A FUNCTIONAL-EQUATION FROM PROBABILITY-THEORY

The functional equation (1) f(x) = PI(j=1)N[f(B(j)x)]gamma(j) has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to cha racterize normal distributions. The aim of the present paper is to stu dy (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author's opinion, are si mpler than those employed by the afore-mentioned authors. We will prov e, for example, that if 0 < beta(j) < 1 and gamma(j) > 0 for 1 less-th an-or-equal-to j less-than-or-equal-to N, SIGMA(j=1)N beta(j)(k)gamma( j) = 1, where k is a natural number, f: R --> [0, +infinity), (1) hold s for x is-an-element-of R and f(k)(0) exists then either f = 0 or the re exists a real constant c such that f(x) = exp(cx(k)) for all x is-a n-element-of R.