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

In this article, the operator\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\diamondsuit _B^k $$ \end{document} is introduced and named as the Bessel diamond operator iteratedk times and is defined by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\diamondsuit _B^k = [(B_{x_1 } + B_{x_2 } + \cdots + B_{x_p } )^2 - (B_{x_{p + 1} } + \cdots + B_{x_{p + q} } )^2 ]^k $$ \end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$p + q = n,B_{x_i } = \tfrac{{\partial ^2 }}{{\partial x_i^2 }} + \tfrac{{2v_i }}{{x_i }}\tfrac{\partial }{{\partial x_i }}$$ \end{document} where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$2v_i = 2\alpha _i + 1,\alpha _i > - \tfrac{1}{2}[8],x_i > 0$$ \end{document},i = 1, 2, ...,nk is a non-negative integer andn 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\diamondsuit _B^k $$ \end{document} 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.