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

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 $\zeta (3,M+1)$.