One-parameter Orthogonality Relations for Basic Hypergeometric Series

The second order hypergeometric q-difference operator is studied for the value c = −q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space l(Z). The operator has deficiency indices (1, 1) and we describe as explicitly as possible the spectral resolutions of the self-adjoint extensions. This gives rise to one-parameter orthogonality relations for sums of two 2φ1-series. In particular, we find that the Ismail-Zhang q-analogue of the exponential function satisfies certain orthogonality relations.