Characterization of the Bernoulli-polynomials, exp andΨ by Nörlund's multiplication formula

P. Schroth

doi:10.1007/bf01849786

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 12: Issue 3

Characterization of the Bernoulli-polynomials, exp andΨ by Nörlund's multiplication formula

Author:

P. Schroth

P. Schroth Institut C für Mathematik Technische Universität Braunschweig Federal Republic of Germany

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Pages:: 191–204

Online Publication Date:: 14 Jul 2005

Publication Date:: 01 Sep 1981

Article Category:: Research Article

DOI:: https://doi.org/10.1007/bf01849786

Keywords:: Primary 39A30, 10A40, 33A10, 33A15; Secondary 39A10; separation of variables

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The system of functional equations

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\forall p\varepsilon N_ + \forall (x,y)\varepsilon D:f(x,y) = \frac{1}{p}\sum\limits_{k = 0}^{p - 1} {f(x + ky,py)}$$ \end{document}

is suited to characterize the functions

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(x,y) \mapsto y^m B_m \left( {\frac{x}{y}} \right),m\varepsilon N,$$ \end{document}

B_m means them-th Bernoulli-polynomial,

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(x,y) \mapsto \exp (x)y(\exp (y) - 1)^{ - 1}$$ \end{document}

(for these functionsD =R ×R₊) and

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(x,y) \mapsto \log y + \Psi \left( {\frac{x}{y}} \right)(D = R_ + \times R_ + )$$ \end{document}

as those continuous solutions of this system which allow a certain separation of variables and take on some prescribed function values.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)