Some Identities Involving Degenerate <i>q</i>-Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

Abstract

This paper intends to define degenerate q-Hermite polynomials, namely degenerate q-Hermite polynomials by means of generating function. Some significant properties of degenerate q-Hermite polynomials such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying the differential equations arising from the generating functions of special numbers and polynomials. Based on the results so far, we find the differential equations for the degenerate q-Hermite polynomials. We also provide some identities for the degenerate q-Hermite polynomials using the coefficients of this differential equation. Finally, we use a computer to view the location of the zeros in degenerate q-Hermite equations. Numerical experiments have confirmed that the roots of the degenerate q-Hermit equations are not symmetric with respect to the imaginary axis.

Keywords

• differential equations
• heat equation
• Hermite polynomials
• degenerate q-Hermite polynomials
• generating functions
• complex zeros