Computational implementations for analyzing the q-analogues of a combinatorial number family by their derivative formulas, interpolation functions and p-adic q-integrals
AbstractThe main purpose of this paper is to provide new computational implementations for describing and analyzing the q-analogues of combinatorial numbers and polynomials which are capable of being an essential tool in solving many scientific and engineering problems. By the provided procedure in the Wolfram language, some members of the aforementioned family are illustrated by their numerical tables and two-and three-dimensional plots. In addition to this computational analysis, we also elaborately investigate some properties pertaining to the aforementioned q-analogues by obtaining their computation formulas, derivative formulas, generating functions and interpolation functions. Especially , by applying the Mellin transform to exponential generating functions for the aforementioned q-analogues, we construct their interpolation functions at negative integers. Moreover, by applying p-adic q-integration to for the afore-mentioned q-analogues, we obtain p-adic q-integral formulas and combinatorial sums involving q-Bernoulli numbers and polynomials. Eventually, this paper is concluded by presenting some comments and observations on the findings.