Abstract:

Let t = (t1,...,tn) be a point of ℝn. We shall write . We put, by the definition, Wα(u, m) = (m-2u)(α - n)/4[π(n - 2)/22(α + n - 2)/2G{cyrillic}(α/2)]J(α - n)/2(m2u)1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. Wα(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate (□ + m2)Wα + 2(u, m) = Wα(u, m), where (□ + m2) is the ultrahyperbolic operator. Then we express Wα(u, m) as a linear combination of Rα(u) of differntial orders; Rα(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W-2k(u, m) = (□ + m2)kδ, k = 0, 1,...; W0(u, m) = δ; and (□ + m2)kW2k(u, m) = δ. Finally we prove that Wα(u, m = 0) = Rα(u). Several of these results, in the particular case μ = 1, were proved earlier by a completely different method. © 2015 Wiley Periodicals, Inc., A Wiley Company.