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Lerch Transcendent

The Lerch transcendent is generalization of the Hurwitz zeta function and polylogarithm function. Many sums of reciprocal powers can be expressed in terms of it. It is classically defined by

Phi(z,s,a)=sum_(k=0)^infty(z^k)/((a+k)^s),

(1)

for |z|<1 and a!=0, -1, .... It is implemented in this form as HurwitzLerchPhi[z, s, a] in the Wolfram Language.

The slightly different form

Phi^*(z,s,a)=sum_(k=0)^infty(z^k)/([(a+k)^2]^(s/2))

(2)

sometimes also denoted Phi^~(z,s,a), for |z|<1 (or |z|=1 and ) and a!=0, -1, -2, ..., is implemented in the Wolfram Language as LerchPhi[z, s, a]. Note that the two are identical only for .

A series formula for Phi(z,s,a) valid on a larger domain in the complex z-plane is given by

(1-z)Phi(z,s,a) =sum_(n=0)^infty((-z)/(1-z))^nsum_(k=0)^n(-1)^k(n; k)(a+k)^(-s),

(3)

which holds for all complex s and complex z with R[z]<1/2 (Guillera and Sondow 2005).

The Lerch transcendent can be used to express the Dirichlet beta function

A special case is given by

Phi(z,n,1)=(Li_n(z))/z,

(6)

(Guillera and Sondow 2005), where Li_n(z) is the polylogarithm.

Special cases giving simple constants include

where K is Catalan's constant, zeta(3) is Apéry's constant, and A is the Glaisher-Kinkelin constant (Guillera and Sondow 2005).

It gives the integrals of the Fermi-Dirac distribution

where Gamma(z) is the gamma function and Li_n(z) is the polylogarithm and Bose-Einstein distribution

Double integrals involving the Lerch transcendent include

int_0^1int_0^1(x^(u-1)y^(v-1))/(1-xyz)[-ln(xy)]^sdxdy =Gamma(s+1)(Phi(z,s+1,v)-Phi(z,s+1,u))/(u-v) int_0^1int_0^1((xy)^(u-1))/(1-xyz)[-ln(xy)]^sdxdy =Gamma(s)Phi(z,s+2,u),

(15)

where Gamma(z) is the gamma function. These formulas lead to a variety of beautiful special cases of unit square integrals (Guillera and Sondow 2005).

It also can be used to evaluate Dirichlet L-series.

See also

Bose-Einstein Distribution, Dirichlet Beta Function, Dirichlet L-Series, Fermi-Dirac Distribution, Hurwitz Zeta Function, Jacobi Theta Functions, Legendre's Chi-Function, Polylogarithm, Unit Square Integral

Related Wolfram sites

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/

Explore with Wolfram|Alpha

References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function Psi(z,s,v)=sum_(n=0)^(infty)(v+n)^(-s)z^n." §1.11 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 27-31, 1981.Gradshteyn, I. S. and Ryzhik, I. M. "The Lerch Function Phi(z,s,v)." §9.55 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1029, 2000.Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.Tyagi, S. "Double Exponential Method for Riemann Zeta, Lerch and Dirichlet L-Functions." https://arxiv.org/abs/2203.02509. 7 Mar 2022.

Referenced on Wolfram|Alpha

Lerch Transcendent

Cite this as:

Weisstein, Eric W. "Lerch Transcendent." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LerchTranscendent.html

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