A Legendre polynomial integral

Abstract

Let { P n ( x ) } \{ {P_n}(x)\} be the usual Legendre polynomials. The following integral is apparently new. \[ ∫ 0 1 P n ( 2 x − 1 ) log ⁡ 1 x d x = ( − 1 ) n n ( n + 1 ) for n ⩾ 1. \int _0^1{P_n}(2x - 1)\log \frac {1}{x}dx = \frac {{{{( - 1)}^n}}}{{n(n + 1)}}\quad {\text {for}}\;n \geqslant 1. \] It has an application in the construction of Gauss quadrature formulas on (0, 1) with weight function log ⁡ ( 1 / x ) \log (1/x) .