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arXiv · 2006.09922

A functional identity for Mahler measures of non-tempered polynomials

Detchat Samart
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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 $m(P_{a,c})$.

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Detchat Samart. 2020-06-17. A functional identity for Mahler measures of non-tempered polynomials. https://doi.org/10.1080/10652469.2020.1799357

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