  In this article, the operator $\Diamond_{B}^{k}$ is introduced and named as
the Bessel diamond operator iterated $k$ times and is defined by $
\Diamond_{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}} =
\frac{\partial^{2}}{\partial x_{i}^{2}} + \frac{2v_{i}}{x_{i}}
\frac{\partial}{\partial x_{i}}, $ where $2v_{i} = 2\alpha_{i} + 1$, $
\alpha_{i} > - {1/2} $ [8], $x_{i} > 0$, $i = 1, 2, ..., n, k$ is a
non-negative integer and $n$ is the dimension of $\mathbb{R}_{n}^{+}$. In this
work we study the elementary solution of the Bessel diamond operator and the
elementary solution of the operator $\Diamond_{B}^{k}$ is called the Bessel
diamond kernel of Riesz. Then, we study the Fourier--Bessel transform of the
elementary solution and also the Fourier--Bessel transform of their
convolution.

Analysis of PDEs