## Recent Trends in Special Numbers and Special Functions and Polynomials

View this Special IssueResearch Article | Open Access

# Explicit Formulas for Meixner Polynomials

**Academic Editor:**Hari M. Srivastava

#### 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.

#### 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. *Suppose is the generating function, in which there is no free term . From this generating function we can write the following condition: The expression is* composita* [19] and it is denoted by . Below we show some required rules and operations with compositae.

Theorem 2. *Suppose is the generating function, and is composita of , and is constant. For the generating function composita is equal to *

Theorem 3. *Suppose is the generating function, and is composita of , and is constant. For the generating function composita is equal to *

Theorem 4. *Suppose , are generating functions, and , are their compositae. Then for the composition of generating functions composita is equal to *

Theorem 5. *Suppose , are generating functions, and is composita of . Then for the composition of generating functions coefficients of generating functions are *

Theorem 6. *Suppose , are generating functions. Then for the product of generating functions coefficients of generating functions are *

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

Then Carlitz [21] defined a class of polynomials associated with the Bessel polynomials by

The is defined by the explicit formula [22] and by the following generating function:

Using the notion of composita, we can obtain an explicit formula (9) from the generating function (10).

For the generating function composita is given as follows (see [19]):

We represent as the composition of generating functions , where

Then, using rules (2) and (3), we obtain composita of :

Coefficients of the generating function are equal to

Then, using (5), we obtain the explicit formula 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]: where

Using (16), we obtain the first few Meixner polynomials of the first kind:

The Meixner polynomials of the first kind are defined by the following generating function [22]:

Using the notion of composita, we can obtain an explicit formula from the generating function (19).

First, we represent the generating function (19) as a product of generating functions , where the functions and are expanded by binomial theorem:

Coefficients of the generating functions and are, respectively, given as follows:

Then, using (6), we obtain a new explicit formula for the Meixner polynomials of the first kind:

#### 4. Meixner Polynomials of the Second Kind

The Meixner polynomials of the second kind are defined by the following recurrence relation [23, 24]: where

Using (23), we get the first few Meixner polynomials of the second kind:

The Meixner polynomials of the second kind are defined by the following generating function [22]:

Using the notion of composita, we can obtain an explicit formula from the generating function (26).

We represent the generating function (26) as a product of generating functions , where

Next we represent as a composition of generating functions and we expand by binomial theorem:

Coefficients of generating function are

The composita for the generating function is given as follows (see [15]):

Then composita of the generating function equals

Using (5), we obtain coefficients of the generating function :

Therefore, we get the following expression:

Next we represent as a composition of generating functions , where

We also represent as a composition of generating functions , where

The composita for the generating function is given as follows (see [19]): where is the Stirling number of the first kind.

Then composita of generating function equals

The composita for the generating function is given as follows (see [19]):

Therefore, composita of generating function is equal to

Using rules (4) and (2), we obtain composita of generating function :

Coefficients of generating function are defined by

Then, using rule (5), we obtain coefficients of generating function :

After some transformations, we obtain the following expression:

Therefore, using (6), we obtain a new explicit formula for the Meixner polynomials of the second kind:

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was partially supported by the Ministry of Education and Science of Russia, Government Order no. 3657 (TUSUR).

