One-parameter orthogonality relations for basic hypergeometric series (Q1432902)

The second order hypergeometric \(q\)-difference equation is \[ (c-abz)f(qz)+\big(-(c+q)+(a+b)z\big)f(z)+(q-z)f(z/q)=0,\tag{1} \] having \[ f(z)={}_{2}\varphi_{1}\Big(a,b,c ;q,z \Big) \] as a solution, where \(_{2}\varphi_{1}\) is the basic hypergeometric series. The \(q\)-analogue of the exponential function \({\mathcal E}_{q}(z;t)\) can be expressed in terms of a \(_{2}\varphi_{1}\) and the function \({\mathcal E}_{q^2}\) is an eigenfunction of the second order hypergeometric \(q\)-difference operator. The author describes the solutions of (1), their interrelations and he is particularly interested in the case \(c=-q\) as this case corresponds to \({\mathcal E}_{q^2}\). In this case, and for specific values of the remaining parameters, the second order hypergeometric \(q\)-difference operator can be realized as an unbounded symmetric operator on the Hilbert space \(\ell^{2}(\mathbb{Z})\). The corresponding operator is not essentially self-adjoint, but it has deficiency indices \((1,1)\). The author describes the self-adjoint extensions, which depend on one extra parameter, and studies the corresponding spectral decompositions. This gives rise to one-parameter orthogonality relations for sums of two \(_{2}\varphi_{1}\)-series. In particular, it is shown that a linear combination of two \({\mathcal E}_{q^2}\) functions satisfies certain orthogonality relations. The author also presents the link with the recurrence relation for the big \(q\)-Jacobi functions or the associated dual \(q\)-Hahn polynomials, and this leads to a quadratic transformation in which a \(_{2}\varphi_{1}\)-series in base \(q\) is given as a \(_{3}\varphi_{2}\)-series in base \(q^2\). In particular, this gives a new expression for \({\mathcal E}_{q}\) as a sum of two \(_{3}\varphi_{2}\)-series in base \(q\).