Variant of Ramanujan polynomials and a conjecture of Maji & Sarkar

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.