Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials

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Kim D., ŞİMŞEK Y., So J. S.

SYMMETRY-BASEL, vol.12, no.1, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 12 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.3390/sym12010009
  • Journal Name: SYMMETRY-BASEL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, INSPEC, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Keywords: generating function, Bernoulli numbers and polynomials of the second kind, Euler numbers and polynomials, Stirling numbers, Combinatorial numbers and polynomials, Changhee numbers and polynomials, p-adic integrals, APOSTOL-TYPE NUMBERS, GENERATING-FUNCTIONS, DIFFERENTIAL-EQUATIONS, EXPLICIT FORMULAS, FAMILIES, EULER, CONSTRUCTION, BERNOULLI, SUMS
  • Akdeniz University Affiliated: Yes


The purpose of this paper is to construct generating functions for negative order Changhee numbers and polynomials. Using these generating functions with their functional equation, we prove computation formulas for combinatorial numbers and polynomials. These formulas include Euler numbers and polynomials of higher order, Stirling numbers, and negative order Changhee numbers and polynomials. We also give some properties of these numbers and polynomials with their generating functions. Moreover, we give relations among Changhee numbers and polynomials of negative order, combinatorial numbers and polynomials and Bernoulli numbers of the second kind. Finally, we give a partial derivative of an equation for generating functions for Changhee numbers and polynomials of negative order. Using these differential equations, we derive recurrence relations, differential and integral formulas for these numbers and polynomials. We also give p-adic integrals representations for negative order Changhee polynomials including Changhee numbers and Deahee numbers.