The Laguerre Gram matrix of Gonçalves' sharpened Strichartz inequality: a closed factorisation, and the spectral conjecture through S=60

Abstract

Let Lₙ be the Laguerre polynomial normalised by Lₙ(0)=1, so that the Lₙ are orthonormal for e⁻ˣ dx on (0,∞), and let Q(a,b,c,d)=∫₀^∞ Lₐ(x/2)L_b(x/2)L_c(x/2)L_d(x/2) e⁻ˣ dx, and let Q_S=[Q(a,S-a,c,S-c)]_(a,c=0)^(S). Gonçalves introduced Q_S to prove a sharpened Strichartz inequality for radial data in ℝ², showed that it is positive semi-definite, doubly stochastic and strictly positive, and closed his paper with two numerical conjectures about it: that the eigenvalues of Q_S are exactly λ(n)=C(2n, n)² 16⁻ⁿ for n ≤ ⌊ S/2⌋, each simple, together with 0 of multiplicity ⌈ S/2⌉; and that the least entry of Q_S is Q(⌊ S/2⌋,⌈ S/2⌉,S,0). He reports verifying both for S ≤ 30. No formula for Q(a,b,c,d) is given there, and we have found none in the literature. We give one on the diagonal slice: writing X_S[a,j]=C(S-2j, a-j), H_S[i,j]=C(2(S-i-j), S-i-j) and boldsymbol(Δ)_S=diag((-4)ⁱC(S-i, i)) for 0 ≤ i,j ≤ ⌊ S/2⌋, 2^(2S) Q_S = X_S boldsymbol(Δ)_SH_Sboldsymbol(Δ)_S X_Sᵗᵒᵖ, an identity we derive from the two-variable generating function of the Laguerre linearisation coefficients. It compresses the characteristic polynomial to char_(Q_S)(X)=X^(⌈ S/2⌉) · char_(4^(-S)M_S²)(X) with M_S=boldsymbol(Δ)_SH_S, so that Gonçalves' eigenvalue conjecture at S becomes a statement about a matrix of half the size whose eigenvalues are the square roots (-1)ⁿC(2n, n)2^(S-2n). Instances of the identity and of the compression are machine-checked, and so is the eigenvalue conjecture in full, with multiplicities, for every S with 31 ≤ S ≤ 60, thirty values beyond the range reported in the source; likewise the minimum-entry conjecture over the same range, together with the observation — stated nowhere, and not new mathematics — that the minimum is attained exactly on the 4- or 8-element symmetry orbit of (⌊ S/2⌋,0). Every statement below is formally verified in Lean 4 except where the text says otherwise.

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