A functional equation from probability theory

Author:
John A. Baker

Journal:
Proc. Amer. Math. Soc. **121** (1994), 767-773

MSC:
Primary 39B22; Secondary 62E10

MathSciNet review:
1186127

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Abstract | References | Similar Articles | Additional Information

Abstract: The functional equation

() |

has been used by Laha and Lukacs (Aequationes Math.

**16**(1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author's opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if and for , where

*k*is a natural number, , (1) holds for and exists then either or there exists a real constant

*c*such that for all .

**[1]**John A. Baker,*Functional equations, tempered distributions and Fourier transforms*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 57–68. MR**979965**, 10.1090/S0002-9947-1989-0979965-5**[2]**Witold Jarczyk,*A recurrent method of solving iterative functional equations*, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 1206, Uniwersytet Śląski, Katowice, 1991. With Polish and Russian summaries. MR**1135795****[3]**Marek Kuczma, Bogdan Choczewski, and Roman Ger,*Iterative functional equations*, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. MR**1067720****[4]**M. Laczkovich,*Nonnegative measurable solutions of difference equations*, J. London Math. Soc. (2)**34**(1986), no. 1, 139–147. MR**859155**, 10.1112/jlms/s2-34.1.139**[5]**R. G. Laha and E. Lukacs,*On a functional equation which occurs in a characterization problem*, Aequationes Math.**16**(1977), no. 3, 259–274. MR**0471023****[6]**E. Vincze,*Bemerkung zur Charakterisierung des Gauss’schen Fehlergesetzes*, Magyar Tud. Akad. Mat. Kutató Int. Közl.**7**(1962), 357–361 (German, with Russian summary). MR**0152773****[7]**Eberhard Zeidler,*Nonlinear functional analysis and its applications. I*, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR**816732**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1186127-5

Keywords:
Functional equation,
probability

Article copyright:
© Copyright 1994
American Mathematical Society