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Quantum Jacobi forms and sums of tails identities

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Amanda Folsom
Amanda Folsom
Amanda Folsom
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Elizabeth Pratt
Elizabeth Pratt
Elizabeth Pratt
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Noah Solomon
Noah Solomon
Noah Solomon
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Andrew R. Tawfeek
Andrew R. Tawfeek
Andrew R. Tawfeek
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Abstract

Our results are on the interconnected topics of quantum Jacobi sums of tails identities, quantum Jacobi and mock Jacobi properties of two-variable q-hypergeometric series and partial theta functions, and two-variable q-hypergeometric generating functions for certain L-values by way of related asymptotic expansions. More specifically, we establish five two-variable quantum Jacobi sums of tails identities. As corollaries, we recover known one-variable quantum sum of tails identities due to Zagier, Andrews-Jiménez-Urroz-Ono, and more. Further, justifying the “quantum Jacobi” description of our two-variable sums of tails identities, we establish the quantum Jacobi and mock Jacobi properties of a number of two-variable q-hypergeometric series and partial Jacobi theta functions which appear in our two-variable sums of tails identities, inspired by related results of Zagier and Rolen-Schneider in the one-variable quantum modular setting. Finally, by establishing related asymptotic expansions, we realize generating functions for certain L-values in terms of two-variable q-hypergeometric series and Jacobi partial theta functions, inspired by earlier work in this direction by Andrews–Jiménez-Urroz–Ono for the Riemann ζ\zeta -function and Dirichlet and Hecke L-functions.
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A. Folsom et al. Res. Number Theory (2022) 8:8
https://doi.org/10.1007/s40993-021-00304-7
RESEARCH
Quantum Jacobi forms and sums of tails
identities
Amanda Folsom1* , Elizabeth Pratt1, Noah Solomon1and Andrew R. Tawfeek2
*Correspondence:
afolsom@amherst.edu
1Department of Mathematics
and Statistics, Seeley Mudd
Building, 31 Quadrangle Dr,
Amherst, MA 01002, USA
Full list of author information is
available at the end of the article
The authors are grateful for
support from National Science
Foundation Grant DMS-190179
(PI = first author).
Abstract
Our results are on the interconnected topics of quantum Jacobi sums of tails identities,
quantum Jacobi and mock Jacobi properties of two-variable q-hypergeometric series
and partial theta functions, and two-variable q-hypergeometric generating functions
for certain L-values by way of related asymptotic expansions. More specifically, we
establish five two-variable quantum Jacobi sums of tails identities. As corollaries, we
recover known one-variable quantum sum of tails identities due to Zagier,
Andrews-Jiménez-Urroz-Ono, and more. Further, justifying the “quantum Jacobi”
description of our two-variable sums of tails identities, we establish the quantum Jacobi
and mock Jacobi properties of a number of two-variable q-hypergeometric series and
partial Jacobi theta functions which appear in our two-variable sums of tails identities,
inspired by related results of Zagier and Rolen-Schneider in the one-variable quantum
modular setting. Finally, by establishing related asymptotic expansions, we realize
generating functions for certain L-values in terms of two-variable q-hypergeometric
series and Jacobi partial theta functions, inspired by earlier work in this direction by
Andrews–Jiménez-Urroz–Ono for the Riemann ζ-function and Dirichlet and Hecke
L-functions.
1 Introduction and statement of results
Early sums of tails identities include the following, found in Ramanujan’s “Lost” Notebook
[24, p14]:
∞
n=0
((−q;q)∞−(−q;q)n)=(−q;q)∞−1
2+∞
n=1
qn
1−qn+1
2
∞
n=0
qn(n+1)
2
(−q;q)n
,
∞
n=01
(q;q2)∞−1
(q;q2)n+1=1
(q;q2)∞−1
2+∞
n=1
q2n
1−q2n+1
2
∞
n=0
qn(n+1)
2
(−q;q)n
,
where here and throughout, the q-Pochhammer symbol is defined for n∈N0∪∞by
(a;q)n:=n−1
j=0(1 −aqj).These are referred to as “sums of tails” identities, as their left-
hand sides are sums of differences between infinite products and their nth partial products.
The q-hypergeometric series
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... which was studied in the context of quantum modular forms in [16,23] and [34]; see also the relevant quantum modular results in [8], noting that as a q-series (|q| < 1) [14], ...
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