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arXiv · 2609.08595

Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function

Andrius Grigutis·Yuri Matiyasevich·Lukas Turčinskas
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Abstract

We prove four explicit continued fraction representations for the Lerch transcendent $\Phi(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $\zeta(2, M+1)$ and $\zeta(3, M+1)$.

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Andrius Grigutis, Yuri Matiyasevich, Lukas Turčinskas. 2026-09-08. Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function. https://arxiv.org/abs/2609.08595

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