Multivariate orthogonal polynomials on finite groups and their q-analogues

在这个研究中，我处理了由有限群或它们的q-类似物引起的特殊函数。得到了一类Gelfand对（S_2n, H_n）的推广Gelfand对的球函数的完备描述，多元Krawtchouk多项式正交性质的一个充分必要条件，以及对称群的一个性质公式。进一步将这些结果应用于有限齐次空间作为随机空间，得到了经典Frobenius-Schur定理的推广。

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.