Open Access

Linear differential equations for families of polynomials

Journal of Inequalities and Applications20162016:95

https://doi.org/10.1186/s13660-016-1038-8

Received: 26 January 2016

Accepted: 10 March 2016

Published: 22 March 2016

Abstract

In this paper, we present linear differential equations for the generating functions of the Poisson-Charlier, actuarial, and Meixner polynomials. Also, we give an application for each case.

Keywords

actuarial polynomials Meixner polynomials Poisson-Charlier polynomials

1 Introduction

As is well known, the Poisson-Charlier polynomials \(C_{k}(x;a)\) are Sheffer sequences (see [14]) with \(g(t) = e^{a(e^{t}-1)} \) and \(f(t) = a(e^{t}-1)\), which are given by the generating function
$$\begin{aligned} C(x,t)=e^{-t}(1+t/a)^{x}=\sum_{n\geq0}C_{n}(x;a) \frac{t^{n}}{n!}\quad (a\neq0). \end{aligned}$$
(1)
They satisfy the Sheffer identity
$$C_{n}(x+y;a)=\sum_{k=0}^{n} \binom{n}{k} a^{k-n}C_{k}(y;a) (x)_{n-k}, $$
where \((x)_{n}\) is the falling factorial (see [5]). Moreover, these polynomials satisfy the recurrence relation
$$C_{n+1}(x;a)=a^{-1}xC_{n}(x-1;a)-C_{n}(x;a)\quad \bigl(\mbox{see [5]}\bigr). $$
The first few polynomials are \(C_{0}(x;a) = 1\), \(C_{1}(x;a) = -\frac{(a-x)}{a}\), \(C_{2}(x;a) = \frac {(a^{2}-x-2ax+x^{2})}{a^{2}}\).
The actuarial polynomials \(a_{n}^{(\beta)}(x)\) are given by the generating function of Sheffer sequence
$$\begin{aligned} F(x,t)=e^{\beta t+x(1-e^{t})}=\sum_{n\geq0}a_{n}^{(\beta)}(x) \frac {t^{n}}{n!} \quad\bigl(\mbox{see [5]}\bigr), \end{aligned}$$
(2)
and the Meixner polynomials of the first kind \(m_{n}(x;\beta,c)\) are also introduced in [5] as follows:
$$\begin{aligned} M(x,t)=\sum_{n\geq0}m_{n}(x; \beta,c)\frac {t^{n}}{n!}=(1-t/c)^{x}(1-x)^{-x-\beta}. \end{aligned}$$
(3)
In mathematics, Meixner polynomials of the first kind (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (see [610]). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by
$$m_{n}(x,\beta,c) = \sum_{k=0}^{n} (-1)^{k}{n \choose k} {x\choose k}k!(x-\beta )_{n-k}c^{-k} \quad\bigl(\mbox{see [5]}\bigr). $$
Some interesting identities and properties of the Poisson-Charlier, actuarial, and Meixner polynomials can be derived from umbral calculus (see [1113]). Kim and Kim [12] introduced nonlinear Changhee differential equations for giving special functions and polynomials. Many researchers have studied the Poisson-Charlier, actuarial and Meixner polynomials in the mathematical physics, combinatorics, and other applied mathematics (for example, see [14, 15]).

In this paper, we study linear differential equations arising from the Poisson-Charlier, actuarial, and Meixner polynomials and derive new recurrence relations for those polynomials from our differential equations.

2 Poisson-Charlier polynomials

Recall that the falling polynomials \((x)_{N}\) are defined by \((x)_{N}=(x-1)\cdots(x-N+1)\) for \(N\geq1\) with \((x)_{0}=1\). For brevity, we denote the generating functions \(C(x,t)\) and \(\frac{d^{j}}{dt^{j}}C(x;t)\) by C and \(C^{(j)}\) for \(j\geq0\).

