2005/06/30 by Jesús Guillera, Jesus Guillera, Jonathan Sondow · 4 citations
Mathematics · #Advanced Mathematical Identities #Constant (computer programming) #Digamma function #Euler's formula #Generalization #Infinite product #Logarithm #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Polylogarithm #Product (mathematics) #Riemann zeta function #Series (stratigraphy) #math.CA #math.NT #msc:11M06 #msc:11M35 #msc:11Y60 #msc:33B15 #msc:33B30
paper · pdf · doi:10.1007/s11139-007-9102-0
published as Ramanujan J. 16 (2008) 247-270 · 21 pages, to appear in The Ramanujan Journal. Added Corollary 3.3, Lemma 3.1, Equations (5) and (33), all or part of Examples 3.5, 3.6, 3.7, 3.9, 3.11, 3.14, 5.12, and references [1], [16], [17], [18]. Omitted the old [4]. Modified Equation (49), Theorem 5.3, and Examples 3.25 and 3.26. Expanded the Abstract and Introduction. Rearranged Sections 2 and 3
arxiv created 2006/08/05 · openalex publication_date 2008/07/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler's constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and of the second author for π and eγ. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.