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 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2016, Volume 16, Issue 3, Pages 310–321 (Mi isu650)

Mathematics

Sobolev orthogonal polynomials generated by Meixner polynomials

I. I. Sharapudinovabc, Z. D. Gadzhievaab

a Dagestan Scientific Center RAS
b Dagestan State Pedagogical University, 45, M.Gadzhieva st., 367032, Makhachkala, Russia

Abstract: The problem of constructing Sobolev orthogonal polynomials $m _{r,n}^{\alpha}(x,q)$ $(n=0,1,\ldots)$, generated by classical Meixner's polynomials is considered. They can by defined using the following equalities $m_{r,k}^{\alpha}(x,q)={x^{[k]}\over k!}$, $x^{[k]}=x(x-1)\cdots(x-k+1)$, $k=0,1,\ldots,r-1$, $m_{r,k+r}^{\alpha}(x,q)=\frac{1}{(r-1)!}\sum\limits_{t=0}^{x-r}(x-1-t)^{[r-1]}m_{k}^{\alpha}(t,q)$, where $m_{k}^{\alpha}(t,q)$ denote Meixner's polynomial of degree $k$, orthonormal on $\Omega=\{0,1,\ldots\}$ with weight $\rho(x)=q^x\frac{\Gamma(x+\alpha+1)}{\Gamma(x+1)}(1-q)^{\alpha+1}$. Polynomials $m _{r,n}^{\alpha}(x,q)$, $(n=0,1,\ldots)$ are orthonormal on $\Omega=\{0,1,\ldots\}$ with respect to the inner product
$$\langle m_{r,n}^{\alpha},m_{r,m}^{\alpha}\rangle= \sum\limits_{k=0}^{r-1}\Delta^km_{r,n}^{\alpha}(0,q)\Delta^km_{r,m}^{\alpha}(0,q)+ \sum\limits_{j=0}^{\infty}\Delta^rm_{r,n}^{\alpha}(j,q)\Delta^r m_{r,m}^{\alpha}(j,q)\rho(j).$$
For $m_{r,n}^{\alpha}(x,q)$ we obtain the explicit formula that contains the Мeixner polynomial $M_{n}^{\alpha-r}(x,q)$:
$$m_{r,k+r}^{\alpha}(x,q)=(\frac{q}{q-1})^r\{h_{k}^{\alpha}(q)\}^{-1/2} [M_{k+r}^{\alpha-r}(x,q)-\sum\limits_{\nu=0}^{r-1}\frac{A_{r,k,\nu}x^{[\nu]}}{\nu!}], k=0,1,\ldots,$$
where $A_{r,k,\nu}=({q-1\over q})^\nu \frac{\Gamma(k+\alpha+1)}{(k+r-\nu)!\Gamma(\nu-r+\alpha+1)}$, $M_n^\alpha(x,q)=\frac{\Gamma (n+\alpha+1)}{n!} \sum_{k=0}^n{n^{[k]}x^{[k]}\over \Gamma (k+\alpha+1)k!}(1-{1\over q})^k$, $h_n^\alpha(q)= {n+\alpha\choose n}q^{-n}\Gamma(\alpha+1)$.

Key words: orthogonal Sobolev polynomial, Meixner polynomials orthogonal on the grid, approximation of discrete functions, mixed series in Meixner polinomials orthogonal on a uniform grid.

DOI: https://doi.org/10.18500/1816-9791-2016-16-3-310-321

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UDC: 517.587

Citation: I. I. Sharapudinov, Z. D. Gadzhieva, “Sobolev orthogonal polynomials generated by Meixner polynomials”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 16:3 (2016), 310–321

Citation in format AMSBIB
\Bibitem{ShaGad16} \by I.~I.~Sharapudinov, Z.~D.~Gadzhieva \paper Sobolev orthogonal polynomials generated by Meixner polynomials \jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. \yr 2016 \vol 16 \issue 3 \pages 310--321 \mathnet{http://mi.mathnet.ru/isu650} \crossref{https://doi.org/10.18500/1816-9791-2016-16-3-310-321} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3557759} \elib{http://elibrary.ru/item.asp?id=26702021} 

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This publication is cited in the following articles:
1. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Sistemy funktsii, ortogonalnykh otnositelno skalyarnykh proizvedenii tipa Soboleva s diskretnymi massami, porozhdennykh klassicheskimi ortogonalnymi sistemami”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 31–60
2. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Sobolev orthogonal functions on the grid, generated by discrete orthogonal functions and the Cauchy problem for the difference equation”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 29–39
3. I. I. Sharapudinov, M. G. Magomed-Kasumov, “Chislennyi metod resheniya zadachi Koshi dlya sistem obyknovennykh differentsialnykh uravnenii s pomoschyu ortogonalnoi v smysle Soboleva sistemy, porozhdennoi sistemoi kosinusov”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 8, 53–60
4. I. I. Sharapudinov, I. G. Guseinov, “Polinomy, ortogonalnye po Sobolevu, porozhdennye polinomami Sharle”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 18:2 (2018), 196–205
5. M. G. Magomed-Kasumov, “Sistema funktsii, ortogonalnaya v smysle Soboleva i porozhdennaya sistemoi Uolsha”, Matem. zametki, 105:4 (2019), 545–552
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