On degenerate central complete Bell polynomials

TL;DR

This paper introduces and studies degenerate versions of central complete and incomplete Bell polynomials, exploring their properties and identities to extend the mathematical framework of Bell polynomials.

Contribution

It presents new degenerate central Bell polynomials and investigates their properties, expanding the theory of Bell polynomials with a focus on degeneracy and centrality.

Findings

Derived properties of degenerate central Bell polynomials

Established identities relating these polynomials to existing Bell polynomials

Extended the mathematical framework of Bell polynomials with degeneracy and centrality

Abstract

In this paper, we study the degenerate central complete and incomplete Bell polynomials which are degenerate versions of the recently introduced central complete and incomplete Bell polynomials and also central analogues for the degenerate complete and incomplete Bell polynomials. We investigate some properties and identities for these polynomials.

Equations

$$\begin{split}&\frac{1}{k!}\Big{(}\sum_{m=1}^{\infty}x_{m}\big{(}(\frac{1}{2})_{m,\lambda}-(-1)^{m}<\frac{1}{2}>_{m,\lambda}\big{)}\frac{t^{m}}{m!}\Big{)}^{k}\\ &\quad\quad\quad\quad\quad\quad=\sum_{n=k}^{\infty}T_{n,k}(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)\frac{t^{n}}{n!},\,\,\,(\textnormal{see}\,\,\,\eqref{05},\eqref{06}),\end{split}$$

$$\begin{split}\textnormal{exp}\Big{(}&\sum_{i=1}^{\infty}x_{i}\big{(}(\frac{1}{2})_{i,\lambda}-(-1)^{i}<\frac{1}{2}>_{i,\lambda}\big{)}\frac{t^{i}}{i!}\Big{)}\\ &\quad\quad\quad\quad\quad=\sum_{n=0}^{\infty}B_{n}^{(c)}(x_{1},x_{2},\cdots,x_{n}|\lambda)\frac{t^{n}}{n!}.\end{split}$$

$$\begin{split}\frac{1}{k!}\big{(}\log(1+t)\big{)}^{k}=\sum_{n=k}^{\infty}S_{1}(n,k)\frac{t^{n}}{n!},\,\,\,\,(\textnormal{see}\,\,\,[1-17]).\end{split}$$

$$\begin{array}{r}{e}_{\lambda }^x(t)=(1+\lambda t{)}^{\frac{x}{\lambda }},(\text{see}[2,14]).\end{array}$$

$$\begin{split}e^{x\big{(}e_{\lambda}(t)-1\big{)}}=\sum_{n=0}^{\infty}B_{n,\lambda}(x)\frac{t^{n}}{n!},\,\,\,\,(\textnormal{see}\,\,\,[15]),\end{split}$$

$$\begin{split}\frac{1}{k!}\Big{(}\sum_{m=1}^{\infty}x_{m}(1)_{m,\lambda}\frac{t^{m}}{m!}\Big{)}^{k}=\sum_{n=k}^{\infty}B_{n,k}(x_{1},\cdots,x_{n-k+1}|\lambda)\frac{t^{n}}{n!},\end{split}$$

$$\begin{split}(x)_{0,\lambda}=1,\,\,\,\,(x)_{n,\lambda}=x(x-\lambda)\cdots\big{(}x-(n-1)\lambda\big{)},\,\,\,\,(n\geq 1).\end{split}$$

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

$$\begin{split}B_{n,k}&(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)=\sum\frac{n!}{i_{1}!i_{2}!\cdots i_{n-k+1}!}\Big{(}\frac{x_{1}(1)_{1,\lambda}}{1!}\Big{)}^{i_{1}}\\ &\times\Big{(}\frac{x_{2}(1)_{2,\lambda}}{2!}\Big{)}^{i_{2}}\times\cdots\times\Big{(}\frac{x_{n-k+1}(1)_{n-k+1,\lambda}}{(n-k+1)!}\Big{)}^{i_{n-k+1}},\end{split}$$

$$\begin{split}\frac{1}{k!}\big{(}e_{\lambda}(t)-1\big{)}^{k}=\sum_{n=k}^{\infty}S_{2,\lambda}(n,k)\frac{t^{n}}{n!},\,\,\,\,(k\geq 0),\,\,\,\,(\textnormal{see}\,\,\,[8]).\end{split}$$

