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Les séries congruo-harmoniques alternées - Partie 1 - Somme et restes partiels

Abstract : For every fixed couple (p;q) of strictly positive integers, the « alternate congruo-harmonic series » parameterized by (p;q) is the convergent series whose general term is (-1)^k/(pk+q). The paper first exposes the general and explicit calculation of the sum S_(p,q) of such a series by means of elementary transcendant functions. It then points the relationships between the family of sums (S_(p,1) )_(p∈N^* ) and the digamma function as well as the more general relationships between the family of sums (S_(p,q) )_((p;q)∈(N^* )^2 ) with the Hurwitz-Lerch function. Finally, an expansion of every partial rest of the series in generalized continued fractions is demonstrated.
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https://hal.archives-ouvertes.fr/hal-03622997
Contributor : David Pouvreau Connect in order to contact the contributor
Submitted on : Tuesday, March 29, 2022 - 1:59:22 PM
Last modification on : Saturday, April 2, 2022 - 3:24:17 AM
Long-term archiving on: : Thursday, June 30, 2022 - 7:12:13 PM

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David Pouvreau. Les séries congruo-harmoniques alternées - Partie 1 - Somme et restes partiels. Quadrature, 2022, 123. ⟨hal-03622997⟩

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