§25.14 Lerch’s Transcendent

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Notes:: See Erdélyi et al. (1953a, Section 1.11).
Permalink:: http://dlmf.nist.gov/25.14
Clarification (effective with 1.0.21):: For (25.14.1), the previous constraint $a\neq 0,-1,-2,\dots,$ was removed and a clarification was added in the text just below.
See also:: Annotations for Ch.25

§25.14(i) Definition
§25.14(ii) Properties

§25.14(i) Definition

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Keywords:: Hurwitz zeta function, Lerch’s transcendent, definition, polylogarithms, relation to Hurwitz zeta function, relation to polylogarithms, relations to other functions
Notes:: See Erdélyi et al. (1953a, Section 1.11).
Referenced by:: Erratum (V1.2.1) for §25.14(i)
Permalink:: http://dlmf.nist.gov/25.14.i
Addition (effective with 1.2.1):: Equations (25.14.3_1), (25.14.3_2), (25.14.3_3) and some text associated with these equations were added.
See also:: Annotations for §25.14 and Ch.25

25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$

; $\Re s>1,|z|=1$ .

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Defines:: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent
Symbols:: $\equiv$ : equals by definition, $\Re$ : real part, : nonnegative integer, : real or complex parameter, : complex variable and : 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, Erratum (V1.0.21) for Equation (25.14.1)
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 is not an integer then $\left|\operatorname{ph}a\right|<\pi$ ; if is a positive integer then $a\neq 0,-1,-2,\dots$ ; if is a non-positive integer then can be any complex number. For other values of , $\Phi\left(z,s,a\right)$ 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 $\operatorname{Li}_{s}\left(z\right)$ (§25.12(ii)) are special cases:

25.14.2 $\zeta\left(s,a\right)=\Phi\left(1,s,a\right),$ $\Re s>1$ , $a\neq 0,-1,-2,\dots$ ,

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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, : real or complex parameter and : complex variable
Keywords:: specialization
Proof sketch:: 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 $\operatorname{Li}_{s}\left(z\right)=z\Phi\left(z,s,1\right),$ $\Re s>1$ , $|z|\leq 1$ .

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Symbols:: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm, $\Re$ : real part, : complex variable and : complex variable
Keywords:: specialization
Proof sketch:: 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

25.14.3_1 ${z^{a}\Phi\left(z,s,a\right)=\Gamma\left(1-s\right)\left(-\ln z\right)^{s-1}+% \sum_{n=0}^{\infty}\zeta\left(s-n,a\right)\frac{\left(\ln z\right)^{n}}{n!}},$ $\left|\ln z\right|<2\pi$ ; $s\not=1,2,3,\ldots$ , $a\not=0,-1,-2,\ldots$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer, : real or complex parameter, : complex variable and : complex variable
Source:: Erdélyi et al. (1953a, (1.11.8), p. 29)
Referenced by:: §25.14(i), Erratum (V1.2.1) for §25.14(i)
Permalink:: http://dlmf.nist.gov/25.14.E3_1
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(i), §25.14 and Ch.25

If a positive integer then

25.14.3_2 ${z^{a}\Phi\left(z,m,a\right)={\sideset{}{{}^{\prime}}{\sum}_{n=0}^{\infty}}% \zeta\left(m-n,a\right)\frac{\left(\ln z\right)^{n}}{n!}+\frac{\left(\ln z% \right)^{m-1}}{(m-1)!}\left(\psi\left(m\right)-\psi\left(a\right)-\ln\left(-% \ln z\right)\right)},$ $\left|\ln z\right|<2\pi$ ; $m=2,3,4,\ldots$ , $a\not=0,-1,-2,\ldots$ .

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer, : nonnegative integer, : real or complex parameter and : complex variable
Source:: Erdélyi et al. (1953a, (1.11.9), p. 30)
Referenced by:: §25.14(i), Erratum (V1.2.1) for §25.14(i)
Permalink:: http://dlmf.nist.gov/25.14.E3_2
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(i), §25.14 and Ch.25

Here the prime signifies that the term for is to be omitted. In the case we have

25.14.3_3 ${a\Phi\left(z,1,a\right)=F\left(a,1;a+1;z\right)},$ $\left|z\right|<1$ .

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, : real or complex parameter and : complex variable
Source:: Erdélyi et al. (1953a, (1.11.10), p. 30)
Referenced by:: §25.14(i), Erratum (V1.2.1) for §25.14(i)
Permalink:: http://dlmf.nist.gov/25.14.E3_3
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(i), §25.14 and Ch.25

For hypergeometric function see 15.2(i).

