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This article is cited in 3 scientific papers (total in 3 papers)
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 $s_{r,k+r}^{\alpha}(x)= U_k^r \left[s_{k+r}^{\alpha}(x) - \sum\limits_{\nu=0}^{r-1} V_{k,\nu}^r x^{[\nu]}\right]$, where for the numbers $U_k^r$, $V_{k,\nu}^r$ we found the explicit expressions, is established.
Key words:
Sobolev orthogonal polynomials, Charlier polynomials, Sobolev-type inner product.
Citation:
I. I. Sharapudinov, I. G. Guseinov, “Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials”, Izv. Saratov Univ. Math. Mech. Inform., 18:2 (2018), 196–205
Linking options:
https://www.mathnet.ru/eng/isu755 https://www.mathnet.ru/eng/isu/v18/i2/p196
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