How to compute multivariate Bessel expansions

Abstract

We develop a constructive method for computing explicitly multivariate Bessel expansions of the type ∑m≥1αm∏i=1kJμi(ζmxi)(ζmxi)μi,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{m\\ge 1} \\alpha _m \\prod _{i=1}^k \\frac{J_{\\mu _i}(\\zeta _m x_i)}{(\\zeta _m x_i)^{\\mu _i}}, \\end{aligned}$$\\end{document}assuming that for a particular value η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} a closed expression for the single-variable Bessel expansion ∑m≥1αmJη(ζmx)(ζmx)η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{m\\ge 1}\\alpha _m \\frac{J_{\\eta }(\\zeta _m x)}{(\\zeta _m x)^\\eta } \\end{aligned}$$\\end{document}as a power series of x2j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^{2j}$$\\end{document}, j∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j\\in \\mathbb {N}$$\\end{document}, is known. Using the method we compute in a closed form a bunch of examples of multivariate Bessel expansions.