Recurrence relations for polynomials orthonormal on Sobolev, generated by Laguerre polynomials

R. M. Gadzhimirzaev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 1, pp. 17-24

Résumé

In this paper we consider the system of polynomials $l_{r,n}^{\alpha}(x)$ ($r$ — natural number, $n=0, 1, \ldots$), orthonormal with respect to the Sobolev inner product (Sobolev orthonormal polynomials) of the following type $\langle f,g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_{0}^{\infty} f^{(r)}(t)g^{(r)}(t)\rho(t)\,dt$ and generated by the classical orthonormal Laguerre polynomials. Recurrence relations are obtained for the system of Sobolev orthonormal polynomials, which can be used for studying various properties of these polynomials and calculate their values for any $x$ and $n$. Moreover, we consider the system of the Laguerre functions $\mu_{n}^{\alpha}(x) = \sqrt{\rho(x)}l_{n}^{\alpha}(x)$, which generates a system of functions $\mu_{r, n}^{\alpha}(x)$ orthonormal with respect to the inner product of the following form $\langle \mu_{r,n}^\alpha,\mu_{r,k}^\alpha\rangle= \sum_{\nu=0}^{r-1}(\mu_{r,n}^\alpha(x))^{(\nu)}|_{x=0} (\mu_{r,k}^\alpha(x))^{(\nu)}|_{x=0}+ \int_{0}^{\infty} (\mu_{r,n}^\alpha(x))^{(r)}(\mu_{r,k}^\alpha(x))^{(r)}\,dx.$ For the generated system of functions $\mu_{r,n}^{\alpha}(x)$, recurrence relations for $\alpha=0$ are also obtained.