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The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the q-Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. It is noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in -conformal field theory.

An integral representation of the Meixner–Pollaczek polynomials is presented in terms of a multidimensional generalization of the Barnes type integral. Motivation consists in the study of the Barnes type integrals from the viewpoint of a finite-difference version of the de Rham theory.

We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type An-1. As a special case, the corresponding constant term conjecture is also proved.

We construct a solution of Cherednik's quantum Knizhnik Zamolodchikov equation associated with the root system of type $C_n$. This solution is given in terms of a restriction of a $q$-Jordan-Pochhammer integral. As its applicaton, we give an explicit expression of a special case of the Macdonald polynomial of the $C_n$ type. Finally we explain the connection with the representation of the Hecke algebra., Comment: 18 pages, AMS-Latex, no figures

Consider the function with variables z1…,zmwhere υ and λj (j = 1,…,m) are complex numbers and.Г is a suitably chosen integral domain. In case m = 2, if we set [z1 z2]n as Г, it is the Selberg integral [22]. Our function can be regarded as an extention of it; so we may call (0.1) a Selberg type integral It is known that (0.1) satisfies a Gauss-Manin system, i.e. a system of rationally holonomic differential equations [3], [21].

We evaluate the intersection numbers of loaded cycles associated with an $n$-fold Selberg-type integral. We proceed inductively using high-dimensional local systems.

Fix a complex numberqwith|q|< 1. LetT1…,Tnbe n-commutingq-difference operators defined byfor a functionf(x), x=(x1,…,xn)ε (C*)n. Consider a system of linearq-difference equations in several variables for a matrix valued functionon(C*)nas follows