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Metadata 
bibkey: meijer2016a112302
authors: Johannes W. Meijer
year: 2016
title: "OEIS A112302, Lerch recurrence limit conjecture"
doi: null
url: https://oeis.org/A112302
claim: "With Phi the Lerch transcendent, LP(0)=1 and LP(n)=(1/n)*sum(Phi(1/2,n-k,1)*LP(k),k=0..n-1) for n>0; the comment conjectures that LP(n) tends to the real Somos constant represented by A112302."
strata_touched:
  - D5/S1/Recurrence/LerchSomosLimit
license: "citation-only; OEIS source governed by https://oeis.org/LICENSE"
triage: anchor

Meijer’s Lerch recurrence conjecture

The exact anchor is Johannes W. Meijer’s COMMENTS paragraph of June 27, 2016 in OEIS A112302:

With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.

The citation year is the comment’s date. A112302 names the real Somos constant through its decimal expansion; its individual digits are not the limit target. Revision 121, dated April 20, 2026 at 00:17:02, retains the conjecture wording without changing its 2016 date.

The entry’s FORMULA section credits Jonathan Sondow (April 7, 2013) for product_{n>=1}n^{1/2^n}, and Jwalin Bhatt (April 2, 2025) for exp(sum_{n>=2}log(n)/2^n). These are entry-level contribution credits, not claims of earliest discovery. SomosConstant reindexes the logarithmic series by n=j+1 and includes the zero term log(1).

D5/L/weisstein2026lerch fixes the classical Lerch normalization from Weisstein’s Lerch Transcendent, equations (1) and (6): Phi(1/2,s,1)=2 Li_s(1/2). D5/L/coffey2015somosseries records the earlier known identity sum_{k>=1}(2 Li_k(1/2)-1)/k=log(sigma_2), from Proposition 5(b), equation (1.25) of its original preprint. The identity evaluates the constant in the LP proof; no LP-convergence theorem is attributed to Coffey.

MathWorld’s Somos article, equations (6) and (10), and section 3 of Sondow–Hadjicostas give the product and logarithmic sum. Guillera–Sondow’s Somos examples give integral/product representations. Their nonlinear quadratic recurrence g_n=n*g_(n-1)^2 is distinct from LP; these background identities alone do not establish Meijer’s limit.

Verified locator

  • https://oeis.org/A112302.
  • Complete official text: https://oeis.org/search?q=id:A112302&fmt=text; revision 121, 6,191 bytes, SHA-256 ed64c8e83d66f7f508c074805f27c16b379eb02e047c0be5cf0c22a12a830318. Exact conjecture: COMMENTS block beginning “From Johannes W. Meijer, Jun 27 2016”. Constant credits: FORMULA lines identified above. No DOI is assigned to this comment.
  • https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html, equations (6) and (10); https://arxiv.org/abs/math/0610499v1, section 3; https://arxiv.org/abs/math/0506319v3, Somos integral and product examples.
  • Coffey’s original locator and bibliography are in D5/L/coffey2015somosseries. Only citation information and the short conjecture quotation are retained.