An Identity Relating a Theta Function to a Sum of Lambert Series

Andrews, George E.; Lewis, Richard P.; Liu, Zhi Guo

doi:10.1112/blms/33.1.25

Cited by 30 publications

(13 citation statements)

References 8 publications

(4 reference statements)

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“…Before proving Theorem 3, we recall the result from Lemma 4 in [10] (this result was previously given by Andrews, Lewis and Liu in [4], using a different labeling for the parameters): if…”

Section: Proof Of the Main Identitiesmentioning

confidence: 99%

A new summation formula for WP-Bailey pairs

McLaughlin

2011

APPL ANAL DISCRETE M

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Let (αn(a, k), βn(a, k)) be a WP-Bailey pair. Assuming the limits exist, letbe the derived WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive new basic hypergeometric summation and transformation formulae involving derived WP-Bailey pairs. We then use these formulae to derive new identities for various theta series/products which are expressible in terms of certain types of Lambert series.

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“…Before proving Theorem 3, we recall the result from Lemma 4 in [10] (this result was previously given by Andrews, Lewis and Liu in [4], using a different labeling for the parameters): if…”

Section: Proof Of the Main Identitiesmentioning

confidence: 99%

A new summation formula for WP-Bailey pairs

McLaughlin

2011

APPL ANAL DISCRETE M

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“…Remark: The proof that the sum of Lambert series above combine to give the stated infinite product was first given by Andrews, Lewis and Liu in [4] (using a different labeling for the parameters) in a different context, so they did not have our reciprocity result for the basic hypergeometric series f (a, k, z, q).…”

Section: Thenmentioning

confidence: 99%

Some new transformations for Bailey pairs and WP-Bailey pairs

Laughlin

2010

Open Mathematics

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We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series. k c j β j (a, c), with c = kyz/aq. Andrews [1] also described a second method for deriving new WP-Bailey pairs from existing pairs, but this second method will not concern us in the present paper.These two constructions allow a "tree" of WP-Bailey pairs to be generated from a single WP-Bailey pair. The implications of these two branches

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“…Besides Ikehara [3] proved a Tauberian theorem known as Wiener-Ikehara theorem for Dirichlet series, which provides another proof for PNT with use of non-vanishing property of Riemann zeta function. Following the early applications to PNT above, many studies have been done dealing with Lambert series and Dirichlet series in number theory and in other branches of mathematics [4][5][6][7][8][9][10][11][12]. Furthermore, motivated by wide-range usage of zeta and related functions authors have recently introduced new families of generalized Riemann zeta functions and investigated corresponding properties.…”

Section: Introductionmentioning

confidence: 99%