1 Introduction
As is well known, the Poisson-Charlier polynomials
\(C_{k}(x;a)\) are Sheffer sequences (see [1‐4]) with \(g(t) = e^{a(e^{t}-1)} \) and \(f(t) = a(e^{t}-1)\), which are given by the generating function They satisfy the Sheffer identity where \((x)_{n}\) is the falling factorial (see [5]). Moreover, these polynomials satisfy the recurrence relation The first few polynomials are \(C_{0}(x;a) = 1\), \(C_{1}(x;a) = -\frac{(a-x)}{a}\), \(C_{2}(x;a) = \frac {(a^{2}-x-2ax+x^{2})}{a^{2}}\).
$$\begin{aligned} C(x,t)=e^{-t}(1+t/a)^{x}=\sum_{n\geq0}C_{n}(x;a) \frac{t^{n}}{n!}\quad (a\neq0). \end{aligned}$$
(1)
$$C_{n}(x+y;a)=\sum_{k=0}^{n} \binom{n}{k} a^{k-n}C_{k}(y;a) (x)_{n-k}, $$
$$C_{n+1}(x;a)=a^{-1}xC_{n}(x-1;a)-C_{n}(x;a)\quad \bigl(\mbox{see [5]}\bigr). $$
The actuarial polynomials \(a_{n}^{(\beta)}(x)\) are given by the generating function of Sheffer sequence and the Meixner polynomials of the first kind \(m_{n}(x;\beta,c)\) are also introduced in [5] as follows:
$$\begin{aligned} F(x,t)=e^{\beta t+x(1-e^{t})}=\sum_{n\geq0}a_{n}^{(\beta)}(x) \frac {t^{n}}{n!} \quad\bigl(\mbox{see [5]}\bigr), \end{aligned}$$
(2)
$$\begin{aligned} M(x,t)=\sum_{n\geq0}m_{n}(x; \beta,c)\frac {t^{n}}{n!}=(1-t/c)^{x}(1-x)^{-x-\beta}. \end{aligned}$$
(3)
Anzeige
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 [6‐10]). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by Some interesting identities and properties of the Poisson-Charlier, actuarial, and Meixner polynomials can be derived from umbral calculus (see [11‐13]). 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]).
$$m_{n}(x,\beta,c) = \sum_{k=0}^{n} (-1)^{k}{n \choose k} {x\choose k}k!(x-\beta )_{n-k}c^{-k} \quad\bigl(\mbox{see [5]}\bigr). $$
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
$$a_{i}(N,x)=(x-i+1)a_{i-1}(N-1,x)-a_{i}(N-1,x)\quad (1 \leq i\leq N-1). $$
Anzeige
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 This shows that the generating function \(C^{(N+1)}\) is given by Comparing with \(C^{(N+1)}= (\sum_{i=0}^{N+1} a_{i}(N+1,x)(t+a)^{-i} )C\), we complete the proof. □
$$\begin{aligned} C^{(N+1)}&= \Biggl(-\sum_{i=0}^{N} a_{i}(N,x)i(t+a)^{-i-1} \Biggr)C + \Biggl(\sum _{i=0}^{N} a_{i}(N,x) (t+a)^{-i} \Biggr) \bigl(-1+x(t+a)^{-1}\bigr)C\\ &= \Biggl(\sum_{i=1}^{N+1}(x-i+1)a_{i-1}(N,x) (t+a)^{-i} -\sum_{i=0}^{N} a_{i}(N,x) (t+a)^{-i} \Biggr)C. \end{aligned}$$
$$\begin{aligned} &\Biggl(-a_{0}(N,x)+\sum_{i=1}^{N} \bigl((x-i+1)a_{i-1}(N,x) -a_{i}(N,x) \bigr) (t+a)^{-i}\\ &\quad{}+(x-N)a_{N}(N,x) (t+a)^{-N-1} \Biggr)C. \end{aligned}$$
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 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 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. □
$$a_{i}(N+1,x)=(x-i+1)a_{i-1}(N,x)-a_{i}(N,x),\quad 0\leq i\leq N+1, $$
$$A_{i}(x;t)=\frac{(x+1-i)t}{1+t}A_{i-1}(x) $$
Theorem 3
The linear differential equations
have a solution
\(C(x,t)=e^{-t}(1+t/a)^{x}\), where
\((x)_{i}=x(x-1)\cdots(x+1-i)\)
with
\((x)_{0}=1\).
