Zeros of Ramanujan Type Entire Functions

TL;DR

This paper investigates the zeros of generalized polynomials and entire functions related to Ramanujan's work, establishing conditions under which these functions have only real zeros, extending classical results in special functions.

Contribution

It introduces new classes of polynomials and entire functions that generalize known q-special functions and proves they have only real zeros, broadening understanding of zero distributions.

Findings

Generalized polynomials have only real zeros.

Entire functions extend Ramanujan's functions with real zeros.

Results include q-Laguerre, q-Bessel, and hypergeometric series generalizations.

Abstract

In this work we establish some polynomials and entire functions have only real zeros. These polynomials generalize q-Laguerre polynomials $L_{n}^{(\alpha)}(x;q)$, while the entire functions are generalizations of Ramanujan's entire function $A_{q}(z)$, q-Bessel functions $J_{\nu}^{(2)}(z;q)$, $J_{\nu}^{(3)}(z;q)$ and confluent basic hypergeometric series.