Remarks on combinatorial sums associated with special numbers and polynomials with their generating functions

Remarks on combinatorial sums associated with special numbers and polynomials

Abstract

The purpose of this article is to give some novel identities and inequalities associated with combinatorial sums involving special numbers and polynomials. In particular, by using the method of generating functions and their functional equations, we derive not only inequalities, but also some other formulas, identities, and relations for the parametrically generalized polynomials and for other known special numbers and polynomials. Our identities, relations, inequalities and combinatorial sums are related to the Bernoulli numbers and polynomials of negative order, the Euler numbers and polynomials of negative order, the Stirling numbers, the Daehee numbers, the Changhee numbers, the Bernoulli polynomials, the Euler polynomials, the parametrically generalized polynomials, and other well-known special numbers and polynomials. Moreover, using Mathematica with the help of the Wolfram programming language, we illustrate some plots of the parametrically generalized polynomials under some of their randomly selected special conditions. Finally, we give some remarks and observations on our results.