NEW FAMILIES OF SPECIAL NUMBERS FOR COMPUTING NEGATIVE ORDER EULER NUMBERS AND RELATED NUMBERS AND POLYNOMIALS

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ŞİMŞEK Y.

APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS, cilt.12, sa.1, ss.1-35, 2018 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 12 Sayı: 1
  • Basım Tarihi: 2018
  • Doi Numarası: 10.2298/aadm1801001s
  • Dergi Adı: APPLICABLE ANALYSIS AND DISCRETE MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.1-35
  • Anahtar Kelimeler: Fibonacci numbers, Bernoulli numbers, Euler numbers, Central factorial numbers, Array polynomials, Stirling numbers, Generating functions, Functional equations, Binomial coefficients, Combinatorial sum, Q-BERNOULLI NUMBERS, GENERATING-FUNCTIONS, COMBINATORIAL SUMS
  • Akdeniz Üniversitesi Adresli: Evet

Özet

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem [15] "Aufgabe 1088. El. Math., 49 (1994), 126-127". Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.