Multivariate orthogonal polynomials on finite groups and their q-analogues

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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.