An orthogonal class of p-Legendre polynomials on variable interval

The work incorporates a generalization of the Legendre polynomial by introducing a parameter p>0 in its generating function. The coefficients thus generated, constitute a class of the polynomials which are termed as the p-Legendre polynomials. It is shown that this class turns out to be orthogonal with respect to the weight function: (1-p\ x) ^p+1{2p-1} (1+p\ x) ^p+1{2p-1} over the interval (-1p, 1p). Among the other properties derived, include the Rodrigues formula, normalization, recurrence relation and zeros. A graphic depiction for p=0. 5, 1, 2, and 3 is shown. The p-Legendre polynomials are used to estimate a function using the least squares approach. The approximations are graphically depicted for p=0. 7, 1, 2.