interval of orthogonality is [−1, 1], and has Q = 1 − x2. They can then be standardized into the Jacobi polynomials . There are several important subclasses of these: Gegenbauer, Legendre, and two types of Chebyshev. Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is , and has Q = x. They can then be standardized into the Associated Laguerre polynomials . The plain Laguerre polynomials are a subclass of these. Every Hermite-like polynomial sequence can have its domain shifted and/or scaled so that its interval of orthogonality is , and has Q = 1 and L(0) = 0. They can then be standardized into the Hermite polynomials . Because all polynomial sequences arising from a differential equation in the manner described above are trivially equivalent to the classical polynomials, the actual classical polynomials are always used. Jacobi polynomial The Jacobi-like polynomials, once they have had their domain shifted and scaled so that the interval of orthogonality is [−1, 1], still have two parameters to be determined. They are and in the Jacobi polynomials, written . We have and . Both and are required to be greater than −1. (This puts the root of L inside the interval of orthogonality.) When and are not equal, these polynomials are not symmetrical about x = 0. The differential equation is Jacobi's equation. For further details, see Jacobi polynomials. Gegenbauer polynomials When one sets the parameters and in the Jacobi polynomials equal to each other, one obtains the Gegenbauer or ultraspherical polynomials. They are written , and defined as We have and . The parameter is required to be greater than −1/2. (Incidentally, the standardization given in the table below would make no sense for α = 0 and n ≠ 0, because it would set the polynomials to zero. In that case, the accepted standardization sets instead of the value given in the table.) Ignoring the above considerations, the parameter is closely related to the derivatives of : or, more generally: All the other classical Jacobi-like polynomials (Legendre, etc.) are special cases of the Gegenbauer polynomials, obtained by choosing a value of and choosing a standardization. For further details, see Gegenbauer polynomials. Legendre polynomials The differential equation is This is Legendre's equation. The second form of the differential equation is: The recurrence relation is A mixed recurrence is Rodrigues' formula is