### A Bochner theorem for Dunkl polynomials.

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We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0): $\genfrac{}{}{0pt}{}{|{t}^{k}{d}^{p}\left({L}_{n}^{\alpha}\left(t\right){e}^{-t/2}\right)|\le {2}^{-min(\alpha ,k)}{4}^{k}(n+1)...(n+k)(n+p+max(\alpha -k,0)}{n)}$ for all natural numbers k, p, n ≥ 0 and t ≥ 0. Also, we give (as the main result of this paper) a technique to estimate the order in k and p in bounds similar to the previous ones, which will be used to see that the estimate on k and p in the previous bounds is sharp and to give an estimate on k and p in other bounds on the Laguerre polynomials proved by Szegö.

We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture...

A new approach is presented for constructing recurrence relations for the modified moments of a function with respect to the Gegenbauer polynomials.

Let ${{A}_{k}}_{k=0}^{+\infty}$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$betheJacobipolynomialsanddefinethefunctions$$H\u2099(\alpha ,z)...$