Geodesic
Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 2, pp. 196-205 
I. I. Sharapudinov; I. G. Guseinov. Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 2, pp. 196-205. http://geodesic.mathdoc.fr/item/ISU_2018_18_2_a5/
@article{ISU_2018_18_2_a5,
     author = {I. I. Sharapudinov and I. G. Guseinov},
     title = {Polynomials orthogonal with respect to {Sobolev} type inner product generated by {Charlier} polynomials},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {196--205},
     year = {2018},
     volume = {18},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2018_18_2_a5/}
}
TY  - JOUR
AU  - I. I. Sharapudinov
AU  - I. G. Guseinov
TI  - Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2018
SP  - 196
EP  - 205
VL  - 18
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ISU_2018_18_2_a5/
LA  - ru
ID  - ISU_2018_18_2_a5
ER  - 
%0 Journal Article
%A I. I. Sharapudinov
%A I. G. Guseinov
%T Polynomials orthogonal with respect to Sobolev type inner product generated by Charlier polynomials
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2018
%P 196-205
%V 18
%N 2
%U http://geodesic.mathdoc.fr/item/ISU_2018_18_2_a5/
%G ru
%F ISU_2018_18_2_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Iserles A., Koch P. E., Norsett S. P., Sanz-Serna J. M., “On polynomials orthogonal with respect to certain Sobolev inner products”, J. Approx. Theory, 65:2 (1991), 151–175 | DOI | MR

[2] Marcellan F., Alfaro M., Rezola M. L., “Orthogonal polynomials on Sobolev spaces: old and new directions”, J. Comput. Appl. Math., 48:1–2 (1993), 113–131 | DOI | MR

[3] Meijer H. G., “Laguerre polynomials generalized to a certain discrete Sobolev inner product space”, J. Approx. Theory, 73:1 (1993), 1–16 | DOI | MR

[4] Kwon K. H., Littlejohn L. L., “The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for positive integers $k$”, Ann. Numer. Anal., 2 (1995), 289–303 | MR

[5] Kwon K. H., Littlejohn L. L., “Sobolev orthogonal polynomials and second-order differential equations”, Rocky Mountain J. Math., 28 (1998), 547–594 | DOI | MR

[6] Marcellan F., Xu Y., On Sobolev orthogonal polynomials, 25 Mar 2014, 40 pp., arXiv: 1403.6249v1 [math.CA] | MR

[7] Sharapudinov I. I., Gadzhieva Z. D., “Sobolev orthogonal polynomials generated by Meixner polynomials”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 16:3 (2016), 310–321 (in Russian) | DOI | MR

[8] Sharapudinov I. I., Mixed Series in Orthogonal Polynomials, Izd-vo DNC RAN, Makhachkala, 2004, 176 pp.

[9] Sharapudinov I. I., Polynomials Orthogonal on Grids, Izd-vo Dag. gos. ped. un-ta, Makhachkala, 1997, 252 pp.

[10] Bateman H., Erdelyi A., Higher Transcendental Functions, v. 2, McGraw-Hill Book Company, New York, 1953, 396 pp. | MR

[11] Shirjaev A. N., Probability-1, MTsNMO, M., 2007, 552 pp. | MR