Variant of Ramanujan polynomials and a conjecture of Maji & Sarkar

2025/07/16 by Mrityunjoy Charan, Charan, Mrityunjoy, Jaban Meher +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.12080

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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,ℓ(z) for positive integers k and ℓ. 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,ℓ(z) lie on the unit circle.

Citations

• Transformation formulas for the higher power of odd zeta values and generalized Eisenstein series
• Period polynomials, derivatives of L-functions, and zeros of polynomials
• Unimodularity of zeros of self-inversive polynomials

Related

• Zeros of Ramanujan-type Polynomials
• Variations of the Ramanujan polynomials and remarks on ζ(2j+1)/π^2j+1
• Unimodularity of zeros of self-inversive polynomials
• Bijections behind the Ramanujan Polynomials
• Zeros of Ramanujan Type Entire Functions