论文标题
Hausdorff力矩问题的棕色和Tutte组合数：精确解决方案
Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution
论文作者
论文摘要
我们研究了W. G. Brown（1964）和W. T. Tutte（1980）引入的组合序列$ a（m，n）$出现在凸多面体的枚举中。他们的公式为$$ a（m，n）= \ frac {2（2m+3）！} {（m+2）！ m！} \，\ frac {（4n+2m+1）！} {n！ （3n + 2m + 3）！} $$，带有$ n，m = 0，1，2，\ ldots $，我们将其视为Hausdorff Moments，其中$ m $是一个参数，$ n $列举了时刻。我们精确地解决了相应的Hausdorff瞬间问题：$ a（m，n）= \ int_ {0}^{r} x^{n} w_ {m}（x）d x $在天然支持$（0，r）$，$ r = 4^{4} {4} {4}/3^{3} $上，使用inverse MellIn of verseMellIn tronsement。我们明确地提供重量功能$ w_ {m}（x）$，就meijer g-功能$ g_ {4，4}^{4，0} $，或等效地，广义的超细几点函数$ {_ {_ {_ {3} f_ {2} f_ {2}} $（for_ {2}} $} （对于$ m \ geq 2 $）。对于$ m = 0，1 $，我们证明$ w_ {m}（x）$是非负且可正常化的，因此它们是概率分布。对于$ m \ geq 2 $，$ w_ {m}（x）$是签名的功能在支撑的四肢上消失。通过完全根据Meijer g表示，我们揭示了一个整体关系，该关系直接基于$ a（m，n）$的普通生成函数直接提供$ w_m（x）$作为输入。所有结果均以分析和图形为单位进行研究。
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.
