Sampling and Moments of Reciprocal Quadratic Mellin Integrals

RESUMO

We study logarithmic Mellin integrals attached to a reciprocal quadratic rational function. When the rational kernel has algebraic coefficients, positive-integer samples of an entire deformation have a generating function which, after division by $\pi$, is algebraic for every $a\in\mathbb Q$ with $0<\vert{}a\vert{}<1$. For the principal half-weight example we determine the minimal quartic, complete finite branch locus, minimal differential equation, polynomial recurrence, and coefficient asymptotics. Throughout the saddle region $c>1$, $-2\sqrt c<b<0$, positive hyperbolic formulas for real $0<a<1$ give determinate Hausdorff moment sequences, strict total positivity, and monotone quotient limits. For real $\vert{}a\vert{}<1$, a general coefficient asymptotic in the same region shows that one maximum controls both the sampling radius and the high logarithmic moments. On the fixed-critical locus, two-term diagonal asymptotics give a normalized product tending to $\pi$; division by the displayed correction $1+\gamma/M$ gives an $O(M^{-2})$ approximation.