§25.14 Lerch’s Transcendent

If $s$ is not an integer then ; if $s$ is a positive integer then $a\ne 0,-1,-2,\mathrm{\dots }$; if $s$ is a non-positive integer then $a$ can be any complex number. For other values of $z$, $\mathrm{\Phi }(z,s,a)$ is defined by analytic continuation. This is the notation used in Erdélyi et al. (1953a, p. 27). Lerch (1887) used $𝔎(a,x,s)=\mathrm{\Phi }({\mathrm{e}}^{2\pi \mathrm{i}x},s,a)$.

The Hurwitz zeta function $\zeta (s,a)$ (§25.11) and the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (§25.12(ii)) are special cases:

If $s=m$ a positive integer then

Here the prime signifies that the term for $n=m-1$ is to be omitted. In the case $s=1$ we have

For hypergeometric function $F$ see 15.2(i).

With the conditions of (25.14.1) and $m=1,2,3,\mathrm{\dots }$,

For these and further properties see Erdélyi et al. (1953a, pp. 27–31).

with . This Mellin–Barnes integral representation is used in Olde Daalhuis (2024) to obtain large $\left|z\right|$ asymptotic approximations for $\mathrm{\Phi }(-z,s,a)$. In the special case $s=m$ an integer these asymptotic approximations simplify

The first sum is zero in the case that $m$ is a non-positive integer. In the case that $m$ is a positive integer we have the additional constraint $a\ne 1,2,3,\mathrm{\dots }$. The coefficients $b_n$ are the Taylor coefficients of $\csc \left(\pi (t-a)\right)$ about $t=0$.

The small and large $a$ asymptotics is discussed in Cai and López (2019), Ferreira and López (2004), Paris (2016), and the asymptotics as $\mathrm{\Re }s\to -\mathrm{\infty }$ is discussed in Navas et al. (2013).