The ultraspherical background of the Jacobi phase expansion, in closed form
Abstract
The large-degree phase of the Jacobi polynomial Pn(α,β)(cosθ) contains, at each order in Nn−1, Nn=n+(α+β+1)/2, an additive constant κr that the phase equation does not determine. These constants are antisymmetric in (α2,β2) and vanish for α=β. We obtain three results. First, on the diagonal α=β the logarithm of the midpoint amplitude is computed in closed form: if Ln denotes the coefficient of (2Nn)−n in that logarithm, then ΥA(t)=∑n≥1Lntn/(n−1)! equals ΥA(t)=(et(cosh2αt−cosh t))/(sinh 2t). The proof is algebraic: the identity is established for half-odd-integer α, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all α by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to α2 at the Legendre point α=β=0 is computed in closed form to all orders. An exact reduction of an Appell F3 function to a Gauss function shows that it is a parameter derivative of a zero-balanced 2F1 at argument $2$; the real part of its expansion is a trigamma series, and the imaginary part is the unique formal solution of an explicit central-difference equation. Third, we deduce that the coefficients wm of an explicit even power series attached to the imaginary part satisfy ν2(wm)=−2ν2((2m)!) for every m≥0 (ν2 the $2$-adic valuation). This determines, for every r, the exact power of $2$ in the denominator of the coefficient of α2 in the polynomial that represents κr. We also give a factorisation of the amplitude into products of pairs of Gauss functions, further valuation formulas, and a numerical illustration for the zeros.
Related
- Equal-Phase Monotonicity for Weighted Jacobi-Radau Functions and Jacobi Lebesgue Constants
- The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition
- Absolute deviations of the binomial about a prescribed centre: the tail expansion in closed form
- Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros
- The lower envelope of ultraspherical polynomials
BibTeX
@misc{area2026ultraspherical,
title = {{The ultraspherical background of the Jacobi phase expansion, in closed form}},
author = {Iván Area},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29531},
eprint = {2609.29531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29531},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The ultraspherical background of the Jacobi phase expansion, in closed form
AU - Iván Area
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.29531
UR - https://arxiv.org/abs/2609.29531
AB - The large-degree phase of the Jacobi polynomial P_n^(α,β)(cosθ) contains, at each order in Nn^−1, Nn=n+(α+β+1)/2, an additive constant κ_r that the phase equation does not determine. These constants are antisymmetric in (α^2,β^2) and vanish for α=β. We obtain three results. First, on the diagonal α=β the logarithm of the midpoint amplitude is computed in closed form: if L_n denotes the coefficient of (2Nn)^−n in that logarithm, then Υ_A(t)=∑_n≥1L_nt^n/(n−1)! equals Υ_A(t)=(e^t(cosh2αt−cosh t))/(sinh 2t). The proof is algebraic: the identity is established for half-odd-integer α, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all α by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to α^2 at the Legendre point α=β=0 is computed in closed form to all orders. An exact reduction of an Appell F_3 function to a Gauss function shows that it is a parameter derivative of a zero-balanced _2F_1 at argument $2$; the real part of its expansion is a trigamma series, and the imaginary part is the unique formal solution of an explicit central-difference equation. Third, we deduce that the coefficients w_m of an explicit even power series attached to the imaginary part satisfy ν_2(w_m)=−2ν_2((2m)!) for every m≥0 (ν_2 the $2$-adic valuation). This determines, for every r, the exact power of $2$ in the denominator of the coefficient of α^2 in the polynomial that represents κ_r. We also give a factorisation of the amplitude into products of pairs of Gauss functions, further valuation formulas, and a numerical illustration for the zeros.
ER -