bibkey: weisstein2026lerch
authors: Eric W. Weisstein
year: 2026
title: Lerch Transcendent
doi: null
url: https://mathworld.wolfram.com/LerchTranscendent.html
claim: "Equation (1) defines Phi(z,s,a)=sum_{j>=0} z^j/(a+j)^s for |z|<1 and a outside the nonpositive integers; equation (6) gives Phi(z,n,1)=Li_n(z)/z."
strata_touched:
  - D5/S1/Recurrence/LerchSomosLimit
license: "citation-only; MathWorld text and images are not reproduced"
triage: anchor

The source definition of the Lerch kernel

Equation (1) specializes at z=1/2, a=1, and a natural exponent s to the real series with j-th term (1/2)^j/(j+1)^s. Its summation starts at zero. Equation (6) gives Phi(1/2,s,1)=2*Li_s(1/2). Both equations are used only to identify the source’s meaning.

The article distinguishes the classical Phi from its modified form Phi-star in equation (2), and states that they agree when the real part of a is positive. The source specialization a=1 meets that condition. The definition used here is explicitly equation (1).

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Verified locator

url: https://mathworld.wolfram.com/LerchTranscendent.html

Article title: Lerch Transcendent. The Cite this as block credits Weisstein, Eric W., MathWorld–A Wolfram Resource. Exact locators: equations (1), (2), and (6). The original HTML read on September 15, 2026 has SHA-256 ac8f5c428db528c6b6f237fb7ed46034f4c244d185c7a2d0fb1baced157c5945. No DOI is assigned here to this web article.