Abstract
In a letter dated March 3, 1971, L. Carlitz defined a sequence of polynomials, Φ n (a,b; x, y; z), generalizing the Al-Salam & Carlitz polynomials, but closely related thereto. He concluded the letter by stating: “It would be of interest to find properties of Φ n (a, b; x, y; z) when all the parameters are free.” In this paper, we reproduce the Carlitz letter and show how a study of Carlitz’s polynomials leads to a clearer understanding of the general 3Φ2 (a, b, c; d; e; q, z).
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Al-Salam, W., Carlitz, L.: Some q-orthogonal polynomials. Math. Nach. 30, 47–61 (1965)
Andrews, G.E.: Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, 618–622 (1972)
Andrews, G.E.: The Theory of Partitions. Encyl. of Math. and Its Appl., vol. 2, Addison-Wesley, Reading, (1976) (reissued: Cambridge University Press, Cambridge (1985, 1998))
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyl. of Math. and Its Appl., vol. 35, Cambridge University Press, Cambridge (1990)
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Dedicated to my friend, Richard Askey.
2000 Mathematics Subject Classification Primary—33D20.
G. E. Andrews: Partially supported by National Science Foundation Grant DMS 0200047.
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Andrews, G.E. Carlitz and the general 3φ2 . Ramanujan J 13, 311–318 (2007). https://doi.org/10.1007/s11139-006-0254-0
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DOI: https://doi.org/10.1007/s11139-006-0254-0