Some properties and applications of (\(\tau, \beta\))-generalized Legendre functions \(^{\tau, \beta}P_{\nu}^{\mu }(z)\) (Q461334)

This paper deals with the generalized associated Legendre function of the first kind in the form \[ ^{\tau, \beta}P_{\nu}^{\mu }(z) =\frac{1}{\Gamma(1-\mu)}\left (\frac{z+1}{z-1}\right )^{\mu/2} {_{2}F}_1^{\tau, \beta}\left (-\nu,\nu+1;1-\mu;\frac{1-z}{2}\right ), \] where \(|1-z|<2\), \(\mu\neq 1,2,\ldots\), and \({_{2}F}_1^{\tau, \beta}\) is the Wright hypergeometric function. By using multiple properties that \({_{2}F}_1^{\tau, \beta}\) satisfies, the recurrence relations as well as differential equations for the \(^{\tau, \beta}P_{\nu}^{\mu }\) are derived.