Lerch transcendent

In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887.<ref>Template:Citation</ref> The Lerch transcendent, is given by:

<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty

\frac { z^n} {(n+\alpha)^s}</math>.

It only converges for any real number <math>\alpha > 0</math>, where <math>|z| < 1</math>, or <math>\mathfrak{R}(s) > 1</math>, and <math>|z| = 1</math>.Template:Sfn

Special cases

The Lerch transcendent is related to and generalizes various special functions.

The Lerch zeta function is given by:

<math>L(\lambda, s, \alpha) = \sum_{n=0}^\infty

\frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}=\Phi(e^{2\pi i\lambda}, s,\alpha)</math>

The Hurwitz zeta function is the special case<ref name="guillera-sondow-248-249">Template:Harvnb</ref>

<math>\zeta(s,\alpha) = \sum_{n=0}^\infty \frac{1}{(n+\alpha)^s} = \Phi(1,s,\alpha)</math>

The polylogarithm is another special case:<ref name="guillera-sondow-248-249" />

<math>\textrm{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s} =z\Phi(z,s,1)</math>

The Riemann zeta function is a special case of both of the above:<ref name="guillera-sondow-248-249" />

<math>\zeta(s) =\sum_{n=1}^\infty \frac{1}{n^s} = \Phi(1,s,1)</math>

The Dirichlet eta function:<ref name="guillera-sondow-248-249" />

<math>\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} = \Phi(-1,s,1)</math>

The Dirichlet beta function:<ref name="guillera-sondow-248-249" />

<math>\beta(s) = \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)^s} = 2^{-s}\Phi(-1,s,\tfrac12)</math>

The Legendre chi function:<ref name="guillera-sondow-248-249" />

<math>\chi_s(z) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^s}= \frac {z}{2^s} \Phi(z^2,s,\tfrac12)</math>

The inverse tangent integral:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\textrm{Ti}_s(z)= \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)^s}=\frac{z}{2^s}\Phi(-z^2,s,\tfrac12) </math>

The polygamma functions for positive integers n:<ref>The polygamma function has the series representation

<math>\psi^{(m)}(z) = (-1)^{m+1}\, m! \sum_{k=0}^\infty \frac{1}{(z+k)^{m+1}}</math>

which holds for integer values of Template:Math and any complex Template:Mvar not equal to a negative integer.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\psi^{(n)}(\alpha)= (-1)^{n+1} n!\Phi (1,n+1,\alpha)</math>

The Clausen function:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\text{Cl}_2(z)= \frac{ie^{-iz}}2 \Phi(e^{-iz},2,1)-\frac{ie^{iz}}2 \Phi(e^{iz},2,1)</math>

Integral representations

The Lerch transcendent has an integral representation:

<math>

\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> The proof is based on using the integral definition of the gamma function to write

<math>\Phi(z,s,a)\Gamma(s)

= \sum_{n=0}^\infty \frac{z^n}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x} = \sum_{n=0}^\infty \int_0^\infty t^s z^n e^{-(n+a)t} \frac{dt}{t}</math> and then interchanging the sum and integral. The resulting integral representation converges for <math>z \in \Complex \setminus [1,\infty),</math> Re(s) > 0, and Re(a) > 0. This analytically continues <math>\Phi(z,s,a)</math> to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

A contour integral representation is given by

<math>

\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points <math>t = \log(z) + 2k\pi i</math> (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.<ref>Template:Harvnb</ref>

Other integral representations

A Hermite-like integral representation is given by

<math>

\Phi(z,s,a)= \frac{1}{2a^s}+ \int_0^\infty \frac{z^t}{(a+t)^s}\,dt+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt </math> for

<math>\Re(a)>0\wedge |z|<1 </math>

and

<math>

\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{\log^{s-1}(1/z)}{z^a}\Gamma(1-s,a\log(1/z))+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt </math> for

<math>\Re(a)>0. </math>

Similar representations include

<math>

\Phi(z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \tanh\pi t }\,dt, </math>

and

<math>\Phi(-z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \sinh\pi t }\,dt,</math>

holding for positive z (and more generally wherever the integrals converge). Furthermore,

<math>\Phi(e^{i\varphi},s,a)=L\big(\tfrac{\varphi}{2\pi}, s, a\big)= \frac{1}{a^s} + \frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}\big(e^{i\varphi}-e^{-t}\big)}{\cosh{t}-\cos{\varphi}}\,dt,</math>

The last formula is also known as Lipschitz formula.

Identities

For λ rational, the summand is a root of unity, and thus <math>L(\lambda, s, \alpha)</math> may be expressed as a finite sum over the Hurwitz zeta function. Suppose <math display="inline">\lambda = \frac{p}{q}</math> with <math>p, q \in \Z</math> and <math>q > 0</math>. Then <math>z = \omega = e^{2 \pi i \frac{p}{q}}</math> and <math>\omega^q = 1</math>.

