Total Nonnegativity of Gessel Arrays: Euler–Hurwitz Factorization, Exact Orders, and Bessel Limits

This preprint proves total nonnegativity of the two-parameter coefficient triangles arising from Gessel’s Narayana Lagrange inversion formula for all positive real parameters, and characterizes exactly which minors are strictly positive. The proof uses an Euler–Hurwitz factorization. Three consecutive Bessel channels admit arbitrary nonnegative columnwise mixing with an explicit minor lower bound and positivity criterion; four channels have a negative minor. The paper also characterizes Euler companions and classifies affine Toeplitz masks preserving Polya-frequency sequences at every finite or infinite order. For nonnegative masks supported on three consecutive diagonals, the exact total-nonnegativity order is expressed through associated ultraspherical continuants. The obstruction is controlled either by the initial block or by the limiting Toeplitz tail, depending on the parameter regime. A fixed-degree even expansion and a joint degree–shift limit identify a Dirichlet inverse-square Sturm–Liouville operator whose eigenvalues are Bessel cross-product zeros. The second-order correction, its singular-endpoint behavior, a sharp parameter threshold for negativity at every positive scaled shift, and an exact rounding rule for the first negative staircase are derived. The uniform continuum error estimate is restricted to compact positive scaled-shift intervals; uniqueness of a sign-change point in the scaled shift is not asserted. Additional results give Jacobi row representations with adjacent-degree interlacing for independent positive parameters, an endpoint Narayana column identity, and a refined interval-chain law with exact moments and a normal limit. The symmetric Narayana case and classical spectral ingredients are distinguished from the asymmetric extensions and cited explicitly. The deposit contains the manuscript PDF, LaTeX source, and a computation supplement. One standard-library Python program reproduces exact rational identities, finite minor enumerations, obstruction examples, and rational Bessel-series certificates. A second program, requiring NumPy, compares finite Jacobi spectra with independent continuum shooting and checks correction formulas and endpoint examples at doubled resolutions. Exact output agrees identically with the archived JSON; spectral output agrees within its stated floating-point tolerances. The finite and numerical checks complement the analytic proofs and do not establish the infinite-matrix claims or uniform asymptotic remainders by themselves.