Combinatorial sums and binomial identities associated with the Beta-type polynomials

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ŞİMŞEK Y.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.47, no.5, pp.1144-1155, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 47 Issue: 5
  • Publication Date: 2018
  • Doi Number: 10.15672/hjms.2017.507
  • Journal Name: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.1144-1155
  • Keywords: Combinatorial sums, Binomial identities, Generating functions, Functional equations, Beta polynomials, Beta function, Gamma function, Bernstein basis functions, Catalan numbers, Harmonic numbers, Young inequality, CATALAN NUMBERS, BERNSTEIN TYPE, SERIES
  • Akdeniz University Affiliated: Yes

Abstract

In this paper, we first provide some functional equations of the generating functions for beta-type polynomials. Using these equations, we derive various identities of the beta-type polynomials and the Bernstein basis functions. We then obtain some novel combinatorial identities involving binomial coefficients and combinatorial sums. We also derive some generalizations of the combinatorics identities which are related to the Gould's identities and sum of binomial coefficients. Next, we present some remarks, comments, and formulas including the combinatorial identities, the Catalan numbers, and the harmonic numbers. Moreover, by applying the classical Young inequality, we derive a combinatorial inequality related to beta polynomials and combinatorial sums. We also give another inequality for the Catalan numbers.