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.
Highlights
The applications of the Classical Orthogonal polynomials in physical sciences are immense
Apart from bilinear and even trilinear generating functions for Jacobi polynomials we find in Chap. 2 of the monograph of Srivastava and Manocha [15] more general generating functions from type (10.6) where the upper indices are not pure parameters
A main aim was to find for the Jacobi polynomials an alternative definition in comparison to the Rodrigues definition but in analogy to the alternative definition of Hermite polynomials where this alternative definition was very successful and led in the past to a basic definition of the Laguerre 2D and Hermite 2D polynomials
Summary
The applications of the Classical Orthogonal polynomials in physical sciences are immense. Very readable is the monograph of Rainville [10] about special functions including the general theory of orthogonal poynomials and more detailed representations about the Jacobi polynomials and of ultraspherical polynomials or equivalently the Gegenbauer polynomials and their special cases of Legendre polynomials. In the work of Luke [11] we find a large representative chapter about orthogonal polynomials with the general theory and with detailed consideration of Jacobi polynomials and of their special cases. A representation of the Classical Orthogonal polynomials including the Jacobi polynomials (with notation Pn ( x;α , β ) ) and, in particular, detailed about the Chebyshev polynomials and the Legendre poynomials and with a representation of the general theory is the monograph of Suyetin [13] (see [14] of the same author about polynomials of two variables). A detailed representation about generating functions for polynomials and special functions and with a lot of
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