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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.
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 $_6psi_6$ summ
Motivated by Alladis recent multi-dimensional generalization of Sylvesters classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different
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 lim
We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar four-point correlation functions given by conformal fishnet Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of pow
We consider a family of solutions of $q-$difference Riccati equation, and prove the meromorphic solutions of $q-$difference Riccati equation and corresponding second order $q-$difference equation are concerning with $q-$gamma function. The growth and