Poly-Genocchi polynomials and its applications

Abstract

In this paper, we discussed some new properties on the newly defined family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the first time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is efficient in solving various problems.

Keywords

• poly-genocchi polynomials
• shifted genocchi polynomials sequence
• catalan numbers
• determinant form and recurrence relation
• fractional differential equation