$$\begin{array}{r}{B}_{n,k}(\underset{n-k+1-times}{\underbrace{1,1,\cdots ,1}}|\lambda )=S_{2,\lambda }(n,k),(n,k\ge 0).\end{array}$$

$$\begin{array}{r}{B}_{n,k}(\alpha x_1,\alpha x_2,\cdots ,\alpha x_{n-k+1}|\lambda )={\alpha }^k{B}_{n,k}(x_1,x_2,\cdots ,x_{n-k+1}|\lambda ),\end{array}$$

$$\begin{array}{r}{B}_{n,k}(\alpha x_1,{\alpha }^2{x}_2,\cdots ,{\alpha }^{n-k+1}{x}_{n-k+1}|\lambda )={\alpha }^n{B}_{n,k}(x_1,x_2,\cdots ,x_{n-k+1}|\lambda ),\end{array}$$

$$\begin{split}\textnormal{exp}\Big{(}\sum_{i=1}^{\infty}x_{i}(1)_{i,\lambda}\frac{t^{i}}{i!}\Big{)}=\sum_{n=0}^{\infty}B_{n}(x_{1},x_{2},\cdots,x_{n}|\lambda)\frac{t^{n}}{n!}.\end{split}$$

$$\begin{array}{r}{B}_n(x_1,x_2,\cdots ,x_n|\lambda )=\sum _{k=0}^n{B}_{n,k}(x_1,x_2,\cdots ,x_{n-k+1}|\lambda ).\end{array}$$

$$\begin{array}{rl}{B}_n(x,x,\cdots ,x|\lambda )& =\sum _{k=0}^n{x}^k{B}_{n,k}(1,1,\cdots ,1|\lambda )\\ & =\sum _{k=0}^n{x}^k{S}_{2,\lambda }(n,k)=B_{n,\lambda }(x),(n\ge 0).\end{array}$$

$$\begin{split}\frac{1}{k!}\big{(}e_{\lambda}^{\frac{1}{2}}(t)-e_{\lambda}^{-\frac{1}{2}}(t)\big{)}^{k}=\sum_{n=k}^{\infty}T_{2,\lambda}(n,k)\frac{t^{n}}{n!},\,\,\,\,(\textnormal{see}\,\,\,[11]),\end{split}$$

$$\begin{split}T_{2,\lambda}(n,k)=\sum_{m=0}^{n}\Big{(}\frac{1}{k!}\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\big{(}l-\frac{k}{2}\big{)}^{m}\Big{)}\lambda^{n-m}S_{1}(n,m),\end{split}$$

$$\begin{split}e^{x\big{(}e_{\lambda}^{\frac{1}{2}}(t)-e_{\lambda}^{-\frac{1}{2}}(t)\big{)}}=\sum_{n=0}^{\infty}B_{n,\lambda}^{(c)}(x)\frac{t^{n}}{n!}.\end{split}$$

$$\begin{array}{r}{B}_{n,\lambda }^{(c)}(x)=\sum _{k=0}^n{T}_{2,\lambda }(n,k)x^k,(n\ge 0).\end{array}$$

$$\begin{split}\frac{1}{k!}&\Big{(}\sum_{m=1}^{\infty}x_{m}\big{(}(\frac{1}{2})_{m,\lambda}-(-1)^{m}<\frac{1}{2}>_{m,\lambda}\big{)}\frac{t^{m}}{m!}\Big{)}^{k}\\ &=\sum_{n=k}^{\infty}T_{n,k}(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)\frac{t^{n}}{n!},\end{split}$$

$$\begin{split}T_{n,k}&(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)=\sum\frac{n!}{i_{1}!i_{2}!\cdots i_{n-k+1}!}\\ &\times\Big{(}\frac{x_{1}\big{(}(\frac{1}{2})_{1,\lambda}+<\frac{1}{2}>_{1,\lambda}\big{)}}{1!}\Big{)}^{i_{1}}\Big{(}\frac{x_{2}\big{(}(\frac{1}{2})_{2,\lambda}-<\frac{1}{2}>_{2,\lambda}\big{)}}{2!}\Big{)}^{i_{2}}\times\cdots\\ &\times\Big{(}\frac{x_{n-k+1}\big{(}(\frac{1}{2})_{n-k+1,\lambda}+<\frac{1}{2}>_{n-k+1,\lambda}\big{)}}{(n-k+1)!}\Big{)}^{i_{n-k+1}},\end{split}$$

