Skip to main content
SpringerLink
Log in

On Functions of the Second Kind in Orthogonal Polynomial Theory

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

A given weight function \(p(x)\) on an interval \([a, b]\) defines uniquely, subject to normalization, a sequence of orthogonal polynomials \(P_n(x)\) and their corresponding sequence of functions of the second kind

$$\begin{aligned} Q_n(x)=\int _a^b\frac{P_n(t)p(t)dt}{x-t},\quad \ x\in \mathbf{C } \backslash [a,b]. \end{aligned}$$

corresponding to them. This paper focuses on the reverse problem of characterizing orthogonal polynomials by means of functions of the second kind, together with the properties of such functions of the second kind. Functions of the second kind for orthogonal polynomials are also of particular interest in that they differ only slightly from the second solution of the differential equation satisfied by the orthogonal polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexits, G.: Konvergenzprobleme der Orthogonalreihen, Budapest (1960)

  2. Bitsadze, A.: Foundations of Theory of Analytic Functions of Complex Argument. Nauka, Moscow (1972)

    Google Scholar 

  3. Chihara, T.: An Introduction to Orhtogonal Polynomials, vol. 13. Gordon and Breach Science Publishers, New York (1978)

    Google Scholar 

  4. Grinshpun, Z.: On Fejer–Shohat problem. Vestnik Leningrad Univ. Math. Mech. Astron. 19, 21–23 (1966)

    Google Scholar 

  5. Grinshpun, Z.: Characteristic Properties of Orthogonal Polynomials in Terms of Functions of the Second Kind, Functional Analysis, Differential Equations and Their Application, pp. 38–43. Kazakh Gos. Univers, Alma-Ata (1982)

  6. Grinshpun, Z.: Special linear combinations of orthogonal polynomials. J. Math. Anal. Appl. 299, 1–18 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nikiforov, A., Uvarov, V.: Special Functios of Mathematical Physics. Birkhäuser, Basel-Boston (1988)

  8. Suetin, M.: Classic Orthogonal Polynomials. Nauka, Moscow (1976)

    Google Scholar 

  9. Szegö, G.: Orthogonal Polynomials. American Mathematical Society, New York (1975)

    MATH  Google Scholar 

  10. Totik, V.: Orthogonal Polynomials: Surveys in Approximation Theory, Vol. 1, pp. 70–125 (2005)

  11. Uvarov, V.: Relation between polynomials orthogonal with different weights (Russian). Dokl. Akad. Nauk SSSR 126, 33–36 (1959)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bell Laboratories, Alcatel-Lucent, rm 2B-318, 600-700 Mountain Ave, Murray Hill, NJ, USA

    Edward Grinshpun

  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan,  52900, Israel

    Zinoviy Grinshpun

Authors
  1. Edward Grinshpun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zinoviy Grinshpun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zinoviy Grinshpun.

Additional information

Communicated by Doron Lubinsky.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grinshpun, E., Grinshpun, Z. On Functions of the Second Kind in Orthogonal Polynomial Theory. Comput. Methods Funct. Theory 13, 65–74 (2013). https://doi.org/10.1007/s40315-012-0006-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-012-0006-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation