On the B-convolutions of the Bessel diamond kernel of Riesz

Abstract In this article, the operator ♢ B k is introduced and named as the Bessel diamond operator iterated k -times and is defined by ♢ B k = [ ( B x 1 + B x 2 + ⋯ + B x p ) 2 - ( B x p + 1 + ⋯ + B x p + q ) 2 ] k where p + q = n , B x i = ∂ 2 ∂ x i 2 + 2 v i x i ∂ ∂ x i ,  2 v i = 2 α i + 1 ,  α i > - 1 2 [8] , x i > 0 , i = 1 , 2 , … , n , k is a nonnegative integer and n is the dimension of the R n + . In this work, we study the elementary solution of the operator ♢ B k and this elementary solution is called the Bessel diamond kernel of Riesz. Then, we study the B -convolution of this elementary solution.