TITLE:
Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials
AUTHORS:
Alfred Wünsche
KEYWORDS:
Orthogonal Polynomials, Lie Algebra SU(1, 1) and Lie Group SU(1, 1), Lowering and Raising Operators, Jacobi Polynomials, Ultraspherical Polynomials, Gegenbauer Polynomials, Chebyshev Polynomials, Legendre Polynomials, Stirling Numbers, Hypergeometric Function, Operator Identities, Vandermond’s Convolution Identity, Pöschl-Teller Potentials
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.7 No.2,
February
16,
2017
ABSTRACT: Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.