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 (-1√p, 1√p). 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 $p=0.7, 1, 2.$