# Explicit Formulas for Meixner Polynomials

Authors: Dmitry V. Kruchinin; Yuriy V. Shablya
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.


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## Body

## 1. Introduction

There are many authors who have studied polynomials and their properties (see [1–10]). The polynomials are applied in many areas of mathematics, for instance, continued fractions, operator theory, analytic functions, interpolation, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory, combinatorics, stochastic processes, sorting, and data compression.The research area of obtaining explicit formulas for polynomials has received much attention from Srivastava [11, 12], Cenkci [13], Boyadzhiev [14], and Kruchinin [15–17].The main purpose of this paper is to obtain explicit formulas for the Meixner polynomials of the first and second kinds.In this paper we use a method based on a notion of composita, which was presented in [18].Definition 1.
SupposeF(t)=∑n>0f(n)tn is the generating function, in which there is no free term f(0)=0. From this generating function we can write the following condition: (1)Ftk=∑n>0F(n,k)tn.
The expressionF(n,k) iscomposita [19] and it is denoted by FΔ(n,k). Below we show some required rules and operations with compositae.Theorem 2.
SupposeF(t)=∑n>0f(n)tn is the generating function, and FΔ(n,k) is composita of F(t), and α is constant. For the generating function A(t)=αF(t) composita is equal to (2)AΔ(n,k)=αkFΔ(n,k).Theorem 3.
SupposeF(t)=∑n>0f(n)tn is the generating function, and FΔ(n,k) is composita of F(t), and α is constant. For the generating function A(t)=F(αt) composita is equal to (3)AΔ(n,k)=αnFΔ(n,k).Theorem 4.
SupposeF(t)=∑n>0f(n)tn, R(t)=∑n>0r(n)tn are generating functions, and FΔ(n,k), RΔ(n,k) are their compositae. Then for the composition of generating functions A(t)=R(F(t)) composita is equal to (4)AΔ(n,k)=∑m=knFΔ(n,m)GΔ(m,k).Theorem 5.
SupposeF(t)=∑n>0f(n)tn, R(t)=∑n⩾0r(n)tn are generating functions, and FΔ(n,k) is composita of F(t). Then for the composition of generating functions A(t)=R(F(t)) coefficients of generating functions A(t)=∑n≥0a(n)tn are (5)a(n)=∑k=1nFΔ(n,k)r(k),a0=r0.Theorem 6.
SupposeF(t)=∑n>0f(n)tn, R(t)=∑n⩾0r(n)tn are generating functions. Then for the product of generating functions A(t)=F(t)R(t) coefficients of generating functions A(t)=∑n≥0a(n)tn are (6)an=∑k=0nfkrn-k.In this paper we consider an application of this mathematical tool for the Bessel polynomials and the Meixner polynomials of the first and second kinds.


## 2. Bessel Polynomials

Krall and Frink [20] considered a new class of polynomials. Since the polynomials connected with the Bessel function, they called them the Bessel polynomials. The explicit formula for the Bessel polynomials is (7)yn(x)=∑k=0n(n+k)!(n-k)!k!x2k.Then Carlitz [21] defined a class of polynomials associated with the Bessel polynomials by (8)pn(x)=xnyn-11x.Thepn(x) is defined by the explicit formula [22] (9)pn(x)=∑k=1n(2n-k-1)!(k-1)!(n-k)!12n-kxkand by the following generating function: (10)∑n=0∞pn(x)n!tn=ex(1-1-2t).Using the notion of composita, we can obtain an explicit formula (9) from the generating function (10).For the generating functionC(t)=(1-1-4t)/2 composita is given as follows (see [19]): (11)CΔ(n,k)=kn2n-k-1n-1.We representex(1-1-2t) as the composition of generating functions A(B(t)), where (12)A(t)=x(1-1-2t)=2xCt2,B(t)=et=∑n=0∞1n!tn.Then, using rules (2) and (3), we obtain composita of A(t): (13)AΔ(n,k)=2xk12nCΔ(n,k)=2k-nxkkn2n-k-1n-1.Coefficients of the generating functionB(t)=∑n=0∞b(n)tn are equal to (14)b(n)=1n!.Then, using (5), we obtain the explicit formula (15)pn(x)=n!∑k=1nAΔ(n,k)b(k)=∑k=1n(2n-k-1)!(k-1)!(n-k)!2k-nxk,which coincides with the explicit formula (9).


