 Equivalently, formula_105 has formula_106 zeros in formula_104.The parity and normalization implicate the values at the boundaries formula_108 to be 
At the origin formula_110 one can show that the values are given by

The shifted Legendre polynomials are defined as
Here the "shifting" function is an affine transformation that bijectively maps the interval to the interval , implying that the polynomials are orthogonal on