Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

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.

A modeling method for vibration analysis of cracked beam with arbitrary boundary condition

This paper establishes a cracked Timoshenko beams model to investigate the vibration behavior based on the ultraspherical polynomials.Timoshenko beam theory is applied to model the free vibration analysis of the cracked beam and the numerical results are obtained by using ultraspherical orthogonal polynomials.The boundary conditions of both ends of the cracked beam are modeled as the elastic spring and the beam is divided into two parts by the crack section,and continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of fracture mechanics theory.Ignoring the influence of boundary conditions,displacements admissible functions of cracked Timoshenko beam can be set up as ultraspherical orthogonal polynomials.The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method(FEM).In addition,the effects of flexibility coefficient on the natural frequencies are also investigated by using the proposed method.Numerical examples are given for free vibration analysis of cracked beams with various boundary conditions,which may be provided as reference data for future study.