MATRIX REPRESENTATIONS FOR A CERTAIN CLASS OF COMBINATORIAL NUMBERS ASSOCIATED WITH BERNSTEIN BASIS FUNCTIONS AND CYCLIC DERANGEMENTS AND THEIR PROBABILISTIC ANDASYMPTOTIC ANALYSES

Applicable Analysis and Discrete Mathematics

article

Irem Kucukoglu¹  Yilmaz Simsek² 

[1] Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University TR-07425 Antalya;Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya

关键词: Generating functions;    Combinatorial numbers and polynomials;    Special numbers and polynomials;    Integral transform;    Probability distribution functions;   

DOI  :  10.2298/AADM201017009K

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

In this paper, we mainly concerned with an alternate form of the generatingfunctions for a certain class of combinatorial numbers and polynomials. Wegive matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula,recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functionsfor these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, butthe two-variable Hermite polynomials and the number of cyclic derangements.We also present some applications of these relationships. By applying Laplacetransform and Mellin transform respectively to the aforementioned functions,we give not only an infinite series representation, but also an interpolationfunction of these combinatorial numbers. We also provide a contour integralrepresentation of these combinatorial numbers. In addition, we constructexponential generating functions for a new family of numbers arising fromthe linear combination of the numbers of cyclic derangements in the wreathproduct of the finite cyclic group and the symmetric group of permutations ofa set. Finally, we analyse the aforementioned functions in probabilistic andasymptotic manners, and we give some of their relationships with not onlythe Laplace distribution, but also the standard normal distribution. Then,we provide an asymptotic power series representation of the aforementionedexponential generating functions.

【 授权许可】

Unknown