The ultraspherical background of the Jacobi phase expansion, in closed form

Abstract

The large-degree phase of the Jacobi polynomial $P_n^{(\alpha ,\beta )}(\cos \theta )$ contains, at each order in $\Nn^{-1}$, $\Nn=n+(α+β+1)/2$, an additive constant ${\kappa }_r$ that the phase equation does not determine. These constants are antisymmetric in $({\alpha }^2,{\beta }^2)$ and vanish for $\alpha =\beta$. We obtain three results. First, on the diagonal $\alpha =\beta$ the logarithm of the midpoint amplitude is computed in closed form: if $L_n$ denotes the coefficient of $(2\Nn)^{-n}$ in that logarithm, then ${{\rm Y}}_A(t)={\sum }_{n\ge 1}{L}_n{t}^n/(n-1)!$ equals \[ Υ_A(t)=\frac{e^{t}\bigl(\cosh2αt-\cosh t\bigr)}{\sinh 2t}. \] The proof is algebraic: the identity is established for half-odd-integer $\alpha$, where the expansion terminates, by means of two classical recurrences for Bessel polynomials, and is extended to all $\alpha$ by a degree argument. Second, the derivative of the logarithm of the amplitude with respect to ${\alpha }^2$ at the Legendre point $\alpha =\beta =0$ 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 ${}_2{F}_1$ 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 ${\nu }_2(w_m)=-2{\nu }_2((2m)!)$ for every $m\ge 0$ (${\nu }_2$ the $2$-adic valuation). This determines, for every $r$, the exact power of $2$ in the denominator of the coefficient of ${\alpha }^2$ in the polynomial that represents ${\kappa }_r$. 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.