Several characterizations of bivariate quantum-Hermite-Appell Polynomials and the structure of their zeros

Abstract

This paper investigated the fundamental characteristics and uses of a new class of bivariate quantum-Hermite-Appell polynomials. The series representation and generating relation for these polynomials were derived. Also, a determinant representation for these polynomials was derived. Further, important mathematical characteristics were derived, such as $ q $-recurrence relations and $ q $-difference equations. These polynomials' numerical features were methodically examined, providing information on their computational possibilities and the framework of their zeros. A coherent framework was established by extending the study to related families, such as quantum-Hermite Bernoulli, quantum-Hermite Euler, and quantum-Hermite Genocchi polynomials. These discoveries enhance the knowledge of quantum polynomials and their relationships to classical and contemporary special functions.

Keywords

• special functions
• quantum calculus
• explicit form
• operational connection
• structure of zeros
• determinant form