling numbers

Related Papers:

Abstract

1. Introduction

2. New approach to $\lambda$ -Stirling numbers

3. $\lambda$ -analogues of $r$ -Stirling numbers

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. New approach to $\lambda$ -Stirling numbers

3. $\lambda$ -analogues of $r$ -Stirling numbers

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

AIMS Mathematics

New approach to λ \lambda -Stirling numbers

Related Papers:

Abstract

1. Introduction

2. New approach to λ \lambda -Stirling numbers

3. λ \lambda -analogues of r r -Stirling numbers

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. New approach to λ \lambda -Stirling numbers

3. λ \lambda -analogues of r r -Stirling numbers

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

New approach to $\lambda$ -Stirling numbers

2. New approach to $\lambda$ -Stirling numbers

3. $\lambda$ -analogues of $r$ -Stirling numbers

2. New approach to $\lambda$ -Stirling numbers

3. $\lambda$ -analogues of $r$ -Stirling numbers