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

by Daeyeoul Kim 1, Yilmaz Simsek 2 and Ji Suk So 1,*
1
Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Korea
2
Department of Mathematics, Faculty of Science, University of Akdeniz, TR-07058 Antalya, Turkey
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(1), 9; https://doi.org/10.3390/sym12010009
Received: 2 December 2019 / Revised: 16 December 2019 / Accepted: 17 December 2019 / Published: 19 December 2019
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. View Full-Text
MDPI and ACS Style

Kim, D.; Simsek, Y.; So, J.S. Identities and Computation Formulas for Combinatorial Numbers Including Negative Order Changhee Polynomials. Symmetry 2020, 12, 9.