$$C^{(N)}= \Biggl(\sum_{i=0}^{N}(x)_{i} \binom{N}{i}(-1)^{N-i}(t+a)^{-i} \Biggr)C\quad (n=0,1,\ldots) $$
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 Since \(\frac{1}{(1+t)^{i+1}}=\sum_{j\geq0}\binom{i+j}{i}(-1)^{j}t^{j}\), we obtain By comparing coefficients of \(t^{k}\) we complete the proof. □
$$C^{(N)}= \Biggl(\sum_{i=0}^{N}(x)_{i} \binom{N}{i}(-1)^{N-i}(t+a)^{-i} \Biggr) \sum _{\ell\geq0}C_{\ell}(x;a)\frac{t^{\ell}}{\ell!}. $$
$$C^{(N)}=\sum_{k\geq0}\sum _{i=0}^{N}\sum_{m=0}^{k}(x)_{i} \binom{N}{i} \binom{k}{m}(-1)^{N-i+m}(i+m-1)_{m}a^{-i-m}C_{k-m}(x;a) \frac{t^{k}}{k!}. $$
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 which shows that the generating function \(F^{(N+1)}\) is given by Comparing with \(F^{(N+1)}= (\sum_{i=0}^{N+1} b_{i}(N+1,x)e^{it} )C\), we complete the proof. □
$$\begin{aligned} F^{(N+1)}&= \Biggl(\sum_{i=0}^{N} b_{i}(N,x)ie^{it} \Biggr)F + \Biggl(\sum _{i=0}^{N} b_{i}(N,x)e^{it} \Biggr) \bigl(\beta-xe^{t}\bigr)F\\ &= \Biggl(\sum_{i=0}^{N}( \beta+i)a_{i}(N,x)e^{it} -x\sum_{i=1}^{N+1} b_{i-1}(N,x)e^{it} \Biggr)F, \end{aligned}$$
$$\begin{aligned} \Biggl(\beta b_{0}(N,x)+\sum_{i=1}^{N} \bigl(-xa_{i-1}(N,x) +(\beta+i)b_{i}(N,x) \bigr)e^{it}-xb_{N}(N,x)e^{(N+1)t} \Biggr)F. \end{aligned}$$
Lemma 6
Proof
By Lemma 5 we have that 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 with \(B_{0}(x;t)=\frac{1}{1-\beta t}\). By induction on i we derive that 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 Since \(\frac{1}{(1+t)^{i+1}}=\sum_{j\geq0}\binom{i+j}{i}(-1)^{j}t^{j}\), we obtain that Thus, by finding the coefficients of \(t^{N}\) we complete the proof. □
$$b_{i}(N+1,x)=-xb_{i-1}(N,x)+(\beta+i)b_{i}(N,x),\quad 0 \leq i\leq N+1, $$
$$B_{i}(x;t)=\frac{-xt}{1-(\beta+i)t}B_{i-1}(x) $$
$$B_{i}(x,t)=\frac{(-xt)^{i}}{(1-\beta t)(1-(\beta+1)t)\cdots(1-(\beta +i)t)}=\frac{(-xt)^{i}}{(1-\beta t)^{i+1}}\prod _{j=0}^{i}\frac {1}{1-jt/(1-\beta t)}. $$
$$B_{i}(x,t)=(-x)^{i}\sum_{j\geq i}S(j,i) \frac{t^{j}}{(1-\beta t)^{j+1}}. $$
$$B_{i}(x,t)=(-x)^{i}\sum_{j\geq i} \sum_{\ell\geq0}\binom{j+\ell}{j}\beta ^{\ell}S(j,i)t^{J+\ell}. $$
Theorem 7
The linear differential equations
have a solution
\(F(x,t)=e^{\beta t+x(1-e^{t})}\).