## 3. Meixner Polynomials of the First Kind

The Meixner polynomials of the first kind are defined by the following recurrence relation [23, 24]: (16)mn+1x;β,c=c-1ncn·(x-bn)mn(x;β,c)-λnmn-1(x;β,c),where (17)m0(x;β,c)=1,m1(x;β,c)=x-b0,bn=(1+c)n+βc1-c,λn=cn(n+β-1)1-c2.Using (16), we obtain the first few Meixner polynomials of the first kind: (18)m0(x;β,c)=1;m1(x;β,c)=x(c-1)+βcc;m2(x;β,c)=x2c-12+x((2β+1)c2-2βc-1)+(β+1)βc2c2.The Meixner polynomials of the first kind are defined by the following generating function [22]: (19)∑n=0∞mn(x;β,c)n!tn=1-tcx1-t-x-β.Using the notion of composita, we can obtain an explicit formulamn(x;β,c) from the generating function (19).First, we represent the generating function (19) as a product of generating functions A(t)B(t), where the functions A(t) and B(t) are expanded by binomial theorem: (20)A(t)=1-tcx=∑n=0∞xn-1nc-ntn,B(t)=1-t-x-β=∑n=0∞-x-βn-1ntn.Coefficients of the generating functionsA(t)=∑n=0∞a(n)tn and B(t)=∑n=0∞b(n)tn are, respectively, given as follows: (21)a(n)=xn-1nc-n,b(n)=-x-βn-1n.Then, using (6), we obtain a new explicit formula for the Meixner polynomials of the first kind: (22)mnx;β,c=n!∑k=0na(k)b(n-k)=-1nn!∑k=0nxk-x-βn-kc-k.


## 4. Meixner Polynomials of the Second Kind

The Meixner polynomials of the second kind are defined by the following recurrence relation [23, 24]: (23)Mn+1(x;δ,η)=(x-bn)Mn(x;δ,η)-λnMn-1(x;δ,η),where (24)M0(x;δ,η)=1,M1(x;δ,η)=x-b0,bn=(2n+η)δ,λn=(δ2+1)n(n+η-1).Using (23), we get the first few Meixner polynomials of the second kind: (25)M0(x;δ,η)=1;M1(x;δ,η)=x-δη;M2(x;δ,η)=x2-x2δ(1+η)+η((η+1)δ2-1).The Meixner polynomials of the second kind are defined by the following generating function [22]: (26)∑n=0∞Mn(x;δ,η)n!tn=1+δt2+t2-η/2·exp⁡xtan-1t1+δt.Using the notion of composita, we can obtain an explicit formulaMn(x;δ,η) from the generating function (26).We represent the generating function (26) as a product of generating functions A(t)B(t), where (27)At=1+δt2+t2-η/2At=1+2δt+δ2+1t2-η/2,B(t)=exp⁡xtan-1t1+δt.Next we representA(t) as a composition of generating functions A1(A2(t)) and we expand A1(t) by binomial theorem: (28)A1(t)=1+t-η/2=∑n=0∞-η2ntn,A2(t)=2δt+(δ2+1)t2.Coefficients of generating functionA1(t)=∑n=0∞a1(n)tn are (29)a1(n)=-η2n.The composita for the generating functionax+bx2 is given as follows (see [15]): (30)a2k-nbn-kkn-k.Then composita of the generating functionA2(t) equals (31)A2Δ(n,k)=2δ2k-nδ2+1n-kkn-k.Using (5), we obtain coefficients of the generating function A(t)=∑n=0∞a(n)tn: (32)a(0)=a1(0),a(n)=∑k=1nA2Δ(n,k)a1(k).Therefore, we get the following expression:(33)a(0)=1,a(n)=∑k=1n2δ2k-nδ2+1n-kkn-k-η2k.Next we representB(t) as a composition of generating functions B1(B2(t)), where (34)B1(t)=et=∑n=0∞‍1n!tn,B2(t)=xtan-1t1+δt.We also representB2(t) as a composition of generating functions xC1(C2(t)), where(35)C1(t)=tan-1(t),C2(t)=t1+δt.The composita for the generating functiontan-1(t) is given as follows (see [19]): (36)-1(3n+k)/2+-1(n-k)/2k!2k+1∑j=kn‍2jj!n-1j-1jk,where jk is the Stirling number of the first kind.Then composita of generating functionC1(t) equals (37)C1Δn,k=-1(3n+k)/2+-1(n-k)/2k!2k+1·∑j=kn2jj!n-1j-1jk.The composita for the generating functionat/(1-bt) is given as follows (see [19]): (38)n-1k-1akbn-k.Therefore, composita of generating functionC2(t) is equal to (39)C2Δ(n,k)=n-1k-1-δn-k.Using rules (4) and (2), we obtain composita of generating function B2(t): (40)B2Δ(n,k)=xk∑m=knC2Δ(n,m)C1Δ(m,k).Coefficients of generating functionB1(t)=∑n=0∞b1(n)tn are defined by (41)b1(n)=1n!.Then, using rule (5), we obtain coefficients of generating function B(t)=∑n=0∞b(n)tn: (42)b(0)=b1(0),b(n)=∑k=1nB2Δ(n,k)b1(k).After some transformations, we obtain the following expression:(43)b(0)=1,bn=∑k=1nxkn∑m=knnmδn-m1+-1m+k-1(m+k)/2-nbn=·∑j=km2j-k-1(j-1)!mjjk.Therefore, using (6), we obtain a new explicit formula for the Meixner polynomials of the second kind: (44)Mn(x;δ,η)=n!∑k=0na(k)b(n-k).



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