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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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