Transcendence of multi-indexed infinite series

We consider the transcendence of the multi-indexed series ∑ n 1 , … , n k = 1 ∞ f ( n 1 , … , n k ) n 1 ⋯ n k and then extend our results to series of the form ∑ n 1 , … , n k = 0 ∞ f ( n 1 , … , n k ) A 1 ( n 1 ) ⋯ A k ( n k ) B 1 ( n 1 ) ⋯ B k ( n k ) where f is a k-periodic function and each A i...

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Bibliographic Details
Published in:Journal of number theory 2011-04, Vol.131 (4), p.705-715
Main Author: Weatherby, Chester
Format: Article
Language:English
Subjects:
Digamma function
Infinite series
Schanuel's Conjecture
Transcendence
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Summary:We consider the transcendence of the multi-indexed series ∑ n 1 , … , n k = 1 ∞ f ( n 1 , … , n k ) n 1 ⋯ n k and then extend our results to series of the form ∑ n 1 , … , n k = 0 ∞ f ( n 1 , … , n k ) A 1 ( n 1 ) ⋯ A k ( n k ) B 1 ( n 1 ) ⋯ B k ( n k ) where f is a k-periodic function and each A i ( x ) , B i ( x ) is a polynomial with algebraic coefficients. We relate these series to special values of the digamma function as well as polygamma functions. We also show some conditional implications on the transcendence of the series via Schanuel's Conjecture.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2010.11.003