Lemma 1

The generating function \(C^{(N)}\) is given by \((\sum_{i=0}^{N}a_{i}(N,x)(t+a)^{-i} )C\), where \(a_{0}(N,x)=(-1)^{N}\), \(a_{N}(N,x)=(x)_{N}\), and
$$a_{i}(N,x)=(x-i+1)a_{i-1}(N-1,x)-a_{i}(N-1,x)\quad (1 \leq i\leq N-1). $$

Proof

Clearly, \(a_{0}(0,x)=1\). For \(N=1\), by (1) we have \(C^{(1)}=(-1+x(t+a)^{-1})C\), which proves the lemma for \(N=1\) (here \(a_{0}(1,x)=-1\) and \(a_{1}(1,x)=x\)). Assume that \(C^{(N)}\) is given by \((\sum_{i=0}^{N} a_{i}(N,x)(t+a)^{-i} )C\). Then
$$\begin{aligned} C^{(N+1)}&= \Biggl(-\sum_{i=0}^{N} a_{i}(N,x)i(t+a)^{-i-1} \Biggr)C + \Biggl(\sum _{i=0}^{N} a_{i}(N,x) (t+a)^{-i} \Biggr) \bigl(-1+x(t+a)^{-1}\bigr)C\\ &= \Biggl(\sum_{i=1}^{N+1}(x-i+1)a_{i-1}(N,x) (t+a)^{-i} -\sum_{i=0}^{N} a_{i}(N,x) (t+a)^{-i} \Biggr)C. \end{aligned}$$
This shows that the generating function \(C^{(N+1)}\) is given by
$$\begin{aligned} &\Biggl(-a_{0}(N,x)+\sum_{i=1}^{N} \bigl((x-i+1)a_{i-1}(N,x) -a_{i}(N,x) \bigr) (t+a)^{-i}\\ &\quad{}+(x-N)a_{N}(N,x) (t+a)^{-N-1} \Biggr)C. \end{aligned}$$
Comparing with \(C^{(N+1)}= (\sum_{i=0}^{N+1} a_{i}(N+1,x)(t+a)^{-i} )C\), we complete the proof. □

In order to obtain an explicit formula for the generating function \(C^{(N)}\), we need the following lemma.

Lemma 2

For all \(0\leq i\leq N\), the coefficient‘s \(a_{i}(N,x)\) in Lemma 1 are given by
$$a_{i}(N,x)=(x)_{i}\binom{N}{i}(-1)^{N-i}. $$

Proof

By Lemma 1 we have that
$$a_{i}(N+1,x)=(x-i+1)a_{i-1}(N,x)-a_{i}(N,x),\quad 0\leq i\leq N+1, $$
with \(a_{0}(0,x)=1\) and \(a_{i}(N,x)=0\) whenever \(i>N\) or \(i<0\). Define \(A_{i}(x;t)=\sum_{N\geq i}a_{i}(N,x)t^{N}\). Then we have
$$A_{i}(x;t)=\frac{(x+1-i)t}{1+t}A_{i-1}(x) $$
with \(A_{0}(x;t)=\frac{1}{1+t}\). By induction on i we derive that \(A_{i}(x,t)=\frac{(x)_{i} t^{i}}{(1+t)^{i+1}}\). Hence, by the fact that \(\frac{1}{(1+t)^{i+1}}=\sum_{j\geq0}\binom{i+j}{i}(-1)^{j}t^{j}\) we obtain that \(a_{i}(N,x)=(x)_{i}\binom{N}{i}(-1)^{N-i}\), as required. □

Thus, by Lemmas 1 and 2 we can state the following result.

Theorem 3

The linear differential equations
$$C^{(N)}= \Biggl(\sum_{i=0}^{N}(x)_{i} \binom{N}{i}(-1)^{N-i}(t+a)^{-i} \Biggr)C\quad (n=0,1,\ldots) $$
have a solution \(C(x,t)=e^{-t}(1+t/a)^{x}\), where \((x)_{i}=x(x-1)\cdots(x+1-i)\) with \((x)_{0}=1\).

