Rahman's biorthogonal functions and superconformal indices

TL;DR

This paper explores biorthogonal functions linked to hypergeometric integrals, which are relevant in quantum field theory and lattice models, providing explicit systems, new summation formulas, and orthogonality measures.

Contribution

It introduces explicit biorthogonal systems based on Rahman's functions and develops new bilateral summation formulas and orthogonality measures.

Findings

Explicit biorthogonal systems using Rahman's functions

New bilateral Jackson and q-Saalschütz summation formulas

Novel continuous and discrete biorthogonality measures

Abstract

We study biorthogonal functions related to basic hypergeometric integrals with coupled continuous and discrete components. Such integrals appear as superconformal indices for three-dimensional quantum field theories and also in the context of solvable lattice models. We obtain explicit biorthogonal systems given by products of two of Rahman's biorthogonal rational ${}_{10}W_9$-functions or their degenerate cases. We also give new bilateral extensions of the Jackson and $q$-Saalsch\"utz summation formulas and new continuous and discrete biorthogonality measures for Rahman's functions.