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Register a new userElliptic extension of Gustafsons $q$-integral of type $G_2$
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The evaluation formula for an elliptic beta integral of type $G_2$ is proved. The integral is expressed by a product of Ruijsenaars elliptic gamma functions, and the formula includes that of Gustafsons $q$-beta integral of type $G_2$ as a special limiting case as $pto 0$. The elliptic beta integral of type $BC_1$ by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.
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We investigate the connection problem for the Jackson integral of type $A_n$. Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of $q$-difference equations.
The Ramanujan $_1psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Baileys very-well-poised $_6psi_6$ summation. Bilateral Jackson integral generalizations of the Dixon--Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_1psi_1$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulas for Jackson integrals relating to other roots systems obtained from the same perspective.
The Dixon--Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The $_1psi_1$ summation is a bilateral generalization of the $q$-binomial theorem. It is shown that a $q$-generalization of the Dixon--Anderson integral, due to Evans, and multi-dimensional generalizations of the $_1psi_1$ summation, due to Milne and Gustafson, can be viewed as having a common origin in the theory of $q$-difference equations as expounded by Aomoto. Each is shown to be determined by a $q$-difference equation of rank one, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.
We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved by combining the $q$-difference equations of the determinant and its asymptotic analysis along the singularities. The elliptic interpolation functions of type $BC_n$ are essentially used in the study of the $q$-difference equations.
The connection formula for the Jackson integral of type $BC_n$ is obtained in the form of a Sears--Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific bilateral multiple series. The coefficients of this expansion are expressed by certain elliptic Lagrange interpolation functions. Analyzing basic properties of the elliptic Lagrange interpolation functions, an explicit determinant formula is provided for a fundamental solution matrix of the associated system of $q$-difference equations.
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