# §25.14 Lerch’s Transcendent

## §25.14(i) Definition

 25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$. ⓘ Defines: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent Symbols: $\equiv$: equals by definition, $\Re$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: definition, infinite series, series representation Source: Erdélyi et al. (1953a, (1.11.1), p. 27) Referenced by: (25.14.2), (25.14.3), §25.14(ii), §25.14, 3rd Erratum (V1.0.21) Permalink: http://dlmf.nist.gov/25.14.E1 Encodings: TeX, pMML, png Clarification (effective with 1.0.21): The previous constraint $a\neq 0,-1,-2,\dots,$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\Phi\left(z,s,a\right)$ has been added in the text immediately below. See also: Annotations for §25.14(i), §25.14 and Ch.25

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

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

 25.14.2 $\zeta\left(s,a\right)=\Phi\left(1,s,a\right),$ $\Re s>1$, $a\neq 0,-1,-2,\dots$, ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent, $\Re$: real part, $a$: real or complex parameter and $s$: complex variable Keywords: specialization Source: Derivable from (25.11.1), (25.14.1). Permalink: http://dlmf.nist.gov/25.14.E2 Encodings: TeX, pMML, png See also: Annotations for §25.14(i), §25.14 and Ch.25
 25.14.3 $\mathrm{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right),$ $\Re s>1$, $|z|\leq 1$. ⓘ Symbols: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent, $\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)$: polylogarithm, $\Re$: real part, $s$: complex variable and $z$: complex variable Keywords: specialization Source: Derivable from (25.12.10) and (25.14.1). Referenced by: (25.12.12) Permalink: http://dlmf.nist.gov/25.14.E3 Encodings: TeX, pMML, png See also: Annotations for §25.14(i), §25.14 and Ch.25

## §25.14(ii) Properties

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

 25.14.4 $\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{% n}}{(a+n)^{s}}.$ ⓘ Symbols: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: recurrence Source: Erdélyi et al. (1953a, (1.11.2), p. 27) Permalink: http://dlmf.nist.gov/25.14.E4 Encodings: TeX, pMML, png See also: Annotations for §25.14(ii), §25.14 and Ch.25
 25.14.5 $\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^% {s-1}e^{-ax}}{1-ze^{-x}}\mathrm{d}x,$ $\Re s>0$, $\Re a>0$, $z\in\mathbb{C}\setminus[1,\infty)$.
 25.14.6 $\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{% s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{arctan}% \left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x,$ $\Re s>0$ if $|z|<1$; $\Re s>1$ if $|z|=1,\Re a>0$.

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