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Register a new userThe $q$-Dixon--Anderson integral and multi-dimensional $_1psi_1$ summations
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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.
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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.
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 colors. This new identity encompasses a handful of classical results as special cases, such as Cauchys identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.
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.
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 power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2,C) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.
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 value distribution of differences on solutions of $q-$difference Riccati equation are also investigated.
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