On Marcel Riesz's Ultrahyperbolic Kernel

Let t = ( t 1 , …, t n ) be a point of ℝ n . We shall write . We put by definition R α ( u ) = u ( α − n )/2 / K n ( α ); here α is a complex parameter, n the dimension of the space, and K n ( α ) is a constant. First we evaluate □ R α ( u ) = R α ( u ), where □ the ultrahyperbolic operator. Then we obtain the following results: R −2 k ( u ) = □ k δ ; R 0 ( u ) = δ; and □ k R 2 k ( u ) = δ , k = 0, 1, …. The first result is the n ‐dimensional ultrahyperbolic correlative of the well‐known one‐dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.