§25.14(ii) Properties

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Keywords:: Lerch’s transcendent, properties
Notes:: See Erdélyi et al. (1953a, Section 1.11).
Referenced by:: Erratum (V1.2.1) for §25.14(ii)
Permalink:: http://dlmf.nist.gov/25.14.ii
Addition (effective with 1.2.1):: Equations (25.14.7), (25.14.8), (25.14.9) and some text associated with these equations were added.
See also:: Annotations for §25.14 and Ch.25

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

25.14.4 $\Phi\left(z,s,a\right)=z^{m}\Phi\left(z,s,a+m\right)+\sum_{n=0}^{m-1}\frac{z^{% n}}{(a+n)^{s}}.$

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Symbols:: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, : nonnegative integer, : nonnegative integer, : real or complex parameter, : complex variable and : 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 $\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^% {s-1}e^{-ax}}{1-ze^{-x}}\,\mathrm{d}x,$ $\Re s>1$ , $\Re a>0$ if

; $\Re s>0$ , $\Re a>0$ if $z\in\mathbb{C}\setminus[1,\infty)$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $\setminus$ : set subtraction, : real variable, : real or complex parameter, : complex variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Erdélyi et al. (1953a, (1.11.3), p. 27)
Referenced by:: Erratum (V1.1.4) for Equation (25.14.5)
Permalink:: http://dlmf.nist.gov/25.14.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §25.14(ii), §25.14 and Ch.25

25.14.6 $\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{% s}}\,\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x\ln z-s\operatorname{% arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\,\mathrm{d}x,$ $\Re a>0$ if $\left|z\right|<1$ ; $\Re s>1$ , $\Re a>0$ if $\left|z\right|=1$ .

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Symbols:: $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : real variable, : real or complex parameter, : complex variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Erdélyi et al. (1953a, (1.11.4), p. 28)
Notes:: For the case $\left|z\right|<1$ see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\Re s>1$ .
Referenced by:: Erratum (V1.1.4) for Equation (25.14.6)
Permalink:: http://dlmf.nist.gov/25.14.E6
Encodings:: TeX, pMML, png
Clarification (effective with 1.1.4):: The constraint which originally read “ $\Re s>0$ if ; $\Re s>1$ if $|z|=1,\Re a>0$ ” has been improved to be “ $\Re a>0$ if $\left|z\right|<1$ ; $\Re s>1$ , $\Re a>0$ if $\left|z\right|=1$ ”.
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.14(ii), §25.14 and Ch.25

Lerch’s transformation formula

25.14.7 $\Phi\left(z,s,a\right)=\frac{\mathrm{i}\left(2\pi\right)^{s-1}}{z^{a}}\Gamma% \left(1-s\right)\left({\mathrm{e}}^{-\pi\mathrm{i}s/2}\Phi\left({\mathrm{e}}^{% -2\pi\mathrm{i}a},1-s,\frac{\ln z}{2\pi\mathrm{i}}\right)-{\mathrm{e}}^{\pi% \mathrm{i}(2a+(s/2))}\Phi\left({\mathrm{e}}^{2\pi\mathrm{i}a},1-s,1-\frac{\ln z% }{2\pi\mathrm{i}}\right)\right).$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, : real or complex parameter, : complex variable and : complex variable
Source:: Erdélyi et al. (1953a, (1.11.7), p. 29)
Referenced by:: §25.14(ii), Erratum (V1.2.1) for §25.14(ii)
Permalink:: http://dlmf.nist.gov/25.14.E7
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(ii), §25.14 and Ch.25

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

25.14.8 $\Phi\left(-z,s,a\right)=\frac{1}{2\pi\mathrm{i}}\int_{\sigma-\mathrm{i}\infty}% ^{\sigma+\mathrm{i}\infty}\frac{\Gamma\left(1+t\right)\Gamma\left(-t\right)z^{% t}}{\left(a+t\right)^{s}}\,\mathrm{d}t,$ $\left|\arg z\right|<\pi$ , $\Re{a}>0$ ,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, : real or complex parameter, : complex variable and : complex variable
Source:: Olde Daalhuis (2024, (1.5))
Referenced by:: §25.14(ii), Erratum (V1.2.1) for §25.14(ii)
Permalink:: http://dlmf.nist.gov/25.14.E8
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(ii), §25.14 and Ch.25

with $\max(-\Re{a},-1)<\sigma<0$ . This Mellin–Barnes integral representation is used in Olde Daalhuis (2024) to obtain large $\left|z\right|$ asymptotic approximations for $\Phi\left(-z,s,a\right)$ . In the special case an integer these asymptotic approximations simplify

25.14.9 $\Phi\left(-z,m,a\right)=\frac{-\pi}{z^{a}}\sum_{n=0}^{m-1}\frac{b_{n}\left(\ln z% \right)^{m-n-1}}{\Gamma\left(m-n\right)}-\sum_{n=1}^{\infty}\frac{\left(-z% \right)^{-n}}{\left(a-n\right)^{m}},$ $\left|\arg z\right|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$ : Lerch’s transcendent, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer, : nonnegative integer, : real or complex parameter and : complex variable
Source:: Olde Daalhuis (2024, (1.6))
Referenced by:: §25.14(ii), Erratum (V1.2.1) for §25.14(ii)
Permalink:: http://dlmf.nist.gov/25.14.E9
Encodings:: TeX, pMML, png
Addition (effective with 1.2.1):: This equation was added.
See also:: Annotations for §25.14(ii), §25.14 and Ch.25

The first sum is zero in the case that is a non-positive integer. In the case that is a positive integer we have the additional constraint $a\not=1,2,3,\ldots$ . The coefficients $b_{n}$ are the Taylor coefficients of $\csc\left(\pi(t-a)\right)$ about .

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

25.13 Periodic Zeta Function 25.15 Dirichlet

-functions

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§25.14 Lerch’s Transcendent

Contents

§25.14(i) Definition

§25.14(ii) Properties