#### References

- R. P. Boas Jr. and R. C. Buck,
*Polynomial Expansions of Analytic Functions*, Springer, 1964. - H. M. Srivastava and H. L. Manocha,
*A Treatise on Generating Functions*, Halsted Press, Ellis Horwood Limited, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1984. - H. M. Srivastava and J. Choi,
*Zeta and Q-Zeta Functions and Associated Series and Integrals*, Elsevier Science Publishers, Amsterdam, The Netherlands, 2012. - H. M. Srivastava, “Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials,”
*Applied Mathematics and Information Sciences*, vol. 5, pp. 390–444, 2011. View at: Google Scholar - Y. Simsek and M. Acikgoz, “A new generating function of (
*q*-) Bernstein-type polynomials and their interpolation function,”*Abstract and Applied Analysis*, vol. 2010, Article ID 769095, 12 pages, 2010. View at: Publisher Site | Google Scholar - H. Ozden, Y. Simsek, and H. M. Srivastava, “A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials,”
*Computers & Mathematics with Applications.*, vol. 60, no. 10, pp. 2779–2787, 2010. View at: Publisher Site | Google Scholar | MathSciNet - R. Dere and Y. Simsek, “Applications of umbral algebra to some special polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 3, pp. 433–438, 2012. View at: Google Scholar | MathSciNet - Y. Simsek, “Complete sum of products of (
*h*,*q*)-extension of Euler polynomials and numbers,”*Journal of Difference Equations and Applications*, vol. 16, no. 11, pp. 1331–1348, 2010. View at: Publisher Site | Google Scholar | MathSciNet - Y. He and S. Araci, “Sums of products of APOstol-Bernoulli and APOstol-Euler polynomials,”
*Advances in Difference Equations*, vol. 2014, article 155, 2014. View at: Publisher Site | Google Scholar | MathSciNet - S. Araci, “Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus,”
*Applied Mathematics and Computation*, vol. 233, pp. 599–607, 2014. View at: Publisher Site | Google Scholar | MathSciNet - H. M. Srivastava and G.-D. Liu, “Explicit formulas for the Norlund polynomials ${B}_{n}^{\left(x\right)}$ and ${b}_{n}^{\left(x\right)}$,”
*Computers & Mathematics with Applications*, vol. 51, pp. 1377–1384, 2006. View at: Google Scholar - H. M. Srivastava and P. G. Todorov, “An explicit formula for the generalized Bernoulli polynomials,”
*Journal of Mathematical Analysis and Applications*, vol. 130, no. 2, pp. 509–513, 1988. View at: Publisher Site | Google Scholar | MathSciNet - M. Cenkci, “An explicit formula for generalized potential polynomials and its applications,”
*Discrete Mathematics*, vol. 309, no. 6, pp. 1498–1510, 2009. View at: Publisher Site | Google Scholar | MathSciNet - K. N. Boyadzhiev, “Derivative polynomials for tanh, tan, sech and sec in explicit form,”
*The Fibonacci Quarterly*, vol. 45, no. 4, pp. 291–303 (2008), 2007. View at: Google Scholar | MathSciNet - D. V. Kruchinin and V. V. Kruchinin, “Application of a composition of generating functions for obtaining explicit formulas of polynomials,”
*Journal of Mathematical Analysis and Applications*, vol. 404, no. 1, pp. 161–171, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. V. Kruchinin and V. V. Kruchinin, “Explicit formulas for some generalized polynomials,”
*Applied Mathematics & Information Sciences*, vol. 7, no. 5, pp. 2083–2088, 2013. View at: Publisher Site | Google Scholar | MathSciNet - D. V. Kruchinin, “Explicit formula for generalized Mott polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 24, no. 3, pp. 327–322, 2014. View at: Google Scholar - D. V. Kruchinin and V. V. Kruchinin, “A method for obtaining expressions for polynomials based on a composition of generating functions,” in
*Proceedings of the AIP Conference on Numerical Analysis and Applied Mathematics (ICNAAM '12)*, vol. 1479, pp. 383–386, 2012. View at: Google Scholar - V. V. Kruchinin and D. V. Kruchinin, “Composita and its properties,”
*Journal of Analysis and Number Theory*, vol. 2, no. 2, pp. 37–44, 2014. View at: Google Scholar - H. L. Krall and O. Frink, “A new class of orthogonal polynomials: the Bessel polynomials,”
*Transactions of the American Mathematical Society*, vol. 65, pp. 100–115, 1949. View at: Publisher Site | Google Scholar | MathSciNet - L. Carlitz, “A note on the Bessel polynomials,”
*Duke Mathematical Journal*, vol. 24, pp. 151–162, 1957. View at: Publisher Site | Google Scholar | MathSciNet - S. Roman,
*The Umbral Calculus*, Academic Press, 1984. View at: MathSciNet - G. Hetyei, “Meixner polynomials of the second kind and quantum algebras representing su(1,1),”
*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 466, no. 2117, pp. 1409–1428, 2010. View at: Publisher Site | Google Scholar | MathSciNet - T. S. Chihara,
*An Introduction to Orthogonal Polynomials*, Gordon and Breach Science, 1978. View at: MathSciNet

#### Copyright

Copyright © 2015 Dmitry V. Kruchinin and Yuriy V. Shablya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.