Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving <i>q</i>-Trigonometric Functions with Applications in Computer Modeling

Abstract

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

Keywords

• <i>q</i>-trigonometric functions
• <i>q</i>-exponential functions
• Frobenius–Euler polynomials
• Apostol Frobenius–Euler polynomials
• generating functions
• combinatorial sums