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.

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 DOI 10.1186/s13660-016-1038-8

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Web End = Linear differential equations for families of polynomials

Taekyun Kim1*, Dae San Kim2, Touk Mansour3 and Jong-Jin Seo4

*Correspondence: [email protected]

1Department of Mathematics, Kwangwoon University, Seoul, 139-701, South KoreaFull list of author information is available at the end of the article

Abstract

In this paper, we present linear dierential 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 Ck(x; a) are Sheer sequences (see []) with g(t) = ea(et) and f (t) = a(et ), which are given by the generating function

C(x, t) = et( + t/a)x = [summationdisplay]

n

Cn(x; a)tnn! (a = ). ()

They satisfy the Sheer identity

Cn(x + y; a) =

n

[summationdisplay]

k=

[parenrightbigg]aknCk(y; a)(x)nk,

where (x)n is the falling factorial (see []). Moreover, these polynomials satisfy the recurrence relation

Cn+(x; a) = axCn(x ; a) Cn(x; a) [parenleftbig]see [][parenrightbig].

The rst few polynomials are C(x; a) = , C(x; a) = (ax)a, C(x; a) = (a

n k

xax+x) a .

The actuarial polynomials a()n(x) are given by the generating function of Sheer sequence

F(x, t) = et+x(et) = [summationdisplay]

n

a()n(x)tnn! [parenleftbig]see [][parenrightbig], ()

and the Meixner polynomials of the rst kind mn(x; , c) are also introduced in [] as follows:

2016 Kim et al. 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.

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 2 of 8

M(x, t) = [summationdisplay]

n

mn(x; , c)tn

n! = ( t/c)x( x)x. ()

In mathematics, Meixner polynomials of the rst kind (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (see []). They are given in terms of binomial coecients and the (rising) Pochhammer symbol by

mn(x, , c) =

n

[summationdisplay]

k=

[parenrightbigg]k!(x )nkck [parenleftbig]see [][parenrightbig].

Some interesting identities and properties of the Poisson-Charlier, actuarial, and Meixner polynomials can be derived from umbral calculus (see []). Kim and Kim [] introduced nonlinear Changhee dierential 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 [, ]).

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

2 Poisson-Charlier polynomials

Recall that the falling polynomials (x)N are dened by (x)N = (x ) (x N + ) for N

with (x) = . For brevity, we denote the generating functions C(x, t) and djdtj C(x; t) by C and C(j) for j .

Lemma The generating function C(N) is given by ([summationtext]N

i= ai(N, x)(t + a)i)C, where

a(N, x) = ()N, aN(N, x) = (x)N, and

ai(N, x) = (x i + )ai(N , x) ai(N , x) ( i N ).

Proof Clearly, a(, x) = . For N = , by () we have C() = ( + x(t + a))C, which proves the lemma for N = (here a(, x) = and a(, x) = x). Assume that C(N) is given by ([summationtext]N

i= ai(N, x)(t + a)i)C. Then

C(N+) = [parenleftBigg]

N

[summationdisplay]

i=

()k[parenleftbigg]n k

[parenrightbigg][parenleftbigg]x k

ai(N, x)i(t + a)i[parenrightBigg]C + [parenleftBigg]

N

i=

[summationdisplay] ai(N, x)(t + a)i[parenrightBigg][parenleftbig] + x(t + a)[parenrightbig]C

= [parenleftBigg]N+[summationdisplay]

i=

ai(N, x)(t + a)i[parenrightBigg]C.

This shows that the generating function C(N+) is given by

[parenleftBigg]a(N, x) +

N

i=

(x i + )ai(N, x)(t + a)i

N

[summationdisplay]

i=

[summationdisplay] (x i + )ai(N, x) ai(N, x)[parenrightbig](t + a)i

+ (x N)aN(N, x)(t + a)N[parenrightBigg]C.

Comparing with C(N+) = ([summationtext]N+

i= ai(N + , x)(t + a)i)C, we complete the proof.

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 3 of 8

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

Lemma For all i N, the coecients ai(N, x) in Lemma are given by

ai(N, x) = (x)i[parenleftbigg]N

i

[parenrightbigg]()Ni.

Proof By Lemma we have that

ai(N + , x) = (x i + )ai(N, x) ai(N, x), i N + ,

with a(, x) = and ai(N, x) = whenever i > N or i < . Dene Ai(x; t) = [summationtext]Ni ai(N, x)tN. Then we have

Ai(x; t) = (x + i)t

+ t Ai(x)

with A(x; t) =

+t . By induction on i we derive that Ai(x, t) =

(x)iti(+t)i+ . Hence, by the fact that

(+t)i+ = [summationtext]j [parenleftbig]i+ji[parenrightbig]()jtj we obtain that ai(N, x) = (x)i[parenleftbig]Ni[parenrightbig]()Ni, as required.