<math>\Phi(\omega, s, \alpha) = \sum_{n=0}^\infty

\frac {\omega^n} {(n+\alpha)^s} = \sum_{m=0}^{q-1} \sum_{n=0}^\infty \frac {\omega^{qn + m}}{(qn + m + \alpha)^s} = \sum_{m=0}^{q-1} \omega^m q^{-s} \zeta \left( s,\frac{m + \alpha}{q} \right) </math>

Various identities include:

<math>\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}</math>

and

<math>\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)</math>

and

<math>\Phi(z,s+1,a)=-\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).</math>

Series representations

A series representation for the Lerch transcendent is given by

<math>\Phi(z,s,q)=\frac{1}{1-z}

\sum_{n=0}^\infty \left(\frac{-z}{1-z} \right)^n \sum_{k=0}^n (-1)^k \binom{n}{k} (q+k)^{-s}.</math> (Note that <math>\tbinom{n}{k}</math> is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.<ref name="benorin">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for<ref>Template:Cite journal</ref>

<math>\left|\log(z)\right| < 2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots</math>

<math>

\Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1} +\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right] </math>

If n is a positive integer, then

<math>

\Phi(z,n,a)=z^{-a}\left\{ \sum_{{k=0}\atop k\neq n-1}^ \infty \zeta(n-k,a)\frac{\log^k (z)}{k!} +\left[\psi(n)-\psi(a)-\log(-\log(z))\right]\frac{\log^{n-1}(z)}{(n-1)!} \right\}, </math> where <math>\psi(n)</math> is the digamma function.

A Taylor series in the third variable is given by

<math>\Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};|x|<\Re(a),</math>

where <math>(s)_{k}</math> is the Pochhammer symbol.

Series at a = −n is given by

<math>

\Phi(z,s,a)=\sum_{k=0}^n \frac{z^k}{(a+k)^s} +z^n\sum_{m=0}^\infty (1-m-s)_{m}\operatorname{Li}_{s+m}(z)\frac{(a+n)^m}{m!};\ a\rightarrow-n </math>

A special case for n = 0 has the following series

<math>

\Phi(z,s,a)=\frac{1}{a^s} +\sum_{m=0}^\infty (1-m-s)_m \operatorname{Li}_{s+m}(z)\frac{a^m}{m!}; |a|<1, </math> where <math>\operatorname{Li}_s(z)</math> is the polylogarithm.

An asymptotic series for <math>s\rightarrow-\infty</math>

<math>\Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [2k\pi i-\log(z)]^{s-1}e^{2k\pi ai}

</math> for <math>|a|<1;\Re(s)<0 ;z\notin (-\infty,0) </math> and

<math>

\Phi(-z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai} </math> for <math>|a|<1;\Re(s)<0 ;z\notin (0,\infty). </math>

An asymptotic series in the incomplete gamma function

<math>

\Phi(z,s,a)=\frac{1}{2a^s}+ \frac{1}{z^a}\sum_{k=1}^\infty \frac{e^{-2\pi i(k-1)a}\Gamma(1-s,a(-2\pi i(k-1)-\log(z)))}

    {(-2\pi i(k-1)-\log(z))^{1-s}}+

\frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}} </math> for <math>|a|<1;\Re(s)<0.</math>

The representation as a generalized hypergeometric function is<ref>Template:Cite journal</ref>

<math>

\Phi(z,s,\alpha)=\frac{1}{\alpha^s}{}_{s+1}F_s\left(\begin{array}{c} 1,\alpha,\alpha,\alpha,\cdots\\ 1+\alpha,1+\alpha,1+\alpha,\cdots\\ \end{array}\mid z\right). </math>

Asymptotic expansion

The polylogarithm function <math>\mathrm{Li}_n(z)</math> is defined as

<math>\mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z).</math>

Let

<math>

\Omega_{a} \equiv\begin{cases} \mathbb{C}\setminus[1,\infty) & \text{if } \Re a > 0, \\ {z \in \mathbb{C}, |z|<1} & \text{if } \Re a \le 0. \end{cases} </math> For <math>|\mathrm{Arg}(a)|<\pi, s \in \mathbb{C}</math> and <math>z \in \Omega_{a}</math>, an asymptotic expansion of <math>\Phi(z,s,a)</math> for large <math>a</math> and fixed <math>s</math> and <math>z</math> is given by

<math>

   \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}}
   +
   \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}}
   +O(a^{-N-s})

</math> for <math>N \in \mathbb{N}</math>, where <math>(s)_n = s (s+1)\cdots (s+n-1)</math> is the Pochhammer symbol.<ref>Template:Cite journal</ref>

Let

<math>f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}.</math>

Let <math>C_{n}(z,a)</math> be its Taylor coefficients at <math>x=0</math>. Then for fixed <math>N \in \mathbb{N}, \Re a > 1</math> and <math>\Re s > 0</math>,

<math>

\Phi(z,s,a) - \frac{\mathrm{Li}_{s}(z)}{z^{a}} = \sum_{n=0}^{N-1} C_{n}(z,a) \frac{(s)_{n}}{a^{n+s}} + O\left( (\Re a)^{1-N-s}+a z^{-\Re a} \right), </math> as <math>\Re a \to \infty</math>.<ref>Template:Cite journal</ref>

Software

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.

References

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• Template:Citation. (See § 1.11, "The function Ψ(z,s,v)", p. 27)
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• Template:Citation. (Includes various basic identities in the introduction.)
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External links

• Template:Citation.
• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
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• {{#invoke:citation/CS1|citation

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• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:LerchTranscendent%7CLerchTranscendent.html}} |title = Lerch Transcendent |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

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