The connection problem associated with a Selberg type integral and the \(q\)-Racah polynomials (Q2868534)

The author considers the special case of Selberg type integral NEWLINE\[NEWLINE\int _{\gamma} \prod _{1\leq i<j\leq m}(t_j-t_i)^g\prod _{1\leq i\leq m}t_i^a(1-t_i)^b(t_i-z)^cdt_1\cdots dt_mNEWLINE\]NEWLINE which satisfies an ordinary differential equation of order \(m+1\) with singular points at \(0\), \(1\) and \(\infty\). The connection problem consists in giving linear relations between the fundamental sets of solutions around the singularities and in writing down the coefficients explicitly. Here \(g\), \(a\), \(b\) and \(c\) are complex numbers and \(\gamma\) a suitable cycle. The connection coefficients are expressed in terms of the \(q\)-Racah polynomials. As an application examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. Such Hermitian forms are related to the correlation functions of non-diagonal type in \(\hat{sl_2}\)-conformal field theory.