Article ID: MTJPAM-D-20-00000

Title: Interpolation functions for new classes special numbers and polynomials via applications of p-adic integrals and derivative operator


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00000; Volume 3 / Issue 1 / Year 2021, Pages 38-61

Document Type: Research Paper

Author(s): Yilmaz Simsek a

aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058, Antalya-TURKEY

Received: 28 January 2020, Accepted: 16 December 2020, Available online: 7 January 2021.

Corresponding Author: Yilmaz Simsek (Email address: ysimsek@akdeniz.edu.tr)

Full Text: PDF

Abstract

The main purpose of this paper is to not only define Apostol type new classes of numbers and polynomials, but also construct generating function for two new classes of special combinatorial numbers and polynomials by applications of p-adic integrals including the Volkenborn integral and the fermionic integral. By using these generating functions, we introduce not only fundamental properties of these combinatorial numbers and polynomials, but also new identities and formulas. In general, identities and formulas obtained in this paper include the newly introduced combinatorial numbers and polynomials, Bernoulli numbers and polynomials, Euler numbers and polynomials, Apostol-Bernoulli numbers and polynomials, Apostol-Euler numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Changhee numbers, the generalized Eulerian type numbers, Eulerian polynomials, Fubini numbers, Dobinski numbers. Moreover, by applying derivative operator to the generating functions for twonew classes of special combinatorial numbers, we construct interpolation functions for these numbers. We also introduce another zeta-type function which interpolates a special case of one of the newly introduced combinatorial numbers at negative integers. Very interesting results are obtained from these interpolation functions, especially a new combinatorial numbers derived. So, 4 open problems are raised involving these new numbers. Finally, we give conclusions for the results of this paper with somecomments and observations.

Keywords: Generating functions, special numbers and polynomials, Bernoulli-type numbers and polynomials, Euler-type numbers and polynomials, Stirling numbers of the second kind, Daehee numbers, Fubini numbers, p-adic integral

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