A functional identity for Mahler measures of non-tempered polynomials

TL;DR

This paper derives a new functional identity for Mahler measures of a specific two-parameter polynomial family, extending previous results and linking some evaluations to special values of L-functions and elliptic integrals.

Contribution

It introduces a novel functional identity for Mahler measures of the polynomial family $P_{a,c}$, extending prior work and providing new evaluations and conditions involving elliptic integrals.

Findings

Derived a functional identity for $m(P_{a,c})$

Evaluated Mahler measures at algebraic parameter values

Established conditions linking elliptic integrals to Mahler measures

Abstract

We establish a functional identity for Mahler measures of the two-parametric family $P_{a,c}(x,y)=a(x+1/x)+y+1/y+c$. Our result extends an identity proven in a paper of Lal\'{i}n, Zudilin and Samart. As a by-product, we obtain evaluations of $m(P_{a,c})$ for some algebraic values of $a$ and $c$ in terms of special values of $L$-functions and logarithms. We also give a sufficient condition for validity of a certain identity between the elliptic integrals of the first and the third kind, which implies several identities for $m(P_{a,c})$.