Rediscovering the contributions of Rodrigues on the representation of special functions (Q813951)

This paper traces the true contribution of Rodrigues to the field of special functions and orthogonal polynomials. In his thesis (published in 1816) Rodrigues gave a representation of the Legendre polynomials (introduced by Legendre in 1785 using a generating function) in the form \[ P_m(x)={(-1)^m\over 2^m m!}{\text{d}^m\over \text{d}x^m}\,(1-x^2)^m, \] a form that was baptized `Rodrigues formula' around 1868 (Hermite, Heine). Along with the Legendre polynomials, there was a formula of the same type for their \(n\)th derivative: Gegenbauer polynomials \(P_{m-n}^{(n,n)}(x)\) and even a representation for the second solution of the differential equation satisfied by this \(n\)th derivative was given. But there appears to have been more: the authors have unearthed quite some results in Rodrigues' thesis that have never been attributed to him and give a historical overview tracing which books and papers do and don't refer to Rodrigues with respect to a variety of matters (Hildebrandt-polynomials, Appell polynomials, Jacobi-transformation etc.). An interesting and well written paper.