A Bessel-zero obstruction to hyperbolic complete monotonicity of noncentral chi-square densities

Abstract

Baricz, Prabhu K, Singh and Vijesh asked for the optimal hyperbolically completely monotone (HCM) range of the noncentral chi-square density. The problem was motivated by the gap between known infinite divisibility and the stronger generalized-gamma-convolution/HCM classification. We prove that the HCM range is exactly the central line. More generally, for $a,b>0$ and $θ\ge 0$, the density $p_{a,b,θ}(x)$ is HCM if and only if $θ=0$. Thus the noncentral chi-square density satisfies $χ_{μ,λ}\in\mathrm{HCM}$ if and only if $λ=0$. The proof uses the leading small-$u$ HCM signs of $p(uv)p(u/v)$. These signs are governed by complete Bell polynomials whose signed generating function is $e^{bt}{}_0F_1(;a;-bθt)$. A positive zero inherited from $J_{a-1}$ rules out nonnegative Taylor coefficients when $θ>0$. Consequently, Poisson shape-mixtures of HCM gamma densities need not be HCM.