A Legendre polynomial integral

Let { P,(x)} be the usual Legendre polynomials. The following integral is apparently new. OfP(2x -1) log-dx = n( + 1) for n > 1. It has an application in the construction of Gauss quadrature formulas on (0, 1) with weight function log (l/x). 1. Motivation. For integrals of the type fJ f(x)w(x) dx, where w(x) is positive in (a, b), Gaussian quadrature formulas of the type b ~~~~n f:f(x)w(x)dx kE hknff(Qkn) k=1 are often useful. The {hkn} and { Skn} are chosen to make the formulas exact when f(x) is a polynomial of degree 2n 1 or less [1]. These formulas are especially useful when w(x) is singular at one or more points in the interval. The method of modified moments [2], [3], [4] provides a stable method for calculating the {hkfl, kn} if the set of polynomials orthogonal on (a, b) with weight function w(x) are known. That is, a set of {Qk}5 such that faQk(X)Qm(x)W(x) dx = 0 if k # m is desired. Any such family of orthogonal polynomials obeys a three-term recurrence relation [5], Q 1 (X) = 0, Q0(X) = 1, XQk(X) = akQk+l(x) + bkQk(X) + CkQk-l(X) k > 1, with ak # 0. For some intervals and weight functions, the orthogonal polynomials are known, and there is no problem. For example, if a = 1, b = + 1, and w(x) = 1, the usual Legendre polynomials {Pk(x)} are an orthogonal set, f1 Pk(X)Pm(x) dx = 0 if k For most intervals and weight functions, the corresponding orthogonal polynomials are not known. If the moments fbxk w(x) dx are known, the {ak, bk, Ck} of the unknown set of orthogonal polynomials can be found [2], but the process is nuReceived February 24, 1978. AMS (MOS) subject classifications(1970). Primary 65D30. ? 1979 American Mathematical Society 0025-571 8/79/0000-0068/$01.75