The solutions of the n-dimensional Bessel diamond operator and the Fourier--Bessel transform of their convolution

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In this article, the operator ᵏ is introduced and named as the Bessel diamond operator iterated k times and is defined by ᵏ = [(Bₓ₁ + Bₓ₂ +... + Bₓₚ)² - (Bₓₚ ₊ ₁ +... + Bₓₚ ₊ q)²]ᵏ, where p + q = n, Bₓᵢ = ∂²∂ xᵢ² + 2vᵢxᵢ ∂∂ xᵢ, where 2vᵢ = 2αᵢ + 1, αᵢ > - 1/2 [8], xᵢ > 0, i = 1, 2,..., n, k is a non-negative integer and n is the dimension of Rₙ⁺. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator 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.