Differential Recursions, Projective Discriminants, and Algebraic Generating Functions
Abstract
We study two hierarchies of bivariate generating functions defined by differential recursions: a self-dual hierarchy and a reduced-symmetry hierarchy. Their rational coefficient functions produce palindromic and anti-palindromic triangles, including OEIS A336110 and A140136. For general differential order p, the reduced hierarchy decomposes into p−1 Horn-type companions. An explicit differential intertwiner of order p−3 relates the last companion to the normalized self-dual generating function. The companions share a principal symbol and a rationally parametrized characteristic curve with a projective S3 action. At p=2 this construction yields the Narayana quadratic. At p=3 the symmetry lifts to S4: two algebraic sectors are edge resolvents of one quartic root configuration, and the nonsymmetric Gross--Witten--Wadia generating function has degree four. For p=4 and p=5 we derive algebraic equations of degrees six and eight, respectively, and find Galois groups S6 and S8 at generic specializations. At p=6 we verify a degree-ten specialization with Galois group S10. Their discriminants exhibit a universal cubic characteristic factor and a squared lower-order factor. A boundary factorization explains the observed algebraic degrees and discriminant monomials. We formulate conjectures on algebraicity and Galois groups in the reduced hierarchy and identify a possible mechanism for transcendence in the higher self-dual hierarchy.
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BibTeX
@misc{voloshyn2026differential,
title = {{Differential Recursions, Projective Discriminants, and Algebraic Generating Functions}},
author = {S. Voloshyn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31189},
eprint = {2609.31189},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31189},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Differential Recursions, Projective Discriminants, and Algebraic Generating Functions
AU - S. Voloshyn
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31189
UR - https://arxiv.org/abs/2609.31189
AB - We study two hierarchies of bivariate generating functions defined by differential recursions: a self-dual hierarchy and a reduced-symmetry hierarchy. Their rational coefficient functions produce palindromic and anti-palindromic triangles, including OEIS A336110 and A140136. For general differential order p, the reduced hierarchy decomposes into p−1 Horn-type companions. An explicit differential intertwiner of order p−3 relates the last companion to the normalized self-dual generating function. The companions share a principal symbol and a rationally parametrized characteristic curve with a projective S_3 action. At p=2 this construction yields the Narayana quadratic. At p=3 the symmetry lifts to S_4: two algebraic sectors are edge resolvents of one quartic root configuration, and the nonsymmetric Gross--Witten--Wadia generating function has degree four. For p=4 and p=5 we derive algebraic equations of degrees six and eight, respectively, and find Galois groups S_6 and S_8 at generic specializations. At p=6 we verify a degree-ten specialization with Galois group S_10. Their discriminants exhibit a universal cubic characteristic factor and a squared lower-order factor. A boundary factorization explains the observed algebraic degrees and discriminant monomials. We formulate conjectures on algebraicity and Galois groups in the reduced hierarchy and identify a possible mechanism for transcendence in the higher self-dual hierarchy.
ER -