A functional identity for Mahler measures of non-tempered polynomials

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 Lalín, 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 $m(P_{a,c})$.

11 pages

References in corpus (6)

• Regulator of modular units and Mahler measures
• Regulators of Siegel units and applications
• On the Mahler measure of a family of genus 2 curves
• Further explorations of Boyd's conjectures and a conductor 21 elliptic curve
• Mahler measures of a family of non-tempered polynomials and Boyd's conjectures
• One Special Identity between the complete elliptic integrals of the first and the third kind