$$\begin{split}T_{n,k}&(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)\\ &=B_{n,k}\Big{(}x_{1}\big{(}(\frac{1}{2})_{1,\lambda}+<\frac{1}{2}>_{1,\lambda}\big{)},x_{2}\big{(}(\frac{1}{2})_{2,\lambda}-<\frac{1}{2}>_{2,\lambda}\big{)},\cdots,x_{n-k+1}\\ &\times\big{(}(\frac{1}{2})_{n-k+1,\lambda}+<\frac{1}{2}>_{n-k+1,\lambda}\big{)}\Big{)}.\end{split}$$

$$\begin{split}B_{n,k}&(x_{1},x_{2},\cdots,x_{n-k+1})=\sum\frac{n!}{i_{1}!i_{2}!\cdots i_{n-k+1}!}\Big{(}\frac{x_{1}}{1!}\Big{)}^{i_{1}}\\ &\times\Big{(}\frac{x_{2}}{2!}\Big{)}^{i_{2}}\times\cdots\times\Big{(}\frac{x_{n-k+1}}{(n-k+1)!}\Big{)}^{i_{n-k+1}},\end{split}$$

$$\begin{split}T_{n,k}&(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)=B_{n,k}\Big{(}x_{1}\big{(}(\frac{1}{2})_{1,\lambda}+<\frac{1}{2}>_{1,\lambda}\big{)},x_{2}\big{(}(\frac{1}{2})_{2,\lambda}\\ &-<\frac{1}{2}>_{2,\lambda}\big{)},\cdots,x_{n-k+1}\big{(}(\frac{1}{2})_{n-k+1,\lambda}+<\frac{1}{2}>_{n-k+1,\lambda}\big{)}\Big{)}.\end{split}$$

$$\begin{split}\sum_{n=k}^{\infty}&T_{n,k}(x,x^{2},\cdots,x^{n-k+1}|\lambda)\frac{t^{n}}{n!}\\ &=\frac{1}{k!}\Big{(}x\big{(}(\frac{1}{2})_{1,\lambda}+<\frac{1}{2}>_{1,\lambda}\big{)}t+x^{2}\big{(}(\frac{1}{2})_{2,\lambda}-<\frac{1}{2}>_{2,\lambda}\big{)}\frac{t^{2}}{2!}+\cdots\Big{)}^{k}\\ &=\frac{1}{k!}\big{(}e_{\lambda}^{\frac{1}{2}}(xt)-e_{\lambda}^{-\frac{1}{2}}(xt)\big{)}^{k}=\frac{1}{k!}e_{\lambda}^{-\frac{k}{2}}(xt)\big{(}e_{\lambda}(xt)-1\big{)}^{k}\\ &=\frac{1}{k!}\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}e_{\lambda}^{(l-\frac{k}{2})}(xt)=\frac{1}{k!}\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\\ &\times e^{\frac{1}{\lambda}(l-\frac{k}{2})\log(1+\lambda xt)}\end{split}$$

$$\begin{split}&=\frac{1}{k!}\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\sum_{m=0}^{\infty}\lambda^{-m}\big{(}l-\frac{k}{2}\big{)}^{m}\frac{1}{m!}\big{(}\log(1+\lambda xt)\big{)}^{m}\\ &=\sum_{n=0}^{\infty}\bigg{(}\sum_{m=0}^{n}\Big{(}\frac{x^{n}}{k!}\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\big{(}l-\frac{k}{2}\big{)}^{m}\Big{)}\lambda^{n-m}S_{1}(n,m)\bigg{)}\frac{t^{n}}{n!}.\end{split}$$

