Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution

Abstract

We investigate the combinatorial sequences $A(M, n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M, n) = \frac{2 (2M+3)!}{(M+2)! M!}\,\frac{(4n+2M+1)!}{n! (3n + 2M + 3)!} $$ with $n, M =0, 1, 2, \ldots$, and we conceive it as Hausdorff moments, where $M$ is a parameter and $n$ enumerates the moments. We solve exactly the corresponding Hausdorff moment problem: $A(M, n) = \int_{0}^{R} x^{n} W_{M}(x) d x$ on the natural support $(0, R)$, $R = 4^{4}/3^{3}$, using the method of inverse Mellin transform. We provide explicitly the weight functions $W_{M}(x)$ in terms of the Meijer G-functions $G_{4, 4}^{4, 0}$, or equivalently, the generalized hypergeometric functions ${_{3}F_{2}}$ (for $M=0, 1$) and ${_{4}F_{3}}$ (for $M \geq 2$). For $M = 0, 1$, we prove that $W_{M}(x)$ are non-negative and normalizable, thus they are probability distributions. For $M \geq 2$, $W_{M}(x)$ are signed functions vanishing on the extremities of the support. By rephrasing this problem entirely in terms of Meijer G representations we reveal an integral relation which directly furnishes $W_M(x)$ based on ordinary generating function of $A(M, n)$ as an input. All the results are studied analytically as well as graphically.