Multivariate orthogonal polynomials on finite groups and their q-analogues

在本研究中，我处理的特殊功能产生有限群或其q-类似物。作为结果，得到了Gelfand对（S_2n，H_n）的推广，Gelfand对的球函数的完整描述，多元Krawtchouk多项式正交性的充要条件，对称群的特征公式。进一步将这些结果应用到有限齐性空间作为随机空间，得到了经典Frobenius-Schur定理的一个推广。

In this study, I dealt with the special functions arising from finite groups or their q-analogues. As results, I obtained the complete description of the spherical functions of a certain Gelfand pair which is generalization of the Gelfand pair(S_2n, H_n), a necessary and sufficient condition of orthogonal properties of multivariate Krawtchouk polynomials, and a formulae of the character of the symmetric groups. Furthermore I applied these results on finite homogenous spaces as stochastic spaces and obtained a generalization of classical Frobenius-Schur theorem.