Polynomial solutions of hypergeometric type difference equations and their classification
Arnold F. Nikiforov, Vasily B. Uvarov · Integral Transforms and Special Functions · 1993
For the most general case of hypergeometric type difference equations on uniform and nonuniform lattices the explicit expression of polynomials solutions in terms of generalized q-hypergeometric series is obtained from the difference analog of the Rodrigues formula. From this expression, the formulas for all particular cases are derived by an appropriate choice of parameters and by taking various limits. Basic hypergeometric series are also used [14]. The consideration of these polynomials gives us the classification of corresponding q-polynomials in accordance with the values of parameters q, μ of the lattice , and with the number of zeroes of the function σ(s) that is the coefficient before the senior difference derivative in the equation of hypergeometric type. All earlier introduced q-polynomials are included in our scheme of classification. This review is an expanded version of the report which was made by one of the authors on the VII simposium sobre polinomios ortogonales y aplicationes (23-27 September 1991, Granada, Spain). Some materials from [1-3] are also used.