Variant of Ramanujan polynomials and a conjecture of Maji & Sarkar

TL;DR

This paper proves a conjecture about the zeros of a two-parameter family of polynomials, showing that all but two real zeros lie on the unit circle, generalizing Ramanujan polynomials.

Contribution

It confirms a conjecture on the zero distribution of generalized Ramanujan polynomials, extending previous results to a broader family.

Findings

All zeros except two are on the unit circle.

The polynomials are reciprocal and generalize Ramanujan polynomials.

The conjecture by Maji & Sarkar is settled.

Abstract

We settle a conjecture proposed by B. Maji and T. Sarkar regarding the location of zeros of a two-parameter family of reciprocal polynomials, $R_{k,\ell}(z)$ for positive integers $k$ and $\ell$. These polynomials are generalizations of Ramanujan polynomials studied by M. R. Murty, C. Smyth, and R. Wang. More specifically, we show that except for two real zeros, all other zeros of $R_{k,\ell}(z)$ lie on the unit circle.