<i>q</i>-Generalized Tangent Based Hybrid Polynomials

Abstract

In this paper, we incorporate two known polynomials to introduce so-called 2-variable q-generalized tangent based Apostol type Frobenius–Euler polynomials. Next we present a number of properties and formulas for these polynomials such as explicit expressions, series representations, summation formulas, addition formula, q-derivative and q-integral formulas, together with numerous particular cases of the new polynomials and their associated formulas demonstrated in two tables. Further, by using computer-aided programs (for example, Mathematica or Matlab), we draw graphs of some particular cases of the new polynomials, mainly, in order to observe in several angles how zeros of these polynomials are distributed and located. Lastly we provide numerous observations and questions which naturally arise amid the present investigation.

Keywords

• <i>q</i>-calculus
• <i>q</i>-generalized tangent polynomials and numbers
• Apostol type <i>q</i>-Frobenius–Euler polynomials and numbers
• generating function
• distribution of zeros