RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



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






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 5 scientific papers (total in 5 papers)

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
c Vladikavkaz Scientific Center RAS

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

Full text: PDF file (230 kB)
References: PDF file   HTML file

Bibliographic databases:

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}


Linking options:
  • http://mi.mathnet.ru/eng/isu650
  • http://mi.mathnet.ru/eng/isu/v16/i3/p310

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  elib
    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  mathnet  crossref
    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  mathnet  crossref
    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  mathnet  crossref  elib
    5. M. G. Magomed-Kasumov, “Sistema funktsii, ortogonalnaya v smysle Soboleva i porozhdennaya sistemoi Uolsha”, Matem. zametki, 105:4 (2019), 545–552  mathnet  crossref  elib
  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
    Number of views:
    This page:209
    Full text:69
    References:23

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019