An orthogonal class of $p$-Legendre polynomials on variable interval

Abstract

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: $(1-\sqrt{p}\ x)^{\frac{p+1}{2p}-1}(1+\sqrt{p}\ x)^{\frac{p+1}{2p}-1}$ over the interval $(-\frac{1}{\sqrt{p}}, \frac{1}{\sqrt{p}}).$ 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 $p=0.7, 1, 2.$