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 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.: Year: Volume: Issue: Page: Find

 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2018, Volume 18, Issue 2, Pages 196–205 (Mi isu755)

Scientific Part
Mathematics

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

I. I. Sharapudinova, I. G. Guseinovba

a Dagestan Scientific Center of RAS, 45, M. Gadzhieva Str., Makhachkala, 367025, Russia
b Dagestan State University, 43-a, M. Gadzhieva Str., Makhachkala, 367000, Russia

Abstract: 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 $s_{r,k+r}^{\alpha}(x)= U_k^r [s_{k+r}^{\alpha}(x) - \sum\limits_{\nu=0}^{r-1} V_{k,\nu}^r x^{[\nu]}]$, where for the numbers $U_k^r$, $V_{k,\nu}^r$ we found the explicit expressions, is established.

Key words: Sobolev orthogonal polynomials, Charlier polynomials, Sobolev-type inner product.

DOI: https://doi.org/10.18500/1816-9791-2018-18-2-196-205

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Bibliographic databases:

UDC: 517.587

Citation: I. I. Sharapudinov, I. G. Guseinov, “Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:2 (2018), 196–205

Citation in format AMSBIB
\Bibitem{ShaGus18} \by I.~I.~Sharapudinov, I.~G.~Guseinov \paper Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials \jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. \yr 2018 \vol 18 \issue 2 \pages 196--205 \mathnet{http://mi.mathnet.ru/isu755} \crossref{https://doi.org/10.18500/1816-9791-2018-18-2-196-205} \elib{http://elibrary.ru/item.asp?id=35085049}