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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(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 Dirichletbeta 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..

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