LerchPhi - Maple Help

LerchPhi

general Lerch Phi function

 Calling Sequence LerchPhi(z, a, v)

Parameters

 z - algebraic expression a - algebraic expression v - algebraic expression

Description

 • The Lerch Phi function is defined as follows:

$\mathrm{LerchPhi}\left(z,a,v\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{z}^{n}}{{\left(v+n\right)}^{a}}$

 This definition is valid for $\left|z\right|<1$ or $\left|z\right|=1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}1<\mathrm{\Re }\left(a\right)$. By analytic continuation, it is extended to the whole complex $z$-plane for each value of $a$ and $v$.
 • If $v$ and $a$ are positive integers, LerchPhi(z, a, v) has a branch cut in the $z$-plane along the real axis to the right of $z=1$, with a branch point at $z=1$.
 • If $a$ is a non-positive integer, LerchPhi(z, a, v) is a rational function of $z$ with a pole of order $1-a$ at $z=1$.
 • LerchPhi(1,a,v) = Zeta(0,a,v).  If $1<\mathrm{\Re }\left(a\right)$, it is also true that limit(LerchPhi(z,a,v),z=1) = Zeta(0,a,v). If $\mathrm{\Re }\left(a\right)\le 1$, this limit does not exist.
 • If $0\le \mathrm{\Re }\left(a\right)$ and $a$ is not an integer, LerchPhi(z, a, v) has an infinite singularity at each non-positive integer v.
 • If the coefficients of the series representation of a hypergeometric function are rational functions of the summation indices, then the hypergeometric function can be expressed as a linear sum of Lerch Phi functions.
 • If the parameters of the hypergeometric functions are rational, we can express the hypergeometric function as a linear sum of polylog functions.

Examples

 > $\mathrm{LerchPhi}\left(3,4,1\right)$
 $\frac{{\mathrm{polylog}}{}\left({4}{,}{3}\right)}{{3}}$ (1)
 > $\mathrm{LerchPhi}\left(0,7,4\right)$
 $\frac{{1}}{{16384}}$ (2)
 > $\mathrm{LerchPhi}\left(4,0,3\right)$
 ${-}\frac{{1}}{{3}}$ (3)
 > $\mathrm{LerchPhi}\left(z,a,1\right)$
 $\frac{{\mathrm{polylog}}{}\left({a}{,}{z}\right)}{{z}}$ (4)
 > $\mathrm{LerchPhi}\left(1,z,1\right)$
 ${\mathrm{\zeta }}{}\left({z}\right)$ (5)
 > $\mathrm{diff}\left(\mathrm{LerchPhi}\left(z,3,4\right),z\right)$
 $\frac{{\mathrm{LerchPhi}}{}\left({z}{,}{2}{,}{4}\right)}{{z}}{-}\frac{{4}{}{\mathrm{LerchPhi}}{}\left({z}{,}{3}{,}{4}\right)}{{z}}$ (6)

References

 Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953.