$$\begin{split}\sum_{m=0}^{n}\Big{(}\frac{x^{n}}{k!}&\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\big{(}l-\frac{k}{2}\big{)}^{m}\Big{)}\lambda^{n-m}S_{1}(n,m)\\ &=\begin{cases}T_{n,k}(x,x^{2},\cdots,x^{n-k+1}|\lambda),\,\,\,\,{\rm if}\,\,\,\,n\geq k,\\ 0,\,\,\,\,{\rm if}\,\,\,\,n<k.\end{cases}\end{split}$$

$$\begin{split}\sum_{m=0}^{n}\Big{(}\frac{1}{k!}&\sum_{l=0}^{k}{k\choose l}(-1)^{k-l}\big{(}l-\frac{k}{2}\big{)}^{m}\Big{)}\lambda^{n-m}S_{1}(n,m)\\ &=\begin{cases}T_{n,k}(1,1,\cdots,1|\lambda),\,\,\,\,{\rm if}\,\,\,\,n\geq k,\\ 0,\,\,\,\,{\rm if}\,\,\,\,n<k.\end{cases}\end{split}$$

$$\begin{array}{r}{T}_{n,k}(1,1,\cdots ,1|\lambda )=T_{2,\lambda }(n,k).\end{array}$$

Full text

1. Introduction and preliminaries

In this article, we consider the degenerate central incomplete Bell polynomials $T_{n,k}(x_{1},x_{2},\cdots,x_{n-k+1}|\lambda)$ given by

[TABLE]

and the degenerate central complete Bell polynomials $B_{n}^{(c)}(x_{1},x_{2},\cdots,x_{n}|\lambda)$ given by

[TABLE]

and investigate some properties and identities for these polynomials. They are degenerate versions of the central complete and incomplete Bell polynomials. They are also viewed as ’central’ analogues for degenerate complete and incomplete Bell polynomials (see   [9]), which are motivated by (1.4) and (1.10). Before we move on to the next section, we will recall the necessary ingredients that are needed for our discussion in this paper.

For $n\geq 0$, the Stirling numbers of the first kind are given by

[TABLE]

For $\lambda\in\mathbb{R}$, we define the degenerate exponential function as follows:

[TABLE]

Note that $\lim_{\lambda\rightarrow 0}e_{\lambda}^{x}(t)=\lim_{\lambda\rightarrow 0}(1+\lambda t)^{\frac{x}{\lambda}}=e^{xt}$. It is known that the degenerate Bell polynomials (also called degenerate Tochard polynomials or degenerate exponential polynomials) are defined by

[TABLE]

where $e_{\lambda}(t)=e_{\lambda}^{1}(t)$.

When $x=1$, $B_{n,\lambda}=B_{n,\lambda}(1)$ are called the degenerate Bell numbers.

The degenerate incomplete Bell polynomials (also called degenerate partial Bell polynomials) are defined by the generating function (see   [9])

[TABLE]

where $k$ is a non-negative integer and $(x)_{m,\lambda}$ is the degenerate falling factorial sequence given by

[TABLE]

Now, we define the degenerate rising factorial sequence as follows:

[TABLE]

Note that $(-x)_{m,\lambda}=(-1)^{m}<x>_{m,\lambda}$.

From (1.4), we note that

[TABLE]

where the summation is over all integers $i_{1},i_{2},\cdots,i_{n-k+1}\geq 0$, such that $i_{1}+i_{2}+\cdots+i_{n-k+1}=k$ and $i_{1}+2i_{2}+\cdots+(n-k+1)i_{n-k+1}=n$.

It is known that the degenerate Stirling numbers of the second kind are defined by

[TABLE]

From (1.4) and (1.8), we note that

[TABLE]

By (1.7), we easily get

[TABLE]

and

[TABLE]

where $\alpha\in\mathbb{R}$.

The degenerate complete Bell polynomials are defined by

[TABLE]

Then, by (1.4) and (1.10), we get

[TABLE]

From (1.11), we note that

[TABLE]

Recently, the degenerate central factorial numbers of the second kind are defined by

[TABLE]

where $k$ is a non-negative integer.

From (1.13), we have

[TABLE]

where $n,k\in\mathbb{Z}$ with $n\geq k\geq 0$, $(\textnormal{see}\,\,\,[11])$.

