Skip to main content
SpringerLink
Log in

Darboux’s identity and its analogs

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Let us choose a positive integern and denoteF(x, y)=\(\sum _{m = 0}^n f^{(n - m)} (x)g^{(m)} (y)\), wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation

$$F(x, y) - F(y, x) = \int_y^x {\{ f^{(n + 1)} (t)g(x + y - t) - f(t)g^{(n + 1)} (x + y - t)\} dt.} $$

are given. We also establish discrete and noncommutative analogs of this identity.

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. G. Darboux, “Sur les développements en série des fonctions d'une seule variable,”J. de Math.,3, No. 2, 291–313 (1876).

    Google Scholar 

  2. E. T. Whittaker and G. N. Watson,A Course of Modern Analysis, 3rd ed., Cambridge Univ. Press, Cambridge (1927).

    MATH  Google Scholar 

  3. O. V. Viskov, “On Darboux’s formula,”Integral Transformations and Special Functions. Information Bulletin,1 (3), 2–5 (1998).

    Google Scholar 

  4. O. V. Viskov, “A noncommutative identity for the Appell polynomials,”Mat. Zametki [Math. Notes],64, No. 2, 307–311 (1998).

    Google Scholar 

  5. R. P. Boas, R. C. Buck, and R. Creighton,Polynomial Expansions of Analytical Functions, Springer-Verlag, Berlin (1964).

    Google Scholar 

  6. G. Szegö,Orthogonal Polynomials, Colloquium Publ., Vol. XXIII, Amer. Math. Soc., Providence, R.I. (1959).

    MATH  Google Scholar 

  7. A. O. Gel’fond,The Calculus of Finite Differences [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, USSR

    O. V. Viskov

Authors
  1. O. V. Viskov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 332–338, September, 2000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Viskov, O.V. Darboux’s identity and its analogs. Math Notes 68, 289–294 (2000). https://doi.org/10.1007/BF02674551

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02674551

Key words

Search

Navigation