Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers

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ŞİMŞEK Y.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, cilt.40, sa.7, ss.2347-2361, 2017 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 40 Sayı: 7
  • Basım Tarihi: 2017
  • Doi Numarası: 10.1002/mma.4143
  • Dergi Adı: MATHEMATICAL METHODS IN THE APPLIED SCIENCES
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.2347-2361
  • Anahtar Kelimeler: Euler numbers and polynomials, generating functions, Stirling numbers, functional equations, central factorial numbers, array polynomials, binomial coefficients, binomial sum, combinatorial sum, ARRAY TYPE POLYNOMIALS, GENERATING-FUNCTIONS, IDENTITIES, BERNOULLI
  • Akdeniz Üniversitesi Adresli: Evet

Özet

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright (c) 2016 John Wiley & Sons, Ltd.