The degenerate central Bell polynomials are given by

[TABLE]

Thus, by (1.13) and (1.15), we get

[TABLE]

When $x=1$, $B_{n,\lambda}^{(c)}=B_{n,\lambda}^{(c)}(1)$ are called the degenerate central Bell numbers.

2. On degenerate central complete and incomplete Bell polynomials

In view of (1.4), we consider the degenerate central incomplete Bell polynomials given by

[TABLE]

where $k$ is a non-negative integer.

For $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$, by (2.1), we get

[TABLE]

where the summation is over all integers $i_{1},i_{2},\cdots,i_{n-k+1}\geq 0$ such that $i_{1}+i_{2}+\cdots+i_{n-k+1}=k$ and $i_{1}+2i_{2}+\cdots+(n-k+1)i_{n-k+1}=n$.

From (2.2), we can derive the following equation (2.3).

For $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$, we have

[TABLE]

Here $B_{n,k}(x_{1},x_{2},\cdots,x_{n-k+1})$ are the incomplete Bell polynomials which are defined by

[TABLE]

where the summation is over all integers $i_{1},i_{2},\cdots,i_{n-k+1}\geq 0$ such that $i_{1}+i_{2}+\cdots+i_{n-k+1}=k$ and $i_{1}+2i_{2}+\cdots+(n-k+1)i_{n-k+1}=n$.

Therefore, by (2.3) and (2.4), we obtain the following lemma.

Lemma 2.1**.**

For $n,k\geq 0$ with $n\geq k$ and $n-k\equiv 0$ $(mod\,\,\,2)$, we have

[TABLE]

Let $n,k\geq 0$ with $n\geq k$ with $n-k\equiv 0$ $(mod\,\,\,2)$. Then, by (2.1), we get

[TABLE]

[TABLE]

Therefore, by comparing the coefficients on both sides of (2.5), we obtain the following theorem.

Theorem 2.2**.**

For $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$, we hvae

[TABLE]

In particular,

[TABLE]

For $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$ and $n\geq k$, by (1.13) and (2.1), we get

[TABLE]

Therefore, by (2.6), we obtain the following corollary.

Corollary 2.3**.**

For $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$ and $n\geq k$, we have

[TABLE]

and

[TABLE]

*where the summation is over all integers $i_{1},i_{2},\cdots,i_{n-k+1}\geq 0$ such that $i_{1}+i_{2}+\cdots+i_{n-k+1}=k$ and $i_{1}+2i_{2}+\cdots+(n-k+1)i_{n-k+1}=n$.

For $n,k\geq 0$ with $n\geq k$ and $n-k\equiv 0$ $(mod\,\,\,2)$, we note from (2.5) that

[TABLE]

Thus, by (2.7), we get

[TABLE]

From (2.2), we have

[TABLE]

and

[TABLE]

where $n,k\geq 0$ with $n-k\equiv 0$ $(mod\,\,\,2)$ and $n\geq k$.

Now, we observe that

[TABLE]

In view of (1.10), we define the degenerate central complete Bell polynomials by

[TABLE]

From (2.10) and (2.11), we have

[TABLE]

By (2.6) and (2.12), we get

[TABLE]

From (2.10), we note that

[TABLE]

where the sum is over all nonnegative integers $m_{1},m_{2},\cdots,m_{n}$ such that $m_{1}+2m_{2}+\cdots+nm_{n}=n$.

Now, for $n\in\mathbb{N}$ with $n\equiv 1$ $(mod\,\,\,2)$, by (2.11) and (2.14), we get

[TABLE]

Therefore, by (2.15), we obtain the following theorem.

Theorem 2.4**.**

For $n\in\mathbb{N}$ with $n\equiv 1$ $(mod\,\,\,2)$, we have

[TABLE]

where the sum is over all nonnegative integers $m_{1},m_{2},\cdots,m_{n}$ such that $m_{1}+2m_{2}+\cdots+nm_{n}=n$.

We observe that

[TABLE]

On the other hand,

[TABLE]

Therefore, by (2.16) and (2.17), we obtain the following theorem.

Theorem 2.5**.**

For $n,k\geq 0$ with $n\geq k$, we have

[TABLE]

By Theorem 2.5, we easily get

[TABLE]

Corollary 2.6**.**

For $n\geq 0$, we have

[TABLE]