We will then identify a natural setting, namely that of q-generalizations of the Dixon Anderson and Selberg integrals both multidimensional generalizations of the Euler beta integral as candidates for permitting analysis from this perspective and as providing new multi-dimensional extensions of (1).
In the limit q [right arrow] 1 this reduces to the Euler beta integral, while taking q [right arrow] 1 in the recurrence (6) implies the beta integral evaluation
In the case n = 1 both (19) and (20) reduce to the Euler beta integral (18).
Finally, integrating Z and Y by using multivariate gamma and multivariate beta integrals
and simplifying the resulting expression, we obtain the desired result.
Among the topics are orthogonal polynomials spanning a non-standard flag, the supersymmetric spectra of two planar integrable quantum systems, solvable rational extension of translationally shaped invariants potentials, explicit higher-dimension Darboux transformations for the time-dependent Schrodinger equation, and elliptic beta integrals
and solvable models of statistical mechanics.
i]-integrals to well known multiple Beta integrals by the following transformation:
Further, we evaluate the multiple Beta integrals thus obtained and finally on reinterpreting the resulting expression thus obtained in terms H-function of 2n variables, we easily arrive at the right-hand side of the main theorem after a little simplification.