As an application of Theorem 3, we obtain the following corollary.

Corollary 4

For all \(k,N\geq0\),
$$C_{k+N}(x;a)=\sum_{i=0}^{N}\sum _{m=0}^{k}(x)_{i}\binom{N}{i} \binom{k}{m}(-1)^{N-i+m}(i+m-1)_{m}a^{-i-m}C_{k-m}(x;a). $$

Proof

By (1) and Theorem 3 we have
$$C^{(N)}= \Biggl(\sum_{i=0}^{N}(x)_{i} \binom{N}{i}(-1)^{N-i}(t+a)^{-i} \Biggr) \sum _{\ell\geq0}C_{\ell}(x;a)\frac{t^{\ell}}{\ell!}. $$
Since \(\frac{1}{(1+t)^{i+1}}=\sum_{j\geq0}\binom{i+j}{i}(-1)^{j}t^{j}\), we obtain
$$C^{(N)}=\sum_{k\geq0}\sum _{i=0}^{N}\sum_{m=0}^{k}(x)_{i} \binom{N}{i} \binom{k}{m}(-1)^{N-i+m}(i+m-1)_{m}a^{-i-m}C_{k-m}(x;a) \frac{t^{k}}{k!}. $$
By comparing coefficients of \(t^{k}\) we complete the proof. □

3 Actuarial polynomials

For brevity, we denote the generating functions \(F(x,t)=e^{\beta t+x(1-e^{t})}\) and \(\frac{d^{j}}{dt^{j}}F(x;t)\) by F and \(F^{(j)}\) for \(j\geq0\).

Lemma 5

The generating function \(F^{(N)}\) is given by \((\sum_{i=0}^{N}b_{i}(N,x)e^{it} )F\), where \(b_{0}(N,x)=\beta^{N}\), \(b_{N}(N,x)=(-x)^{N}\), and \(b_{i}(N,x)=-xb_{i-1}(N-1,x)+(\beta+i)b_{i}(N-1,x)\) (\(1\leq i\leq N-1\)).

Proof

Clearly, \(b_{0}(0,x)=1\). For \(N=1\), by (2) we have \(F^{(1)}=(\beta-xe^{t})F\), which proves the lemma for \(N=1\) (here \(b_{0}(1,x)=\beta\) and \(b_{1}(1,x)=-x\)). Assume that \(F^{(N)}\) is given by \((\sum_{i=0}^{N} b_{i}(N,x)e^{it} )F\). Then
$$\begin{aligned} F^{(N+1)}&= \Biggl(\sum_{i=0}^{N} b_{i}(N,x)ie^{it} \Biggr)F + \Biggl(\sum _{i=0}^{N} b_{i}(N,x)e^{it} \Biggr) \bigl(\beta-xe^{t}\bigr)F\\ &= \Biggl(\sum_{i=0}^{N}( \beta+i)a_{i}(N,x)e^{it} -x\sum_{i=1}^{N+1} b_{i-1}(N,x)e^{it} \Biggr)F, \end{aligned}$$
which shows that the generating function \(F^{(N+1)}\) is given by
$$\begin{aligned} \Biggl(\beta b_{0}(N,x)+\sum_{i=1}^{N} \bigl(-xa_{i-1}(N,x) +(\beta+i)b_{i}(N,x) \bigr)e^{it}-xb_{N}(N,x)e^{(N+1)t} \Biggr)F. \end{aligned}$$
Comparing with \(F^{(N+1)}= (\sum_{i=0}^{N+1} b_{i}(N+1,x)e^{it} )C\), we complete the proof. □

Lemma 6

For all \(0\leq i\leq N\), the coefficients \(b_{i}(N,x)\) in Lemma 5 are given by
$$b_{i}(N,x)=(-x)^{i}\sum_{j=i}^{N} \binom{N}{j}\beta^{N-j}S(j,i), $$
where \(S(n,k)\) are the Stirling numbers (for example, see [16]) of the second kind.

