Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent

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 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, of the second author for π, and of Ser for 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.

[1]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[2]  J. Littlewood,et al.  Collected Papers of Srinivasa Ramanujan , 1929, Nature.

[3]  J. Littlewood,et al.  Collected Papers of Srinivasa Ramanujan , 1929, Nature.

[4]  Helmut Hasse,et al.  Ein Summierungsverfahren für die Riemannsche ζ-Reihe , 1930 .

[5]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[6]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[7]  F. Beukers.  A Note on the Irrationality of ζ(2) and ζ(3) , 1979 .

[8]  Leonard Lewin,et al.  Polylogarithms and Associated Functions , 1981 .

[9]  J. Milnor.  On polylogarithms, Hurwitz zeta functions, and the Kubert identities , 1983 .

[10]  B. Berndt.  Ramanujan's Notebooks , 1985 .

[11]  K. B. Oldham,et al.  An Atlas of Functions. , 1988 .

[12]  V. Rich.  Personal communication , 1989, Nature.

[13]  F. Dyson.  RAMANUJAN'S NOTEBOOKS Parts I and II , 1990 .

[14]  L. Lewin.  Structural Properties of Polylogarithms , 1991 .

[15]  Jonathan Sondow,et al.  Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series , 1994 .

[16]  B. Berndt.  Ramanujan’s Notebooks: Part V , 1997 .

[17]  David Bailey,et al.  On the rapid computation of various polylogarithmic constants , 1997, Math. Comput..

[18]  P. Hadjicostas.  Some Generalizations of Beukers' Integrals , 2002 .

[19]  J. Sondow.  Criteria for irrationality of Euler’s constant , 2002, math/0209070.

[20]  An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma , 2003, math/0306008.

[21]  A conjecture-generalization of Sondow's formula , 2004, math/0405423.

[22]  M. Émery.  On a multiple harmonic power series , 2004, math/0411267.

[23]  R. Chapman.  A proof of Hadjicostas's conjecture , 2004, math/0405478.

[24]  J. Sondow.  Double Integrals for Euler's Constant and In and an Analog of Hadjicostas's Formula , 2002, Am. Math. Mon..

[25]  Jonathan Sondow,et al.  A Faster Product for π and a New Integral for In , 2005, Am. Math. Mon..

[26]  Jonathan Sondow.  A Faster Product for $\pi$ and a New Integral for ln $\frac{\pi}{2}$ , 2005 .

[27]  J. Borwein,et al.  On Two Fundamental Identities For Euler Sums , 2005 .

[28]  Christina Goldschmidt,et al.  Random Recursive Trees and the Bolthausen-Sznitman Coalesent , 2005, math/0502263.

[29]  Steven R. Finch,et al.  Mathematical constants , 2005, Encyclopedia of mathematics and its applications.