Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 020, 10 pages      arXiv:1612.00927      https://doi.org/10.3842/SIGMA.2017.020

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

Satoru Odake and Ryu Sasaki
Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan

Received December 30, 2016, in final form March 23, 2017; Published online March 29, 2017

Abstract
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.

Key words: multi-indexed orthogonal polynomials; Laguerre and Jacobi polynomials; Wronskian formula; determinant formula.

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