New classes of Catalan-type numbers and polynomials with their applications related to p-adic integrals and computational algorithms

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KÜÇÜKOĞLU İ., Simsek B., ŞİMŞEK Y.

TURKISH JOURNAL OF MATHEMATICS, cilt.44, sa.6, ss.2337-2355, 2020 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 44 Sayı: 6
  • Basım Tarihi: 2020
  • Doi Numarası: 10.3906/mat-2008-24
  • Dergi Adı: TURKISH JOURNAL OF MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Sayfa Sayıları: ss.2337-2355
  • Anahtar Kelimeler: Generating function, Bernoulli polynomials, Stirling numbers, Catalan numbers, partial differential equations, computational algorithms, p-adic integral, APOSTOL-TYPE NUMBERS, 2ND KIND, IDENTITIES
  • Akdeniz Üniversitesi Adresli: Evet

Özet

The aim of this paper is to construct generating functions for new classes of Catalan-type numbers and polynomials. Using these functions and their functional equations, we give various new identities and relations involving these numbers and polynomials, the Bernoulli numbers and polynomials, the Stirling numbers of the second kind, the Catalan numbers and other classes of special numbers, polynomials and functions. Some infinite series representations, including the Catalan-type numbers and combinatorial numbers, are investigated. Moreover, some recurrence relations and computational algorithms for these numbers and polynomials are provided. By implementing these algorithms in the Python programming language, we illustrate the Catalan-type numbers and polynomials with their plots under the special conditions. We also give some derivative formulas for these polynomials. Applying the Riemann integral, contour integral, Volkenborn (bosonic p-adic) integral and fermionic p-adic integral to these polynomials, we also derive some integral formulas. With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.