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.
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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 \(C_{0}(x;a) = 1\), \(C_{1}(x;a) = -\frac{(a-x)}{a}\), \(C_{2}(x;a) = \frac {(a^{2}-x-2ax+x^{2})}{a^{2}}\).
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:
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]).
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
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. □
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
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. □
Thus, by Lemmas 1 and 2 we can state the following result.
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 \((x)_{0}=1\).
As an application of Theorem 3, we obtain the following corollary.
Corollary 4
For all \(k,N\geq0\),
Proof
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. □
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. □
Lemma 6
For all \(0\leq i\leq N\), the coefficients \(b_{i}(N,x)\) in Lemma 5 are given by
where \(S(n,k)\) are the Stirling numbers (for example, see [16]) of the second kind.
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. □
Thus, by Lemmas 5 and 6 we can state the following result.
Theorem 7
The linear differential equations
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 \(k,N\geq0\),
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,
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
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
After rearranging the indices of the sums, we obtain
This implies
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 \(k,N\geq0\),
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. □
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.
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Acknowledgements
The present research has been conducted by the Research Grant of Kwangwoon University in 2016.
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Kim, T., Kim, D.S., Mansour, T. et al. Linear differential equations for families of polynomials. J Inequal Appl 2016, 95 (2016). https://doi.org/10.1186/s13660-016-1038-8
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DOI: https://doi.org/10.1186/s13660-016-1038-8