PreprintCC BY 4.0arXiv
A q-recurrence for a finite Apéry limit
Abstract
The Kaneko-Zagier conjecture predicts a correspondence between finite and symmetric multiple zeta values. Under this correspondence, ζ(3) corresponds to an element Z(3) defined by Bernoulli numbers. We prove a conjecture of Tasaka relating Z(3) to the quotient of two solutions of a recurrence. A two-index q-recurrence connects this quotient to a finite harmonic q-series. Using a method of the author, Takeyama, and Tasaka, we obtain the algebraic and analytic limits 3Z(3)/4 and 3ζ(3)/4 at roots of unity.
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BibTeX
@misc{bachmann2026qrecurrence,
title = {{A $q$-recurrence for a finite Apéry limit}},
author = {Henrik Bachmann},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18271},
eprint = {2609.18271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18271},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A q-recurrence for a finite Apéry limit
AU - Henrik Bachmann
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18271
UR - https://arxiv.org/abs/2609.18271
AB - The Kaneko-Zagier conjecture predicts a correspondence between finite and symmetric multiple zeta values. Under this correspondence, ζ(3) corresponds to an element Z(3) defined by Bernoulli numbers. We prove a conjecture of Tasaka relating Z(3) to the quotient of two solutions of a recurrence. A two-index q-recurrence connects this quotient to a finite harmonic q-series. Using a method of the author, Takeyama, and Tasaka, we obtain the algebraic and analytic limits 3Z(3)/4 and 3ζ(3)/4 at roots of unity.
ER -