q-orthogonal polynomials
This chapter and the next one have many things in common. The generating function technique by Rainville is used to prove recurrences for q-Laguerre polynomials. We prove product expansions and bilinear generating functions for q-Laguerre polynomials by u
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q-orthogonal polynomials
9.1 Ciglerian q-Laguerre polynomials 9.1.1 The different Laguerre-philosophies The French philosophy under Appell and Kampé de Fériet laid the foundations for the so-called Hungarian-French-Italian hypergeometric philosophy. In 1936 Ervin Feldheim (Budapest) wrote together with Paul Erdös on interpolation. The Feldheim family tree, just as the Pringsheim and Heine family trees, consists of famous mathematicians, artists and musicians. In 1943 Feldheim [199] wrote down the expansion formulas of BurchnallChaundy [89, (26), (27), (29)–(31), (38), (39), (42), (43)] from 1940. Some of these formulas were expressed as integral formulas. In the second section Feldheim derived the generating function of Laguerre polynomials and different results for orthogonal polynomials. The paper [199] also included formulas for Laguerre polynomials of two variables. Feldheim summarized the relationship between JLH in [200]. In this chapter we will find four q-analogues of the formula min(m,n) (−x)k (α+n+k) m + n (α) Lm+n (x) = (x)L(α+k) Lm−k n−k (x). m k!
(9.1)
k=0
Feldheim [200, p. 134, (43)] has given the rather similar formula
min(m,n) m + n (α) (α+n) (α+m) k m+n+α (−1) Lm−k (x)Ln−k (x). Lm+n (x) = k m
(9.2)
k=0
Feldheim’s formulas have attracted the interest of Carlitz. Letterio Toscano did, in fact, continue the work of Feldheim and Schwatt. He did publish 192 articles in the period 1930–1980; like Carlitz, he did not write any book. At the beginning he wrote in French, then in his native Italian and, in the recent years, alternately in French and English. In Messina he took over the legacy of Pia Nalli, but without T. Ernst, A Comprehensive Treatment of q-Calculus, DOI 10.1007/978-3-0348-0431-8_9, © Springer Basel 2012
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q-orthogonal polynomials
her fame. The reason was that he, in contrast to Feldheim, did not master the real Analysis and could therefore usually publish only in journals of secondary importance. But do not forget that special functions are not real Analysis; there is even a theory of orthogonality in q-integrals! Nevertheless, Toscano’s contributions to the theory of orthogonal polynomials and hypergeometric functions of many variables were certainly no less interesting than those of Feldheim. Toscano dealt with polynomial difference operators (see Chapter 5) and obtained some deep results. There are also many of Toscano’s findings in Chapter 4 (Bernoulli and Euler numbers) and Chapter 5 (Stirling numbers and derivative formulas). In principle Toscano used a difference method similar to that of Sears [460], [461] to prove transformations for special functions. Sears did, however, investigate infinite series. Some polynomials in the so-called Askey tableau can be expressed in the form of Laguerre polynomials. The same goes for the q-case: some polynomials in the socalled q-Askey-Tableau can be expressed in the form of q-Laguerre polynomials. It is our hope to be able to treat also other q-orthogonal polynomials in other books. The work on q-Hermite pol
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