Proof

By Lemma 5 we have that
$$b_{i}(N+1,x)=-xb_{i-1}(N,x)+(\beta+i)b_{i}(N,x),\quad 0 \leq i\leq N+1, $$
with \(b_{0}(0,x)=1\) and \(b_{i}(N,x)=0\) whenever \(i>N\) or \(i<0\). Define \(B_{i}(x;t)=\sum_{N\geq i}b_{i}(N,x)t^{N}\). Then we have
$$B_{i}(x;t)=\frac{-xt}{1-(\beta+i)t}B_{i-1}(x) $$
with \(B_{0}(x;t)=\frac{1}{1-\beta t}\). By induction on i we derive that
$$B_{i}(x,t)=\frac{(-xt)^{i}}{(1-\beta t)(1-(\beta+1)t)\cdots(1-(\beta +i)t)}=\frac{(-xt)^{i}}{(1-\beta t)^{i+1}}\prod _{j=0}^{i}\frac {1}{1-jt/(1-\beta t)}. $$
Hence, since \(\frac{x^{k}}{(1-x)(1-2x)\cdots(1-kx)}=\sum_{n\geq k}S(n,k)x^{n}\) (for example, see [16]), where \(S(n,k)\) are the Stirling numbers of the second kind, we obtain that
$$B_{i}(x,t)=(-x)^{i}\sum_{j\geq i}S(j,i) \frac{t^{j}}{(1-\beta t)^{j+1}}. $$
Since \(\frac{1}{(1+t)^{i+1}}=\sum_{j\geq0}\binom{i+j}{i}(-1)^{j}t^{j}\), we obtain that
$$B_{i}(x,t)=(-x)^{i}\sum_{j\geq i} \sum_{\ell\geq0}\binom{j+\ell}{j}\beta ^{\ell}S(j,i)t^{J+\ell}. $$
Thus, by finding the coefficients of \(t^{N}\) we complete the proof. □

Thus, by Lemmas 5 and 6 we can state the following result.

Theorem 7

The linear differential equations
$$F^{(N)}=\sum_{i=0}^{N} \Biggl((-x)^{i}e^{it}\sum_{j=i}^{N} \binom{N-1}{j-1}\beta ^{N-j}S(j,i) \Biggr)F \quad(N=0,1,\ldots) $$
have a solution \(F(x,t)=e^{\beta t+x(1-e^{t})}\).

Recall that \(F(x,t)=e^{\beta t+x(1-e^{t})}=\sum_{n\geq0}a_{n}^{(\beta )}(x)\frac{t^{n}}{n!}\), which is the generating function for the actuarial polynomials \(a_{n}^{(\beta)}(x)\) (see (2)). As an application of Theorem 7, we obtain the following corollary.

Corollary 8

For all \(k,N\geq0\),
$$a_{N+k}^{(\beta)}(x)=\sum_{i=0}^{N} \sum_{m=0}^{k}b_{i}(N;x) \binom {k}{m}i^{k-m}a_{m}^{(\beta)}(x), $$
where \(b_{i}(N,x)=(-x)^{i}\sum_{j=i}^{N}\binom{N-1}{j-1}\beta^{N-j}S(j,i)\).

Proof

By (2) and Theorem 7 we have \(F^{(N)}= (\sum_{i=0}^{N}b_{i}(N,x)e^{it} ) \sum_{\ell\geq0}a_{\ell}^{(\beta)}(x)\frac{t^{\ell}}{\ell!}\). Thus,
$$F^{(N)}=\sum_{k\geq0}\sum _{i=0}^{N}\sum_{m=0}^{k}b_{i}(N,x) \binom {k}{m}i^{k-m}a_{m}^{(\beta)}(x) \frac{t^{k}}{k!}. $$
By comparing the coefficients of \(t^{N+k}\) we complete the proof. □

4 Meixner polynomials of the first kind

Recall that the rising polynomials \(\langle x\rangle_{N}\) are defined by \(\langle x\rangle_{N}=x(x+1)\cdots(x+N-1)\) with \(\langle x\rangle_{0}=1\). For brevity, we denote the generating functions \(M(x,t)=(1-t/c)^{x}(1-x)^{-x-\beta}\) and \(\frac{d^{j}}{dt^{j}}M(x;t)\) by M and \(M^{(j)}\) for \(j\geq0\), respectively.

