On the Elementary Retarded, Ultrahyperbolic Solution of the Klein‐Gordon Operator, Iterated k Times

Let t = ( t 1 ,…, t n ) be a point of ℝ n . We shall write . We put, by the definition, W α ( u , m ) = ( m −2 u ) (α − n )/4 [π ( n − 2)/2 2 (α + n − 2)/2 Г(α/2)] J (α − n )/2 ( m 2 u ) 1/2 ; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. W α ( u , m ), which is an ordinary function if Re α ≥ n , is an entire distributional function of α. First we evaluate {□ + m 2 } W α + 2 ( u , m ) = W α ( u , m ), where {□ + m 2 } is the ultrahyperbolic operator. Then we express W α ( u , m ) as a linear combination of R α ( u ) of differntial orders; R α ( u ) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W −2 k ( u , m ) = {□ + m 2 } k δ, k = 0, 1,…; W 0 ( u , m ) = δ; and {□ + m 2 } k W 2 k ( u , m ) = δ. Finally we prove that W α ( u , m = 0) = R α ( u ). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method.