A Generalized Radon Transform on the Plane

A new generalized Radon transform Rα, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δα, β, and the Jacobi polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{k}^{(\beta,\,\alpha)}(t)$\end{document}. The transform Rα, β and its dual \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{\alpha,\,\beta}^{\ast}$\end{document} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for Rα, β for functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+})$\end{document} are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform Rα, β is used to represent the solutions of the partial differential equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Lu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f$\end{document} with constant coefficients aj and the Cauchy problem for the generalized wave equation associated with the operator Δα, β. Another application is that, by an invariant property of Rα, β, a new product formula for the Jacobi polynomials of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)}$\end{document} is obtained.