Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials

I. I. Sharapudinov ; I. G. Guseinov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 2, pp. 196-205

Résumé

The problem of constructing of the Sobolev orthogonal polynomials $s_{r,n}^\alpha(x)$ generated by Charlier polynomials $s_n^\alpha(x)$ is considered. It is shown that the system of polynomials $s_{r,n}^\alpha(x)$ generated by Charlier polynomials is complete in the space $W^r_{l_\rho}$, consisted of the discrete functions, given on the grid $\Omega=\{0,1,\ldots\}$. $W^r_{l_\rho}$ is a Hilbert space with the inner product $\langle f,g \rangle$. An explicit formula in the form of $s_{r,k+r}^{\alpha}(x) = \sum\limits_{l=0}^{k} b_l^r x^{[l+r]} $, where $x^{[m]} = x(x-1)\ldots(x-m+1)$, is found. The connection between the polynomials $s_{r,n}^\alpha(x)$ and the classical Charlier polynomials $s_n^\alpha(x)$ in the form of $s_{r,k+r}^{\alpha}(x)= U_k^r \left[s_{k+r}^{\alpha}(x) - \sum\limits_{\nu=0}^{r-1} V_{k,\nu}^r x^{[\nu]}\right]$, where for the numbers $U_k^r$, $V_{k,\nu}^r$ we found the explicit expressions, is established.