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Parameterized partial theta identities and a unified q-difference proof
Abstract
We establish five families of integer-parameter extensions of Ramanujan's partial theta identities. The families follow from a common two-parameter specialization of Andrews' transformation, for which we give an independent proof based on a q-difference recurrence and a boundary estimate. Specializations of the integer parameter recover six identities from Ramanujan's lost notebook. As an application, a residue argument applied to the fifth family gives an integer-parameter extension of Lovejoy's residual identity, from which we construct a corresponding family of conjugate Bailey pairs.
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BibTeX
@misc{su2026parameterized,
title = {{Parameterized partial theta identities and a unified $q$-difference proof}},
author = {Chen-Yang Su},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07245},
eprint = {2609.07245},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07245},
}
RIS
TY - UNPB
TI - Parameterized partial theta identities and a unified q-difference proof
AU - Chen-Yang Su
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07245
UR - https://arxiv.org/abs/2609.07245
AB - We establish five families of integer-parameter extensions of Ramanujan's partial theta identities. The families follow from a common two-parameter specialization of Andrews' transformation, for which we give an independent proof based on a q-difference recurrence and a boundary estimate. Specializations of the integer parameter recover six identities from Ramanujan's lost notebook. As an application, a residue argument applied to the fifth family gives an integer-parameter extension of Lovejoy's residual identity, from which we construct a corresponding family of conjugate Bailey pairs.
ER -