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I am studying some classes of orthogonal polynomials and want to find out which of them belong to the Askey scheme. To give a simple example consider the polynomials $${p_n}(x,r) = \sum\limits_{k = 0}^{\left\lfloor {\frac{n}{2}} \right\rfloor } {{{( - 1)}^k}c(n,k,r){x^{n - 2k}}} $$ with $$c(n,k,r) = \binom{\lfloor{{n}/2}\rfloor}k \prod\limits_{j = 0}^{k - 1} {\left( {\left\lfloor {\frac{{n + 1 - 2j}}{2}} \right\rfloor r - 1} \right)}.$$ For $r = 2$ they coincide with the (probabilists’) Hermite polynomials.

My questions are: Have these polynomials been studied in the literature? Which of them belong to the Askey scheme?

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  • $\begingroup$ It is not an answer to your question. Nevertheless, sometimes I also study families of polynomials which seem to belong to the Askey scheme. Rather then the explicit expression for the polynomials, I found more useful to use coefficients from the three-term recurrence or the orthogonality relation. Do you known these coefficients and/or the orthogonality relation for polynomials $p_{n}(x,r)$? I believe (although I did not try) that at least the former could be established. $\endgroup$
    – Twi
    Mar 9, 2015 at 19:32
  • $\begingroup$ You might consider adding a top-level tag in order to increase the visibility of this question. $\endgroup$
    – Stefan Kohl
    Oct 15, 2015 at 20:21

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