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Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials. (English) Zbl 0862.33006

Levi, Decio (ed.) et al., Symmetries and integrability of difference equations. Papers from the workshop, May 22–29, 1994, Estérel, Canada. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 9, 319-335 (1996).
If one has two families of orthogonal polynomials, say \(\{P_n(x)\}\) and \(\{Q_n (x)\}\), and expresses the elements of one family by elements of the other one, then the coefficients appearing in these representations are called connection coefficients. In the paper a simple approach is given for building up these coefficients recursively. The assumptions on both families of polynomials are rather weak and allow to include classical and semiclassical orthogonal polynomials as well as some other families of polynomials. The new algorithm is applied to classical discrete families of polynomials (Charlier, Meixner, Krawtchouck, Hahn). For this case, the recurrences and explicit expressions for the connection coefficients are derived, and the problem of positivity of these coefficients is solved.
For the entire collection see [Zbl 0852.00027].
Reviewer: H.Stahl (Berlin)

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33E30 Other functions coming from differential, difference and integral equations
39A10 Additive difference equations
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