On the orthogonality of $q$-classical polynomials of the Hahn class

The central idea behind this review article is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the $q$-hypergeometric difference equation on a $q$-linear lattice by means of a qualitative analysis of the $q$-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every possible rational form of the polynomial coefficients in the $q$-Pearson equation, together with various relative positions of their zeros, to describe a desired $q$-weight function supported on a suitable set of points. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known $q$-polynomials of the Hahn class to a larger set of their parameters.