Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions  

作　　者：Alfred Wünsche Alfred Wünsche(Humboldt-Universitä,t Berlin, Institut für Physik, Nichtklassische Strahlung (MPG), Berlin, Germany)

机构地区：[1]Humboldt-Universitä,t Berlin, Institut für Physik, Nichtklassische Strahlung (MPG), Berlin, Germany

出　　处：《Advances in Pure Mathematics》2023年第8期504-536,共16页理论数学进展（英文）

摘　　要：In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

关 键 词：Spherical Bessel Functions Chebyshev Polynomials Legendre Polynomials Hermite Polynomials Derivatives of Delta Functions Normally and Anti-Normally Ordered Operators 

分 类 号：O17[理学—数学]