Research on the holonomic q difference systems associated with the Jackson integrals of Weyl group invariant

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A key reason to consider the Jackson integrals, which admit Weyl group symmetry, is to provide an explanation and an extension of a series of classical basic hypergeometric series and their q-difference equations with respect to their parameters.In order to show the linear independence of the fundamental solutions of the q-difference system, the non-degeneracy of the Wronskian should be proved. For this purpose, Ito and K. Aomoto (Kyoto sangyo university) showed that the Wronskian is expressed as a product of q-gamma functions. Ito and Koike showed that a Vandermonde-type determinant, whose entries are the irreducible characters of the classical groups, is expressed as a power of the difference product. Technically speaking, the proof of the non-degeneracy of the Wronskian is obtained from the Vandermonde-type determinant, which is a limiting case of q-0 for the Wronskian. The formula given by Ito and Koike plays an important role in the result by Ito and Aomoto.Ito and Y. Sanada (Tsuda collage) gave an explicit connection formula, which indicates that the general solution of the q-difference system of the BC1-type Jackson integral is expressed as a linear combination of the fundamental solutions of the system. They showed that the connection formula is equivalent to the classical hypergeometric transformation formula, which was discovered in 1950s by Sears and Slater. Consequently they found the Weyl group symmetry in the transformation formula and gave a very simple proof for the formula. Ito also gave an explicit connection formula for the BCn-type Jackson integral by extending it from the result of Ito and Sanada.Taniguchi gave an elementary construction for the invariant polynomials of type F4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D4, based on the fact that Weyl group of type D4 is included in a normal subgroup of Weyl group of type F4.