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

We prove four explicit continued fraction representations for the Lerch transcendent $Φ(z, s, M+1)$, 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 $ζ(2, M+1)$ and $ζ(3, M+1)$.