A Rigidity Property of the Deformed q-Exponential Function
Abstract
The deformed q-exponential function eq(z,u) = ∑n=0∞ ubinomn2 zn/(q;q)n provides a common framework containing several classical q-exponential functions as particular cases, including the Jackson q-exponentials eq(z) and Eq(z). In this paper we investigate the multiplicative inversion problem eq(z,u)eq(−z,v)=1, and determine all pairs of deformation parameters (u,v) for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that eq(z,u)eq(−z,v)=1 if and only if (u,v)=(1,q) or (u,v)=(q,1). Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.
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BibTeX
@misc{lopez2026rigidity,
title = {{A Rigidity Property of the Deformed q-Exponential Function}},
author = {Ronald Orozco López},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14095},
eprint = {2609.14095},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14095},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Rigidity Property of the Deformed q-Exponential Function
AU - Ronald Orozco López
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14095
UR - https://arxiv.org/abs/2609.14095
AB - The deformed q-exponential function e_q(z,u) = ∑_n=0^∞ u^binomn2 z^n/(q;q)_n provides a common framework containing several classical q-exponential functions as particular cases, including the Jackson q-exponentials e_q(z) and E_q(z). In this paper we investigate the multiplicative inversion problem e_q(z,u)e_q(−z,v)=1, and determine all pairs of deformation parameters (u,v) for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that e_q(z,u)e_q(−z,v)=1 if and only if (u,v)=(1,q) or (u,v)=(q,1). Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.
ER -