$$F^{(N)}=\sum_{i=0}^{N} \Biggl((-x)^{i}e^{it}\sum_{j=i}^{N} \binom{N-1}{j-1}\beta ^{N-j}S(j,i) \Biggr)F \quad(N=0,1,\ldots) $$
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
\(k,N\geq0\),
where
\(b_{i}(N,x)=(-x)^{i}\sum_{j=i}^{N}\binom{N-1}{j-1}\beta^{N-j}S(j,i)\).
$$a_{N+k}^{(\beta)}(x)=\sum_{i=0}^{N} \sum_{m=0}^{k}b_{i}(N;x) \binom {k}{m}i^{k-m}a_{m}^{(\beta)}(x), $$
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, By comparing the coefficients of \(t^{N+k}\) we complete the proof. □
$$F^{(N)}=\sum_{k\geq0}\sum _{i=0}^{N}\sum_{m=0}^{k}b_{i}(N,x) \binom {k}{m}i^{k-m}a_{m}^{(\beta)}(x) \frac{t^{k}}{k!}. $$
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
have a solution
\(M=M(x,t)=(1-t/c)^{x}(1-x)^{-x-\beta}\).
$$M^{(N)}= \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)M \quad(N=0,1,\ldots) $$
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 After rearranging the indices of the sums, we obtain This implies and the induction step is completed. □
$$\begin{aligned} &M^{(N+1)}\\ &=\frac{d}{dt} \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)M \\ &\quad= \Biggl\{ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}i \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i-1}(t-c)^{-(N-i)} \Biggr)M \\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}(N-i) \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M \\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr)\\ &\qquad{}\times \bigl(x(t-c)^{-1}-(x+\beta ) (t-1)^{-1}\bigr)M \Biggr\} . \end{aligned}$$
$$\begin{aligned} &M^{(N+1)}\\ &\quad= \Biggl(\sum_{i=1}^{N+1}(-1)^{i}(i-1) \binom {N}{i-1}(x)_{N+1-i}\langle x+\beta\rangle_{i-1}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i+1}(N-i) \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}x(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M\\ &\qquad{}+ \Biggl(\sum_{i=1}^{N+1}(-1)^{i} \binom{N}{i-1}(x)_{N+1-i}(x+\beta)\langle x +\beta\rangle_{i-1}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M. \end{aligned}$$
$$\begin{aligned} M^{(N+1)}= \Biggl(\sum_{i=0}^{N+1}(-1)^{i} \binom{N+1}{i}(x)_{N+1-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N+1-i)} \Biggr)M, \end{aligned}$$
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 Thus, since \((t-c)^{-s}=(-1)^{s}\sum_{\ell\geq0}\binom{s+\ell-1}{\ell}c^{-s-\ell }t^{\ell}\), we obtain Hence, by finding the coefficients of \(t^{k}\) in the generating function \(M^{(N)}\) we complete the proof. □
$$\begin{aligned} M^{(N)}= \Biggl(\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}(t-1)^{-i}(t-c)^{-(N-i)} \Biggr) \sum_{\ell\geq0}m_{\ell}(x;\beta,c) \frac{t^{\ell}}{\ell!}. \end{aligned}$$
$$\begin{aligned} M^{(N)}={}&(-1)^{N}\sum_{i=0}^{N}(-1)^{i} \binom{N}{i}(x)_{N-i}\langle x+\beta\rangle_{i}\\ &{}\times\sum_{\ell\geq0}\sum_{m\geq0} \sum_{n\geq0} \binom{i+\ell-1}{\ell}\binom{N+m-i-1}{m}m_{n}(x; \beta,c)\frac {c^{-N-m+i}t^{\ell+m+n}}{n!}. \end{aligned}$$
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.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.