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On the Rodrigues’ Formula Approach to Operator Factorization

Robin, William


William Robin


In this paper, we derive general formulae that reproduce well-known instances of recurrence relations for the classical orthogonal polynomials as special cases. These recurrence relations are derived, using only elementary mathematics, directly from the general Rodrigues’ formula for the classical orthogonal polynomials – a ‘first-principles’ derivation – and represent a unified presentation of various approaches to the exact solution of an important class of second-order linear ordinary differential equations. When re-expressed in ladder-operator form, the recurrence relations are seen to represent to a basic development of the work of Jafarizadeh and Fakhri [5] and allow a ‘Schrödinger operator factorization’ of the defining equation of the classical orthogonal polynomials, as well as an operational formula for the solution of this defining equation. The identity between the Rodrigues’ formula and the operational formula is determined and standard examples involving the application of the ladder-operator approach presented. The relationship with previous work is discussed.


Robin, W. (2012). On the Rodrigues’ Formula Approach to Operator Factorization. International Mathematical Forum, 7(45-48), 2333-2351

Journal Article Type Article
Publication Date 2012
Deposit Date Apr 22, 2015
Publicly Available Date Apr 22, 2015
Journal International Mathematical Forum
Electronic ISSN 1312-7594
Publisher Hikari
Peer Reviewed Peer Reviewed
Volume 7
Issue 45-48
Pages 2333-2351
Keywords classical orthogonal polynomials; ladder-operators; operational formula
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