Legendre polynomial

Shifted Legendre polynomials

The shifted Legendre polynomials are defined as $${\displaystyle {\widetilde {P}}_{n}(x)=P_{n}(2x-1)\,.}$$ Here the "shifting" function x ↦ 2x − 1 is an affine transformation that bijectively maps the interval [0, 1] to the interval [−1, 1], implying that the polynomials P̃_n(x) are orthogonal on [0, 1]: $${\displaystyle \int _{0}^{1}{\widetilde {P}}_{m}(x){\widetilde {P}}_{n}(x)\,dx={\frac {1}{2n+1}}\delta _{mn}\,.}$$