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2017•Symmetry Integrability and Geometry Methods and ApplicationsOpen access

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

Satoru Odake, Ryu Sasaki

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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.

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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.

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Available 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 concepts: Laguerre polynomials, Mathematics, Jacobi polynomials, Classical orthogonal polynomials, Orthogonal polynomials, Wilson polynomials, Discrete orthogonal polynomials, Gegenbauer polynomials

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