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

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 $\theta \ge 0$, the density $p_{a,b,\theta }(x)$ is HCM if and only if $\theta =0$. Thus the noncentral chi-square density satisfies ${\chi }_{\mu ,\lambda }\in \mathrm{H}\mathrm{C}\mathrm{M}$ if and only if $\lambda =0$. The proof uses the leading small-$u$ HCM signs of $p(uv)p(u\mathrm{/}v)$. These signs are governed by complete Bell polynomials whose signed generating function is $e^{bt}{}_0{F}_1(;a;-b\theta t)$. A positive zero inherited from $J_{a-1}$ rules out nonnegative Taylor coefficients when $\theta >0$. Consequently, Poisson shape-mixtures of HCM gamma densities need not be HCM.