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

Theorem The linear dierential equations

C(N) = [parenleftBigg]

N

[summationdisplay]

i=

[parenrightbigg]()Ni(t + a)i[parenrightBigg]C (n = , , . . .)

have a solution C(x, t) = et( + t/a)x, where (x)i = x(x ) (x + i) with (x) = .

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

Corollary For all k, N ,

Ck+N(x; a) =

N

[summationdisplay]

i=

(x)i[parenleftbigg]N i

[parenrightbigg]()Ni+m(i + m )maimCkm(x; a).

Proof By () and Theorem we have

C(N) = [parenleftBigg]

N

[summationdisplay]

i=

k

[summationdisplay]

m=

(x)i[parenleftbigg]N i

[parenrightbigg][parenleftbigg] k m

(x)i[parenleftbigg]N i

[parenrightbigg]()Ni(t + a)i[parenrightBigg] [summationdisplay]

C (x; a)t

! .

Since

[parenrightbigg]()Ni+m(i + m )maimCkm(x; a)tk

k! .

By comparing coecients of tk we complete the proof.

(+t)i+ = [summationtext]j [parenleftbig]i+ji[parenrightbig]()jtj, we obtain

C(N) = [summationdisplay]

k

N

[summationdisplay]

i=

k

[summationdisplay]

m=

(x)i[parenleftbigg]N i

[parenrightbigg][parenleftbigg] k m

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 4 of 8

3 Actuarial polynomials

For brevity, we denote the generating functions F(x, t) = et+x(et) and d

j

dtj F(x; t) by F and

F(j) for j .

Lemma The generating function F(N) is given by ([summationtext]N

i= bi(N, x)eit)F, where b(N, x) = N, bN(N, x) = (x)N, and bi(N, x) = xbi(N , x) + ( + i)bi(N , x) ( i N ).

Proof Clearly, b(, x) = . For N = , by () we have F() = ( xet)F, which proves the lemma for N = (here b(, x) = and b(, x) = x). Assume that F(N) is given by ([summationtext]N

i= bi(N, x)eit)F. Then

F(N+) = [parenleftBigg]

N

[summationdisplay]

i=

bi(N, x)ieit[parenrightBigg]F + [parenleftBigg]

N

i=

[summationdisplay] bi(N, x)eit[parenrightBigg][parenleftbig] xet[parenrightbig]F

= [parenleftBigg]

bi(N, x)eit[parenrightBigg]F,

which shows that the generating function F(N+) is given by

[parenleftBigg]b(N, x) +

N

i=

N

[summationdisplay]

i=

( + i)ai(N, x)eit x

N+

[summationdisplay]

i=

[summationdisplay] xai(N, x) + ( + i)bi(N, x)[parenrightbig]eit xbN(N, x)e(N+)t[parenrightBigg]F.

Comparing with F(N+) = ([summationtext]N+

i= bi(N + , x)eit)C, we complete the proof.

Lemma For all i N, the coecients bi(N, x) in Lemma are given by

bi(N, x) = (x)i

N

[summationdisplay]

j=i

[parenrightbigg]NjS(j, i),

where S(n, k) are the Stirling numbers (for example, see []) of the second kind.

Proof By Lemma we have that

bi(N + , x) = xbi(N, x) + ( + i)bi(N, x), i N + ,

with b(, x) = and bi(N, x) = whenever i > N or i < . Dene Bi(x; t) = [summationtext]Ni bi(N, x)tN. Then we have

Bi(x; t) = xt

( + i)t Bi(x)

with B(x; t) =

t . By induction on i we derive that

Bi(x, t) = (xt)i

( t)( ( + )t) ( ( + i)t)

= (xt)i

( t)i+

N j

i

[productdisplay]

j=

jt/( t).

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 5 of 8

Hence, since x

(x)(x)(kx) = [summationtext]nk S(n, k)xn (for example, see []), where S(n, k) are the Stirling numbers of the second kind, we obtain that

Bi(x, t) = (x)i [summationdisplay]

ji

S(j, i) tj

k

( t)j+ .

(+t)i+ = [summationtext]j [parenleftbig]i+ji[parenrightbig]()jtj, we obtain that

Bi(x, t) = (x)i [summationdisplay]

ji

[summationdisplay]

Since

[parenrightbigg] S(j, i)tJ+ .

Thus, by nding the coecients of tN we complete the proof.

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

Theorem The linear dierential equations

F(N) =

N

[summationdisplay]

i=

j + j

[parenleftBigg](x)ieit

N

[summationdisplay]

j=i

N j

[parenrightbigg]NjS(j, i)[parenrightBigg]F (N = , , . . .)

have a solution F(x, t) = et+x(et).

Recall that F(x, t) = et+x(et) = [summationtext]n a()n(x)tnn! , which is the generating function for the actuarial polynomials a()n(x) (see ()). As an application of Theorem , we obtain the following corollary.

