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

(1)

for  and , , .... It is implemented in this form as HurwitzLerchPhi[z, s, a] in the Wolfram Language.

The slightly different form

(2)

sometimes also denoted , for  (or  and ) and , , , ..., is implemented in the Wolfram Language as LerchPhi[z, s, a]. Note that the two are identical only for .

A series formula for  valid on a larger domain in the complex -plane is given by

(3)

which holds for all complex  and complex  with  (Guillera and Sondow 2005).

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

			(4)
			(5)

A special case is given by

(6)

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

Special cases giving simple constants include

			(7)
			(8)
			(9)
			(10)

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

It gives the integrals of the Fermi-Dirac distribution

			(11)
			(12)

where  is the gamma function and  is the polylogarithm and Bose-Einstein distribution

			(13)
			(14)

Double integrals involving the Lerch transcendent include

(15)

where  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.

REFERENCES:

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function ." §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 ." §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.