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We give a simple proof of Ramanujan’s 1 ψ 1 {}_1\psi _1 sum. The present argument depends on the residue theorem and the q q -binomial theorem only. Even the manipulation of the series is not used. We give also a proof of Bailey’s 6 ψ 6 {}_6\psi _6 sum in the same manner.

We consider a complex version of the Gauss hypergeometric integral from the view point of the twisted de Rham theory. In particular, we give a formula to express the complex hypergeometric integral in terms of the hermitian form of the ordinary Gauss hypergeometric integrals.

We study a monodromy representation on the space spanned by the integrals over several chains, which are related with Lauricella's system of differential equations ED and the Jordan-Pochhammer differential equation EJP . As an application, we give a necessary and sufficient condition for the irreducibility of ED and of EJP . Mathematics Subject Classification 2000: 33C60, 33C65, 32S40 Running Head: Lauricella's FD

We study monodromy representations associated with the Gauss hypergeometric fuction by using integrals of a multivalued function over several chains. In particular, we give a necessary and sufficient condition for the irreducibility, a concrete realization of the represen- tation in each of the reducible cases, and a list of finite groups arising from the reducible cases.

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.

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

We solve a connection problem for the equation of rank four in Simpson's even family, which we call Even four. We use an integral representation of solutions, and consider the twisted homology group associated with it. The rank of the homology group is five. Its four dimensional subspace, consisting of regularizable cycles, corresponds to the solution space of Even four. The connection problem is reduced to solving the linear relations among twisted cycles. There are only few equations for which connection problems are solved in this way.

Connection matrices associated with the generalized hypergeometric function 3F2 are determined. Our approach is based on the integral representation, and the theory of twisted homology groups (homology with coefficients in local system). We also consider the related monodromy representation and the monodromy-invariant Hermitian form.

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function.

We evaluate the integral appearing in the solution of the Knizhnik-Zamolodchikov equation associated with the vector representation of the simple Lie algebra g of type Bn, Cn or Dn. This integral, which can be considered as a generalization of the Beta integral, is expressed in terms of an alternating product of the Gamma function.

We evaluate the intersection numbers of loaded cycles and twisted forms associated with an n-fold Selberg-type integral. The result is deeply related with the geometry of the configuration space of n+3 points in the projective line.

An integral representation of the Askey–Wilson polynomials is presented in terms of a q -Selberg type integral. Our motivation consists in the study of q -Selberg type integrals from the viewpoint of de Rham theory or holonomic systems.

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.

We construct an integral representation of eigenfunctions for Macdonald's q -difference operator associated with the root system of type C n . It is given in terms of a restriction of a q -Jordan–Pochhammer integral. Choosing a suitable cycle of the integral, we obtain an integral representation of a special case of the Macdonald polynomial for the root system of type C n .

Employing a multidimensional generalization of the beta integral of Barnes, we derive an expression of the continuous Hahn polynomials. Motivation of the present work is to study such integrals from the viewpoint of a finite-difference version of the de Rham 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

We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.

The hypergeometric system [Formula: see text] and the contiguity operators defined on the space of [Formula: see text] matrices [Formula: see text] induce the systems and the contiguity operators on the Grassmannian manifold [Formula: see text] and on the configuration space [Formula: see text] where [Formula: see text] is the group consisting of diagonal matrices of size [Formula: see text]. The first purpose of this paper is to give a rigorous treatment of these systems and contiguity operators, the second is to derive a condition that the system [Formula: see text] is reducible, and the third is to derive the contiguity relations and the contiguous relations satisfied by the solutions of the system on [Formula: see text] or the solutions of the system on [Formula: see text]

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].

Connection matrices associated with the generalized hypergeometric function n+1 F n are determined by using the intersection number of twisted cycles.

In this paper, we study the finite dimensional unitary representations of the quantum group SUq(2). Then we obtain the Peter-Weyl theorem for SUq(2) and the matrix elements of these unitary representations are explicitly expressed in terms of the little q-Jacobi polynomials which are known as q-analogues of orthogonal polynomials. Using these expressions, the orthogonality relations of these polynomials are obtained in terms of the Haar measure on the quantum group SUq(2).

The big q-Jacobi polynomials and the q-Hahn polynomials are realized as spherical functions on a new quantum SUq(2)-space which can be regarded as the total space of a family of quantum 3-spheres.

Real forms of the quantum universal enveloping algebraU q (sl(2)) and a topological quantum group associated with this algebra are discussed.

Some series of unitary representations of the quantum group SU q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU q (1, 1) and aq-analogue of some classical identities are discussed.

Orthogonal bases for the algebras of functions of Podles' quantum 2-spheres are explicitly determined in terms of bigq-Jacobi polynomials. This gives a group-theoretic interpretation of the symmetric bigq-Jacobi polynomials and the symmetricq-Hahn polynomials.

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

We study the twisted homology group attached to a Selberg type integral under some resonance condition, which naturally appears in the su_2-conformal field theory and the representation of the Iwahori-Hecke algebra. We determine the dimension of the space of the regularizable cycles. The dimension-formula is given in terms of the generalized hypergeometric series., Comment: 17 pages, no figures

We study the structure of the twisted homology groups attached to weighted lines in the plane with resonant singular points, and find possible intersection pairings. As a typical example, we treat homology groups attached to 2-dimensional Selberg-type integrals.

We give a formula representing a duality of Macdonald-Koornwinder polynomials. Using this formula, an integral representation of Macdonald-Koornwinder polynomials is derived, a special case of which is the conjectural formula stated in [22]. We also present the corresponding formula to Heckman and Opdam's Jacobi polynomials of type $BC_m$.

We present a new formula of Cauchy type for the nonsymmetric Macdonald polynomials which are joint eigenfunctions of q-Dunkl operators. This gives the explicit formula for a reproducing kernel on the polynomial ring of n variables., Comment: 19 pages, AMS-TeX Ver.2.1, no figures

The concept of quantum groups is important for the study of the quantum Yang-Baxter equations, Drinfeld [2], Jimbo [4], Manin [5] and others. On the other hand, Woronowicz [10] introduced the concept of compact matrix pseudogroups through the study of the dual object of groups. As pointed out by Rosso in [8], these two concepts are related to each other as quantum Lie algebras and quantum Lie groups. In this talk we want to indicate that the ideas of Kac algebras studied by Takesaki [9] and Enock and Schwartz [3] et al. are helpful for the study of quantum groups. As a result we can give a geometric interpretation for a q-analogue of a certain class of special functions, which has been a long standing problem of q- analogues.