Connection Formulae for a Generalised Ramanujan Entire Function
Abstract
The Ramanujan−q-Airy connection formula relates the convergent series at the origin to the behaviour at infinity of the Ramanujan entire function. We extend this connection to a one-parameter deformation, which embeds the Ramanujan (second-order) q-difference operator in a family of third-order equations. By contour integral as q-Borel inversion, we give behaviour at infinity in terms of divergent local expansions with connection coefficients uniquely determined. Discrete q-Borel−Laplace summation yields a convergent, bilateral power series representation, whose dependence on summation path reflects the q-Stokes phenomenon. We prove remainder estimates establishing the formal, generally divergent, expansion at infinity as an asymptotic description of the entire function.
Related
- An entire function that violates quasiminimality
- Ramanujan Conjecture and Riemann Hypothesis for the unitary groups over function fields
- A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann Ξ-function
- Higher-order sum rules for zeros of the planar Gaussian entire function
- Ramanujan-type congruences for the smallest parts function
BibTeX
@misc{holroyd2026connection,
title = {{Connection Formulae for a Generalised Ramanujan Entire Function}},
author = {Joshua Holroyd},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18255},
eprint = {2609.18255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18255},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Connection Formulae for a Generalised Ramanujan Entire Function
AU - Joshua Holroyd
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18255
UR - https://arxiv.org/abs/2609.18255
AB - The Ramanujan−q-Airy connection formula relates the convergent series at the origin to the behaviour at infinity of the Ramanujan entire function. We extend this connection to a one-parameter deformation, which embeds the Ramanujan (second-order) q-difference operator in a family of third-order equations. By contour integral as q-Borel inversion, we give behaviour at infinity in terms of divergent local expansions with connection coefficients uniquely determined. Discrete q-Borel−Laplace summation yields a convergent, bilateral power series representation, whose dependence on summation path reflects the q-Stokes phenomenon. We prove remainder estimates establishing the formal, generally divergent, expansion at infinity as an asymptotic description of the entire function.
ER -