Theorem 9

The linear differential equations
$$M^{(N)}= \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)M \quad(N=0,1,\ldots) $$
have a solution \(M=M(x,t)=(1-t/c)^{x}(1-x)^{-x-\beta}\).

Proof

We proceed the proof by induction on N. Clearly, the theorem holds for \(N=0\). By (3) we have \(M^{(1)}=(x(t-c)^{-1}-(x+\beta)(t-1)^{-1})M\), which proves the theorem for \(N=1\). Assume that the theorem holds for \(N\geq1\). Then by the induction hypothesis we have
$$\begin{aligned} &M^{(N+1)}\\ &=\frac{d}{dt} \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)M \\ &\quad= \Biggl\{ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}i \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i-1}(t-c)^{-(N-i)} \Biggr)M \\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}(N-i) \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M \\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)\\ &\qquad{}\times \bigl(x(t-c)^{-1}-(x+\beta ) (t-1)^{-1}\bigr)M \Biggr\} . \end{aligned}$$
After rearranging the indices of the sums, we obtain
$$\begin{aligned} &M^{(N+1)}\\ &\quad= \Biggl(\sum_{i=1}^{N+1}(-1)^{i}(i-1) \binom {N}{i-1}(x)_{N+1-i}\langle x+\beta\rangle_{i-1}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}(N-i) \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}x(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=1}^{N+1}(-1)^{i} \binom{N}{i-1}(x)_{N+1-i}(x+\beta)\langle x +\beta\rangle_{i-1}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M. \end{aligned}$$
This implies
$$\begin{aligned} M^{(N+1)}= \Biggl(\sum_{i=0}^{N+1}(-1)^{i} \binom{N+1}{i}(x)_{N+1-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M, \end{aligned}$$
and the induction step is completed. □

From (3) we have \(M^{(N)}=\sum_{k\geq0}m_{k+N}(x;\beta,c)\frac {t^{k}}{k!}\) for all \(N\geq0\). Similarly to the previous section, we have a recurrence relation for the coefficients of \(m_{n}(x;\beta,c)\).

Corollary 10

For all \(k,N\geq0\),
$$\begin{aligned} &m_{k+N}(x;\beta,c)=(-1)^{N}\sum _{i=0}^{N}(-1)^{i}\binom{N}{i}(x)_{N-i} \langle x+\beta\rangle_{i}\sum_{\ell+m+n=k} \frac{k!\binom{i+\ell-1}{\ell}\binom {N+m-i-1}{m}}{n!c^{N-i+m}} m_{n}(x;\beta,c). \end{aligned}$$

Proof

By Theorem 9 we have
$$\begin{aligned} M^{(N)}= \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr) \sum_{\ell\geq0}m_{\ell}(x;\beta,c) \frac{t^{\ell}}{\ell!}. \end{aligned}$$
Thus, since \((t-c)^{-s}=(-1)^{s}\sum_{\ell\geq0}\binom{s+\ell-1}{\ell}c^{-s-\ell }t^{\ell}\), we obtain
$$\begin{aligned} M^{(N)}={}&(-1)^{N}\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}\\ &{}\times\sum_{\ell\geq0}\sum_{m\geq0} \sum_{n\geq0} \binom{i+\ell-1}{\ell}\binom{N+m-i-1}{m}m_{n}(x; \beta,c)\frac {c^{-N-m+i}t^{\ell+m+n}}{n!}. \end{aligned}$$
Hence, by finding the coefficients of \(t^{k}\) in the generating function \(M^{(N)}\) we complete the proof. □

5 Results and discussion

In this paper, the Poisson-Charlier polynomials, actuarial, and Meixner polynomial are introduced. We study linear differential equations arising from the Poisson-Charlier, actuarial, and Meixner polynomials and present some their recurrence relations. Linear differential equations for various families of polynomials are derived. Furthermore, some particular cases of the results are presented.

