Geodesic
Difference equations and Sobolev orthogonal polynomials, generated by Meixner polynomials
Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 2, pp. 58-72 
I. I. Sharapudinov; Z. D. Gadzhieva; R. M. Gadzhimirzaev. Difference equations and Sobolev orthogonal polynomials, generated by Meixner polynomials. Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 2, pp. 58-72. http://geodesic.mathdoc.fr/item/VMJ_2017_19_2_a6/
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     author = {I. I. Sharapudinov and Z. D. Gadzhieva and R. M. Gadzhimirzaev},
     title = {Difference equations and {Sobolev} orthogonal polynomials, generated by {Meixner} polynomials},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {58--72},
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     url = {http://geodesic.mathdoc.fr/item/VMJ_2017_19_2_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The representation of the Cauchy problem's solution for a difference equation with variable coefficients and given initial conditions at $x = 0$ by expanding this solution in a Fourier series on Sobolev polynomials orthogonal on the grid $(0,1,\ldots)$. The representation is based on contraction new polynomials orthogonal on Sobolev and generated by classical Meixner's polynomials. For new polynomials an explicit formula containing Meixner polynomials is obtained. This result allows us to investigate the asymptotic properties of new polynomials orthogonal on Sobolev on the grid $(0,1, \ldots)$ with a given weight. In addition, it allows to solve the problem of the calculation of the polynomials orthogonal on Sobolev, reducing it to use of well known recurrence relations for classical Meixner polynomials.

[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 (1991), 151–175 | DOI | MR | Zbl

[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 | Zbl

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

[4] Kwon K. H., Littlejohn L. L., “The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for positive integers $k$”, Ann. Numer. Anal., 1995, no. 2, 289–303 | MR | Zbl

[5] Kwon K. H., Littlejohn L. L., “Sobolev orthogonal polynomials and second-order differential equations”, Ann. Numer. Anal., 28 (1998), 547–594 | MR | Zbl

[6] Marcellan F., Yuan Xu, On Sobolev orthogonal polynomials, 25 Mar 2014, 40 pp., arXiv: 6249v1 [math.CA] | MR

[7] Sharapudinov I. I., “Priblizhenie diskretnykh funktsii i mnogochleny Chebysheva, ortogonalnye na ravnomernoi setke”, Mat. zametki, 67:3 (2000), 460–470 | DOI | Zbl

[8] Sharapudinov I. I., “Priblizhenie funktsii s peremennoi gladkostyu summami Fure–Lezhandra”, Mat. sb., 191:5 (2000), 143–160 | DOI

[9] Sharapudinov I. I., “Approksimativnye svoistva operatorov ${\cal Y}_{n+2r}(f)$ i ikh diskretnykh analogov”, Mat. zametki, 72:5 (2002), 765–795 | DOI | Zbl

[10] Sharapudinov I. I., “Smeshannye ryady po ultrasfericheskim polinomam i ikh approksimativnye svoistva”, Mat. sb., 194:3 (2003), 115–148 | DOI | Zbl

[11] Sharapudinov I. I., Smeshannye ryady po ortogonalnym polinomam, Dagestan. nauch. tsentr RAN, Makhachkala, 2004, 276 pp.

[12] Sharapudinov I. I., “Smeshannye ryady po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Mat. zametki, 78:3 (2005), 442–465 | DOI | Zbl

[13] Sharapudinov I. I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Lezhandra na klassakh $W^r$”, Mat. sb., 197:3 (2006), 135–154 | DOI | Zbl

[14] Sharapudinov T. I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Vestn. Dagestan. nauch. tsentra RAN, 29 (2007), 12–23 | Zbl

[15] Sharapudinov I. I., “Approksimativnye svoistva srednikh tipa Valle-Pussena chastichnykh summ smeshannykh ryadov po polinomam Lezhandra”, Mat. zametki, 84:3 (2008), 452–471 | DOI | Zbl

[16] Sharapudinov I. I., Muratova G. N., “Nekotorye svoistva $r$-kratno integrirovannykh ryadov po sisteme Khaara”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 68–76 | Zbl

[17] Sharapudinov I. I., Sharapudinov T. I., “Smeshannye ryady po polinomam Yakobi i Chebysheva i ikh diskretizatsiya”, Mat. zametki, 88:1 (2010), 116–147 | DOI | Zbl

[18] Sharapudinov I. I., “Sistemy funktsii, ortogonalnykh po Sobolevu, porozhdennye ortogonalnymi funktsiyami”, Sovremennye problemy teorii funktsii i ikh pril., Materialy 18-i mezhdunar. Saratovskoi zimnei shk., 2016, 329–332 | Zbl

[19] Trefethen L. N., Spectral methods in Matlab, SIAM, Fhiladelphia, 2000 | MR | Zbl

[20] Trefethen L. N., Finite Difference and Spectral Methods for Ordinary and Partial Differential Equation, Cornell Univ., 1996

[21] Magomed-Kasumov M. G., “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem smeshannykh ryadov po sisteme Khaara”, Sovremennye problemy teorii funktsii i ikh pril., Materialy 18-i mezhdunar. Saratovskoi zimnei shk., 2016, 176–178 | Zbl

[22] Sharapudinov I. I., Mnogochleny, ortogonalnye na diskretnykh setkakh, Izd-vo Dag. gos. ped. un-ta, Makhachkala, 1997

[23] Gasper G., “Positivity and special function”, Theory and Appl. Spec. Funct., ed. R. A. Askey, Acad. Press Inc., N. Y., 1975, 375–433 | MR