I was trying to derive the Hobson's relation between the associated Legendre function and Gauss Hypergeometric function. This is given by, $$P^m_n(z)=\frac{1}{\Gamma(1-m)}\left(\frac{z+1}{z-1}\right)^{m/2}{_2F_1}\left(-n,n+1;1-m;\frac{1}{2}-\frac{z}{2}\right) $$ Now the P-symbol for the associated Legendre DE is, $$P^m_n(z)=P\begin{pmatrix} -1 & \infty & 1 & \\ \frac{m}{2} & n+1 & \frac{m}{2} & z\\ -\frac{m}{2}&-n&-\frac{m}{2}& \end{pmatrix}$$ Using homographic transformations one can show that, $$P^m_n(z)=\left(\frac{z+1}{z-1}\right)^{m/2}P\begin{pmatrix} -1 & 1 & \infty & \\ 0 & 0 & n+1 & z\\ -m & m & -n & \end{pmatrix}$$ And further if I transform the regular singular point at -1 to 1 and at 1 to 0, I get, $$P^m_n(z)=\left(\frac{z+1}{z-1}\right)^{m/2}P\begin{pmatrix} 0 & 1 & \infty & \\ 0 & 0 & n+1 & \frac{1}{2}-\frac{z}{2}\\ m & -m & -n & \end{pmatrix}$$ This gives me the relation, $$P^m_n(z)=\left(\frac{z+1}{z-1}\right)^{m/2}{_2F_1}\left(-n,n+1;1-m;\frac{1}{2}-\frac{z}{2}\right) $$ where the gamma function term is missing. Again this relation looks similar to the one with regularized hypergeometric function, $$P^m_n(z)=\left(\frac{z+1}{z-1}\right)^{m/2}{_2\tilde{F}_1}\left(-n,n+1;1-m;\frac{1}{2}-\frac{z}{2}\right)$$ Now I have three questions regarding this.
- What am I doing wrong and how to get the Gamma function term?
- I did transform RSP at -1 to 1 and at 1 to 0, but then had I transformed RSP at -1 to 0 directly I would have gotten a different relation although the P symbol would still be consistent. So why exactly the former transformation is right?
- What is the range of convergence for this definition? Whittaker says $m$ is unrestricted. But using an unrestricted $m$ on both the associated Legendre function and Gauss Hypergeometric function is giving me different results in Mathematica.