Double integrals and infinite products for some classical constants via
analytic continuations of Lerch's transcendent
J. Guillera, and J. Sondow. (2005)cite arxiv:math/0506319Comment: 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.
DOI: 10.1007/s11139-007-9102-0
Abstract
The two-fold aim of the paper is to unify and generalize on the one hand the
double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the
second author for Euler's constant $\gamma$ and its alternating analog
$łn(4/\pi),$ and on the other hand the infinite products of the first author
for $e$, and of the second author for $\pi$ and $e^\gamma.$ 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.
Description
Double integrals and infinite products for some classical constants via
analytic continuations of Lerch's transcendent
cite arxiv:math/0506319Comment: 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
%0 Generic
%1 guillera2005double
%A Guillera, Jesus
%A Sondow, Jonathan
%D 2005
%K classical double infinite integral product
%R 10.1007/s11139-007-9102-0
%T Double integrals and infinite products for some classical constants via
analytic continuations of Lerch's transcendent
%U http://arxiv.org/abs/math/0506319
%X The two-fold aim of the paper is to unify and generalize on the one hand the
double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the
second author for Euler's constant $\gamma$ and its alternating analog
$łn(4/\pi),$ and on the other hand the infinite products of the first author
for $e$, and of the second author for $\pi$ and $e^\gamma.$ 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.
@misc{guillera2005double,
abstract = {The two-fold aim of the paper is to unify and generalize on the one hand the
double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the
second author for Euler's constant $\gamma$ and its alternating analog
$\ln(4/\pi),$ and on the other hand the infinite products of the first author
for $e$, and of the second author for $\pi$ and $e^\gamma.$ 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.},
added-at = {2013-12-23T07:04:46.000+0100},
author = {Guillera, Jesus and Sondow, Jonathan},
biburl = {https://www.bibsonomy.org/bibtex/26d49d65898404152df6754271991e958/aeu_research},
description = {Double integrals and infinite products for some classical constants via
analytic continuations of Lerch's transcendent},
doi = {10.1007/s11139-007-9102-0},
interhash = {1599090c7da7b8b93600e5778c9d7282},
intrahash = {6d49d65898404152df6754271991e958},
keywords = {classical double infinite integral product},
note = {cite arxiv:math/0506319Comment: 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},
timestamp = {2013-12-24T01:12:04.000+0100},
title = {Double integrals and infinite products for some classical constants via
analytic continuations of Lerch's transcendent},
url = {http://arxiv.org/abs/math/0506319},
year = 2005
}