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


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KÜÇÜKOĞLU İ., ŞİMŞEK Y.

APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS, cilt.15, sa.1, ss.45-68, 2021 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 15 Sayı: 1
  • Basım Tarihi: 2021
  • Doi Numarası: 10.2298/aadm201017009k
  • Dergi Adı: APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.45-68
  • Anahtar Kelimeler: Generating functions, Combinatorial numbers and polynomials, Special numbers and polynomials, Integral transform, Probability distribution functions, GENERATING FUNCTION, POLYNOMIALS, BERNOULLI
  • Akdeniz Üniversitesi Adresli: Evet

Özet

In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give 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 functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.