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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


A double Stirling formula

Authors: J. R. Quine and Richard R. Song
Journal: Proc. Amer. Math. Soc. 119 (1993), 373-379
MSC: Primary 11F20; Secondary 11E45, 11M41
MathSciNet review: 1164151
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Abstract: We give an asymptotic expansion as $ M \to \infty ,\;N \to \infty $ of $ \prod {{{(m + n\tau )}^2}} $, where the product is over $ 1 \leqslant \vert m\vert \leqslant M,\;1 \leqslant \vert n\vert \leqslant N,(m,n) \ne (0,0)$. The formula is analogous to the classical Stirling expansion on $ M!$. Of special interest is the constant term in the expansion, which involves the Dedekind eta function $ \eta (z)$. Finding this constant is related to the Kronecker limit formula for the derivative at 0 of the zeta function $ Z(s) = \sum\nolimits_{m,n}'{\vert m + n\tau {\vert^{ - 2s}}} $. We consider instead the zeta function without absolute values.

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Additional Information

PII: S 0002-9939(1993)1164151-5
Keywords: Stirling expansion, Dedekind eta function, Kronecker limit formula, zeta regularized product
Article copyright: © Copyright 1993 American Mathematical Society

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