Some Properties of the Peter-Genocchi Polynomials with Location of Their Zeros

Abstract

Recently, the Changhee-Genocchi polynomials and the Boole-Genocchi polynomials have been considered with their various extensions and many of their applications, and properties have been investigated. Inspired by these developments, in this paper, we introduce the Peter-Genocchi polynomials (or say higher-order Boole-Genocchi polynomials) and then explore some of their fundamental properties and formulas, including some summation formulas, addition formulas, symmetric identities, and an implicit summation formula. Also, for the Peter-Genocchi polynomials, we provide diverse correlations associated with the higher-order Genocchi polynomials, Stirling numbers of both kinds, and higher-order Daehee polynomials. Moreover, we investigate some derivative properties and a differential operator formula for the Peter-Genocchi polynomials. Finally, we provide several graphical representations and a list in a table for certain zero values of the Peter-Genocchi polynomials, enhancing the understanding of the numerical data and facilitating a more intuitive grasp of the concepts discussed.

Keywords

• Genocchi polynomials
• Peter polynomials
• Changhee polynomials
• Stirling numbers of the first kind
• zeros of polynomials

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