Declarations

Acknowledgements

The present research has been conducted by the Research Grant of Kwangwoon University in 2016.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University
(2)
Department of Mathematics, Sogang University
(3)
Department of Mathematics, University of Haifa
(4)
Department of Applied Mathematics, Pukyong National University

References

  1. Dieulefait, C: On the Poisson-Charlier series. An. Soc. Ci. Argentina 128, 10-24 (1939) (in Spanish) MathSciNetMATHGoogle Scholar
  2. Karadzhov, GE: Spectral asymptotics for Toeplitz matrices generated by the Poisson-Charlier polynomials. Proc. Am. Math. Soc. 114(1), 129-134 (1992) MathSciNetView ArticleMATHGoogle Scholar
  3. Khan, MA: On some new generating functions for Poisson-Charlier polynomials of several variables. Math. Sci. Res. J. 15(5), 127-136 (2011) MathSciNetMATHGoogle Scholar
  4. Roman, S: The Umbral Calculus. Pure and Applied Mathematics, vol. 111. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1984) MATHGoogle Scholar
  5. Pender, J: A Poisson-Charlier approximation for nonstationary queues. Oper. Res. Lett. 42(4), 293-298 (2014) MathSciNetView ArticleGoogle Scholar
  6. Atakishiyev, NM, Jafarova, AM, Jafarov, EI: Meixner polynomials and representations of the 3D Lorentz group \(SO(2,1)\). Ann. Math. Stat. 17(2), 14-23 (2014) MathSciNetMATHGoogle Scholar
  7. Dominici, D: Mehler-Heine type formulas for Charlier and Meixner polynomials. Ramanujan J. 39(2), 271-289 (2016) MathSciNetView ArticleMATHGoogle Scholar
  8. Gaboriaud, J, Genest, VX, Lemieux, J, Vinet, L: A superintegrable discrete oscillator and two-variable Meixner polynomials. J. Phys. A 48(41), 415202 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Kruchinin, DV, Shablya, YV: Explicit formulas for Meixner polynomials. Int. J. Math. Math. Sci. 2015, Article ID 620569 (2015) MathSciNetView ArticleGoogle Scholar
  10. Miki, H, Tsujimoto, S, Vinet, L, Zhedanov, A: An algebraic model for the multiple Meixner polynomials of the first kind. J. Phys. A 45(32), 325205 (2012) MathSciNetView ArticleMATHGoogle Scholar
  11. Kim, DS, Kim, T, Kwon, HI, Mansour, T: Barnes-type Narumi of the second kind and Poisson-Charlier mixed-type polynomials. J. Comput. Anal. Appl. 19(5), 837-850 (2015) MathSciNetMATHGoogle Scholar
  12. Kim, DS, Kim, T: A note on nonlinear Changhee differential equations. Russ. J. Math. Phys. (to appear) Google Scholar
  13. Kim, T: Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21(1), 36-45 (2014) MathSciNetView ArticleMATHGoogle Scholar
  14. Aptekarev, AI, Tulyakov, DN: The saturation regime of Meixner polynomials and the discrete Bessel kernel. Mat. Zametki 98(1), 147-151 (2015) (in Russian) MathSciNetView ArticleMATHGoogle Scholar
  15. Truesdell, C: A note on the Poisson-Charlier functions. Ann. Math. Stat. 18, 450-454 (1947) MathSciNetView ArticleMATHGoogle Scholar
  16. Mansour, T, Schork, M: Commutation Relations, Normal Ordering and Stirling Numbers. Chapman & Hall/CRC, Boca Raton (2015) View ArticleMATHGoogle Scholar

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