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### 9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme

9.171 The hypergeometric equation (see 9.151):

$u=P\left\{\begin{array}{ccc}0& \infty \hfill & 1\hfill \\ 0& \alpha \hfill & 0\hfill \\ 1-\gamma & \beta \hfill & \gamma -\alpha -\beta \hfill \end{array}z\right\}$

WH

9.172 The associated Legendre's equation defining the functions $pnm(z)$ for n and m integers (see 8.7001):

1.

$u=P\left\{\begin{array}{ccc}0& \infty \hfill & 1\hfill \\ \frac{1}{2}m& n+1\hfill & \frac{1}{2}m\hfill & \frac{1-z}{2}\hfill \\ -\frac{1}{2}m& -n\hfill & -\frac{1}{2}m\hfill \end{array}\right\}$

WH

2.

$u=P\left\{\begin{array}{ccc}0& \infty \hfill & 1\hfill \\ -\frac{1}{2}n& \frac{1}{2}m\hfill & 0\hfill & \frac{1}{1-{z}^{2}}\hfill \\ \frac{n+1}{2}& -\frac{1}{2}m\hfill & \frac{1}{2}\hfill \end{array}\right\}$

WH

9.173 The function $pnm(1−z22n2)$ satisfies the equation

$u=P\left\{\begin{array}{ccc}4{n}^{2}& \infty \hfill & 0\hfill \\ \frac{1}{2}m& n+1\hfill & \frac{1}{2}\hfill \end{array}$

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