Proof of a conjecture of M. Patrick concerning Jacobi polynomials

Abstract

The squared modulus of every real-valued on R function f from the Laguerre–Pólya class L–P obeys a MacLaurin-type series representation|f(x+iy)|2=∑k=0∞Lk(f;x)y2k,x,y∈R. If f is a polynomial with only real roots, then the sum becomes finite. The coefficients {Lk} are representable as non-linear differential operators acting on f, and by a classical result of Jensen Lk(f;x)≥0 for f∈L–P and x∈R. A conjecture of M. Patrick from 1971 states that for f=Pn(α,β), the n-th Jacobi polynomial, with α≥β>−1, the functions Lk(f;x), 1≤k≤n−1, attain their maxima in [0,1] at x=1. The aim of this paper is to validate Patrick's conjecture. Moreover, we prove a refined version of this conjecture, showing that {Lk(f;x)}k=1n−1 are strictly monotonically increasing functions on the positive semi-axis. Towards our proof of Patrick's conjecture we extend the Sonin–Pólya majorization approach to all coefficient functions {Lk(f;x)}k=0n, f=Pn(α,β).