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Advances in Mathematical Physics
Volume 2009, Article ID 268134, 43 pages
Research Article

Additional Recursion Relations, Factorizations, and Diophantine Properties Associated with the Polynomials of the Askey Scheme

1Dipartimento di Fisica, Università di Roma “La Sapienza”, 00185 Roma, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Roma, 00185 Roma, Italy
3Dipartimento di Fisica, Università Roma Tre, 00146 Roma, Italy

Received 29 July 2008; Accepted 1 December 2008

Academic Editor: M. Lakshmanan

Copyright © 2009 M. Bruschi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper, we apply to (almost) all the “named” polynomials of the Askey scheme, as defined by their standard three-term recursion relations, the machinery developed in previous papers. For each of these polynomials we identify at least one additional recursion relation involving a shift in some of the parameters they feature, and for several of these polynomials characterized by special values of their parameters, factorizations are identified yielding some or all of their zeros—generally given by simple expressions in terms of integers (Diophantine relations). The factorization findings generally are applicable for values of the Askey polynomials that extend beyond those for which the standard orthogonality relations hold. Most of these results are not (yet) reported in the standard compilations.