Corollary For all k, N ,

a()N+k(x) =

N

[summationdisplay]

i=

k

[summationdisplay]

m=

bi(N; x)[parenleftbigg] k m

[parenrightbigg]ikma()m(x),

where bi(N, x) = (x)i [summationtext]N

j=i

N j

NjS(j, i).

Proof By () and Theorem we have F(N) = ([summationtext]N

i= bi(N, x)eit) [summationtext] a() (x)t ! . Thus,

F(N) = [summationdisplay]

k

[parenrightbigg]ikma()m(x)tk k! .

By comparing the coecients of tN+k we complete the proof.

4 Meixner polynomials of the rst kind

Recall that the rising polynomials x N are dened by x N = x(x + ) (x + N ) with x = . For brevity, we denote the generating functions M(x, t) = ( t/c)x( x)x and

djdtj M(x; t) by M and M(j) for j , respectively.

N

[summationdisplay]

i=

k

[summationdisplay]

m=

bi(N, x)[parenleftbigg] k m

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 6 of 8

Theorem The linear dierential equations

M(N) = [parenleftBigg]

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(Ni)

[parenrightBigg]M (N = , , . . .)

have a solution M = M(x, t) = ( t/c)x( x)x.

Proof We proceed the proof by induction on N. Clearly, the theorem holds for N = . By () we have M() = (x(t c) (x + )(t ))M, which proves the theorem for N = .

Assume that the theorem holds for N . Then by the induction hypothesis we have

M(N+)

= d

dt

[parenleftBigg] N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(Ni)

[parenrightBigg]M

= [braceleftBigg][parenleftBigg]

N

[summationdisplay]

i=

()i+i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(Ni)

[parenrightBigg]M

+ [parenleftBigg]

N

[summationdisplay]

i=

()i+(N i)[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(N+i)

[parenrightBigg]M

+ [parenleftBigg]

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(Ni)

[parenrightBigg]

[parenleftbig]x(t c) (x + )(t )[parenrightbig]M

[bracerightBigg].

After rearranging the indices of the sums, we obtain

M(N+)

= [parenleftBigg]N+[summationdisplay]

i=

()i(i )[parenleftbigg] N

i

[parenrightbigg](x)N+i x + i(t )i(t c)(N+i)

[parenrightBigg]M

+ [parenleftBigg]

N

[summationdisplay]

i=

()i+(N i)[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(N+i)

[parenrightBigg]M

+ [parenleftBigg]

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg]x(x)Ni x + i(t )i(t c)(N+i)

[parenrightBigg]M

+ [parenleftBigg]N+[summationdisplay]

i=

()i[parenleftbigg] Ni

[parenrightbigg](x)N+i(x + ) x + i(t )i(t c)(N+i)

[parenrightBigg]M.

This implies

M(N+) = [parenleftBigg]N+[summationdisplay]

i=

()i[parenleftbigg]N + i

[parenrightbigg](x)N+i x + i(t )i(t c)(N+i)

[parenrightBigg]M,

and the induction step is completed.

Kim et al. Journal of Inequalities and Applications (2016) 2016:95 Page 7 of 8

From () we have M(N) = [summationtext]k mk+N(x; , c)tkk! for all N . Similarly to the previous section, we have a recurrence relation for the coecients of mn(x; , c).

Corollary For all k, N ,

mk+N(x; , c) = ()N

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i [summationdisplay]

+m+n=k

k![parenleftbig]i+ [parenrightbig][parenleftbig]N+mi

m

[parenrightbig]

n!cNi+m mn(x;

, c).

Proof By Theorem we have

M(N) = [parenleftBigg]

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i(t )i(t c)(Ni)

[parenrightBigg] [summationdisplay]

m (x; , c)t

! .

Thus, since (t c)s = ()s [summationtext] [parenleftbig]s+ [parenrightbig]cs t , we obtain

M(N) = ()N

N

[summationdisplay]

i=

()i[parenleftbigg]N i

[parenrightbigg](x)Ni x + i

[summationdisplay]

[parenrightbigg]mn(x; , c)cNm+it +m+n n! .

Hence, by nding the coecients of tk 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 dierential 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.

[summationdisplay]

m

[summationdisplay]

n

i +

[parenrightbigg][parenleftbigg]N + m i m

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 nal manuscript.

Author details

1Department of Mathematics, Kwangwoon University, Seoul, 139-701, South Korea. 2Department of Mathematics, Sogang University, Seoul, 121-742, South Korea. 3Department of Mathematics, University of Haifa, Haifa, 3498838, Israel.

4Department of Applied Mathematics, Pukyong National University, Busan, 48513, South Korea.

Acknowledgements

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

Received: 26 January 2016 Accepted: 10 March 2016

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