İrem KÜÇÜKOĞLU, Burçin ŞİMŞEK, Yılmaz ŞİMŞEK

An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

The aim of this paper is to not only provide a definition of a new family of special numbers and polynomialsof higher-order with their generating functions, but also to investigate their fundamental properties in the spirit ofprobabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities andformulas involving the Chu–Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions,and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm whichreturns special values of these numbers and polynomials. In addition, we show that our new identities and formulasare connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present sometheoretical and applied details on probabilistic distributions arising from the aforementioned Chu–Vandermonde-typeconvolution formulas.

PDF

___

[1] Apostol TM. On the Lerch zeta function. Pacific Journal of Mathematics 1951; 1: 161-167.
[2] Bona M. Introduction to Enumerative Combinatorics. New York, NY, USA: The McGraw-Hill Companies Inc., 2007.
[3] Boyadzhiev KN. Apostol–Bernoulli functions, derivative polynomials and Eulerian polynomials. arXiv:0710.1124v1, 2007.
[4] Djordjevic GB, Milovanovic GV. Special classes of polynomials. Leskovac, Serbia: University of Nis, Faculty of Technology, 2014.
[5] Gould HW. Inverse series relations and other expansions involving Humbert polynomials. Duke Mathematical Journal 1965; 32 (4): 697-712.
[6] Gould HW. Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series. Available at http://www.math.wvu.edu/~gould/Vol.4.PDF.
[7] Gould HW. Fundamentals of Series: Table III: Basic Algebraic Techniques. Available at http://www.math.wvu. edu/~gould/Vol.3.PDF.
[8] Hilton P, Pedersen J. Catalan numbers, their generalization, and their uses. Mathematical Intelligencer 1991; 13 (2): 64-75.
[9] Johnson NL, Kemp AW, Kotz S. Univariate Discrete Distributions, Third Edition. New Jersey, USA: Wiley Series in Probability and Statistics, 2005.
[10] Jordan C. Calculus of Finite Differences (2nd ed.). New York, NY, USA: Chelsea Publishing Company, 1950.
[11] Khan S, Nahid T, Riyasat M. Partial derivative formulas and identities involving 2-variable Simsek polynomials. Boletín de la Sociedad Matemática Mexicana 2019 ; 1-13. doi: 10.1007/s40590-019-00236-4
[12] Kim DS, Kim T, Seo J. A note on Changhee numbers and polynomials. Advanced Studies in Theoretical Physics 2013; 7: 993-1003.
[13] Kim DS, Kim T. Differential equations associated with degenerate Changhee numbers of the second kind. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2019; 113 (3): 1785-1793.
[14] Kim DS, Kim T. Daehee numbers and polynomials. Applied Mathematical Sciences 2013; 7: 5969-5976.
[15] Koshy T. Catalan numbers with applications. New York, NY, USA: Oxford University Press, 2009.
[16] Lawrence L. Univariate Distribution Relationships. Negative hypergeometric distribution. Available at http://www. math.wm.edu/~leemis/chart/UDR/UDR.html.
[17] Liu GD, Zhang WP. Applications of an explicit formula for the generalized Euler numbers. Acta Mathematica Sinica, English Series 2008; 24 (2): 343-352.
[18] Lorentz GG. Bernstein Polynomials. New York, NY, USA: Chelsea Publishing Comp., 1986.
[19] Luo QM. Apostol–Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwanese Journal of Mathematics 2006; 10: 917-925.
[20] Luo QM. The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order. Integral Transforms and Special Functions 2009; 20: 377-391.
[21] Lu DQ, Srivastava HM. Some series identities involving the generalized Apostol type and related polynomials. Computers & Mathematics with Applications 2011; 62: 3591-3602.
[22] Luo QM, Srivastava HM. Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials.Journal of Mathematical Analysis and Applications 2005; 308: 290-302.
[23] Luo QM, Srivastava HM. Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind. Applied Mathematics and Computation 2011; 217: 5702-5728.
[24] Ozden H, Simsek Y. Modification and unification of the Apostol-type numbers and polynomials and their applications. Applied Mathematics and Computation 2014; 235: 338-351.
[25] Simsek Y. Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory and Applications 2013; 87: 343-355.
[26] Simsek Y. Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers. Mathematical Methods in the Applied Sciences 2015; 38 (14): 3007-3021.
[27] Simsek Y. Combinatorial identities associated with Bernstein type basis functions. Filomat 2016; 30 (7): 1683-1689.
[28] Simsek Y. Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q -integrals. Turkish Journal of Mathematics 2018; 42: 557-577.
[29] Simsek Y, So JS. Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series. Journal of Inequalities and Applications 2019; 62. doi: 10.1186/s13660-019-2006-x
[30] Simsek Y, So JS. On generating functions for Boole type polynomials and numbers of higher order and their applications. Symmetry 2019; 11 (3): 352. doi: 10.3390/sym11030352
[31] Srivastava HM, Manocha HL. A Treatise on Generating Functions. Chichester, UK: Ellis Horwood Limited Publisher, 1984.
[32] Srivastava HM, Kim T, Simsek Y. q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L-series. Russian Journal of Mathematical Physics 2005; 12: 241-268.
[33] Srivastava HM, Choi J. Zeta and q -zeta functions and associated series and integrals. New York, NY, USA: Elsevier Science Publishers, 2012
[34] Srivastava HM, Kucukoglu I, Simsek Y. Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials. Journal of Number Theory 2017; 181: 117- 146.
[35] Sury B. Sum of the reciprocals of the binomial coefficients. European Journal of Combinatorics 1993; 14: 351-353.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv