The -Hahn class OP’s comprise systems of OP’s , , or , that are eigenfunctions of a second-order -difference operator. Thus
where , , and are independent of , and where the are the eigenvalues. In the -Hahn class OP’s the role of the operator in the Jacobi, Laguerre, and Hermite cases is played by the -derivative , as defined in (17.2.41). A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949). There are 18 families of OP’s of -Hahn class. These families depend on further parameters, in addition to . The generic (top level) cases are the -Hahn polynomials and the big -Jacobi polynomials, each of which depends on three further parameters.
All these systems of OP’s have orthogonality properties of the form
where is given by or . Here are fixed positive real numbers, and and are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. If and are both nonempty, then they are both unbounded to the right. Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval.
Here only a few families are mentioned. They are defined by their -hypergeometric representations, followed by their orthogonality properties. For other formulas, including -difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 3). See also Gasper and Rahman (2004, pp. 195–199, 228–230) and Ismail (2005, Chapters 13, 18, 21).
For see Koekoek et al. (2010, Eq. (3.6.2)).
For see Koekoek et al. (2010, Eq. (3.12.2)).
(Sometimes in the literature is replaced by .)
The measure is not uniquely determined:
(For discrete -Hermite II polynomials the measure is not uniquely determined.)