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Linear differential equations for families of polynomials

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.

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.

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. □

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. □

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. □

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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Keywords

  • actuarial polynomials
  • Meixner polynomials
  • Poisson-Charlier polynomials