On a conjectured inequality for the largest zero of Jacobi polynomials

Abstract

P. Leopardi and the author recently investigated, among other things, the validity of the inequality nθ (α,β) n <(n+1)θ (α,β) n+1 between the largest zero x n = cos θThe domain in the parameter space (α, β) in which the inequality holds for all n ≥ 1, conjectured by us, is shown here to require a small adjustment-the deletion of a very narrow lens-shaped region in the square {−1 < α < −1/2, −1/2 < β < 0}.

New conjectured inequalities for zeros of Jacobi polynomials

“…We have agreement of this expansion with the one in (2) up to, and including, the n −2 term, whereas the inequalities in [2] give agreement only up to the n −1 term. Comparing (3) with (2), we see that the inequality (1) with r fixed holds for all sufficiently large n if…”

“…Inequalities for the largest zero of Jacobi polynomials, recently conjectured by Leopardi and the author [5], and slightly revised in [2], have been extended by us in [3] to all zeros of Jacobi polynomials. They state that for each r = 1, 2, .…”

Luigi Gatteschi’s work on asymptotics of special functions and their zeros

“…If solved for j α,r , it yields a good approximation for the first few zeros of the Bessel function J α . The simplified O(n −5 ) version of (59), with r = 1, has been found useful by one of us (W. G.) to discuss (in [51]) a conjectured inequality involving θ (α,β) n, 1 and θ (α,β) n+1,1 . The final sections of [35] discuss inequalities holding between zeros of Jacobi polynomials and zeros of Bessel functions, some of which sharpening (26), and others extending (26), with the bounds switched, to 8 |α| > 1/2, |β| > 1/2.…”

On conjectured inequalities for zeros of Jacobi polynomials

“…We know from [1] that vertically below K the inequality (2) for r = 1 is false for all n sufficiently large. Therefore, it remains to examine the line segments L − α for α > −.5.…”

“…For r = 1, this was done in [2] and slightly revised in [1]. Since in this case, i.e., for r fixed, the inequality ( (2), this will no longer be true.…”