ABSTRACT The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors , whose zeros all lie on the ‘critical line’ (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their ‘critical polynomial’ factors are then identified in terms of hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation . Normalization yields the rational function whose denominator has singularities on the negative real axis. Moreover, as along the positive real axis, from below. For the Chebyshev polynomials, we obtain the simpler S:2/1 binomial form, and with the nth Catalan number, we deduce that and yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics.

中文翻译：

---

模仿黎曼 zeta 函数的二项式多项式

摘要 Gegenbauer 多项式的（广义）梅林变换具有多项式因子 ，其零点都位于“临界线”（称为临界多项式）上。根据与 HW Gould 的 S:4/3、S:4/2 和 S:3/1 二项式系数形式相关的组合和来识别变换。然后根据超几何函数确定它们的“关键多项式”因子。此外，我们将这些结果扩展到具有函数方程 的关键多项式的单参数族。归一化产生分母在负实轴上具有奇点的有理函数。此外，沿着正实轴，从下方。对于切比雪夫多项式，我们获得了更简单的 S:2/1 二项式形式，并使用第 n 个加泰罗尼亚数推导出来并产生奇数。