give the "official" values of R(x) and W(x). Rodrigues' formula Under the assumptions of the preceding section, Pn(x) is proportional to This is known as Rodrigues' formula, after Olinde Rodrigues. It is often written where the numbers en depend on the standardization. The standard values of en will be given in the tables below. The numbers λn Under the assumptions of the preceding section, we have (Since Q is quadratic and L is linear, and are constants, so these are just numbers.) Second form for the differential equation Let Then Now multiply the differential equation by R/Q, getting or This is the standard Sturm–Liouville form for the equation. Third form for the differential equation Let Then Now multiply the differential equation by S/Q, getting or But , so or, letting u = Sy, Formulas involving derivatives Under the assumptions of the preceding section, let P denote the r-th derivative of Pn. (We put the "r" in brackets to avoid confusion with an exponent.) P is a polynomial of degree n − r. Then we have the following: (orthogonality) For fixed r, the polynomial sequence P, P, P, ... are orthogonal, weighted by . (generalized Rodrigues' formula) P is proportional to (differential equation) P is a solution of , where λr is the same function as λn, that is, (differential equation, second form) P is a solution of There are also some mixed recurrences. In each of these, the numbers a, b, and c depend on n and r, and are unrelated in the various formulas. There are an enormous number of other formulas involving orthogonal polynomials in various ways. Here is a tiny sample of them, relating to the Chebyshev, associated Laguerre, and Hermite polynomials: Orthogonality The differential equation for a particular λ may be written (omitting explicit dependence on x) multiplying by yields and reversing the subscripts yields subtracting and integrating: but it can be seen that so that: If the polynomials f are such that the term on the left is zero, and for , then the orthogonality relationship will hold: for . Derivation from differential equation All of the polynomial sequences arising from the differential equation above are equivalent, under scaling and/or shifting of the domain, and standardizing of the polynomials, to more restricted classes. Those restricted classes are exactly "classical orthogonal polynomials". Every Jacobi-like polynomial sequence can have its domain shifted and/or scaled so that its