Advertisement

Analysis Mathematica

, Volume 25, Issue 1, pp 133–146 | Cite as

Determinants, sums involving binomial coefficients, and moment sequences

  • Zoltán Sasvári
  • Э. Щащвари
Article
  • 44 Downloads

Abstract

In the present paper we investigate some problems connected with the positive definiteness of the sequences\(j \to e^{j^\alpha } (j \in N_0 )\) and\(j \to e^{ - \left| j \right|^\alpha } (j \in Z)\), whereα≥0. For this we need and prove some results about certain determinants and finite sums that might be of independent interest.

Keywords

Nonnegative Integer Independent Interest Positive Semidefinite Positive Definiteness Binomial Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Определители, суммы, содерзашие биномиаляные коЭффициенты, и последователяности моментов

Abstract

Иэучаутся полозителяная определенностя некоторых функции и своиства последователяностеи моментов (именно, последователяностеи моментов тригонометрическои и Гамбургера). Согласно классическому реэулятату Герголяца последователяностяf является последователяностяу моментов Гамбургера в том и толяко том случае, когда она полозителяно определена на полугруппеH 0, т.е. когда матрица (f(j+k)) jk n полозителяно полуопределена для всех полозителяныхn.

Иэучаутся специаляное вазные последователяности
$$f_\alpha (j) = \exp (j^\alpha ), g_\alpha (j) = \exp ( - \left| j \right|^\alpha ).$$
Получены некоторые реэулятаты дляf α p иα α p .

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T. M. Bisgaard, On the positive definiteness of\(n \to e^{pn^\alpha } \),Arkiv för Mat. (submitted).Google Scholar
  2. [2]
    T. M. Bisgaard andZ. Sasvári, On the positive definiteness of certain functions,Math. Nachr.,186(1997), 81–99.MATHMathSciNetGoogle Scholar
  3. [3]
    C. Berg, J. P. R. Christensen, andP. Ressel,Harmonic analysis on semigroups, Springer (New York-Berlin-Heidelberg-Tokyo, 1984).MATHGoogle Scholar
  4. [4]
    I. S. Gradsteyn andI. M. Ryzhik,Table of integrals, series, and products, Academic Press (Boston, 1994).Google Scholar
  5. [5]
    G. Kowalewski,Einführung in die Determinantentheorie, de Gruyter & Co. (Berlin, 1942).MATHGoogle Scholar
  6. [6]
    F. W. J. Olver,Asymptotics and special functions, Academic Press (San Diego, 1974).Google Scholar
  7. [7]
    M. Petkovšek, H. S. Wilf, andD. Zeilberger,A =B, A. K. Peters, Ltd. (Wellesley, MA, 1996).Google Scholar
  8. [8]
    A. P. Prudnikov, Yu. A. Brychkov, andO. I. Marichev,Integrals and series, Vol.1, Gordon and Breach Sci. Publ. (New York, 1986).Google Scholar
  9. [9]
    J. Riordan,Combinatorial identities, John Wiley & Sons (New York-London-Sydney, 1968).MATHGoogle Scholar
  10. [10]
    Z. Sasvári,Positive definite and definitizable functions, Akademie Verlag (Berlin, 1994).MATHGoogle Scholar

Copyright information

© Akadémiai Kiadó 1999

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DresdenDresdenGermany

Personalised recommendations