In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and ∞. The equation is also known as the Papperitz equation.^{[1]}
Definition
The differential equation is given by

\frac{d^2w}{dz^2} + \left[ \frac{1\alpha\alpha'}{za} + \frac{1\beta\beta'}{zb} + \frac{1\gamma\gamma'}{zc} \right] \frac{dw}{dz}

+\left[ \frac{\alpha\alpha' (ab)(ac)} {za} +\frac{\beta\beta' (bc)(ba)} {zb} +\frac{\gamma\gamma' (ca)(cb)} {zc} \right] \frac{w}{(za)(zb)(zc)}=0.
The regular singular points are a, b, and c. The pairs of exponents for each are respectively α; α′, β; β′, and γ; γ′. The exponents are subject to the condition

\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.
Solutions
The solutions are denoted by the Riemann Psymbol (also known as the Papperitz symbol)

w(z)=P \left\{ \begin{matrix} a & b & c & \; \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\}
The standard hypergeometric function may be expressed as

\;_2F_1(a,b;c;z) = P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\ 0 & a & 0 & z \\ 1c & b & cab & \; \end{matrix} \right\}
The Pfunctions obey a number of identities; one of them allows a general Pfunction to be expressed in terms of the hypergeometric function. It is

P \left\{ \begin{matrix} a & b & c & \; \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\} = \left(\frac{za}{zb}\right)^\alpha \left(\frac{zc}{zb}\right)^\gamma P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\ 0 & \alpha+\beta+\gamma & 0 & \;\frac{(za)(cb)}{(zb)(ca)} \\ \alpha'\alpha & \alpha+\beta'+\gamma & \gamma'\gamma & \; \end{matrix} \right\}
In other words, one may write the solutions in terms of the hypergeometric function as

w(z)= \left(\frac{za}{zb}\right)^\alpha \left(\frac{zc}{zb}\right)^\gamma \;_2F_1 \left( \alpha+\beta +\gamma, \alpha+\beta'+\gamma; 1+\alpha\alpha'; \frac{(za)(cb)}{(zb)(ca)} \right)
The full complement of Kummer's 24 solutions may be obtained in this way; see the article hypergeometric differential equation for a treatment of Kummer's solutions.
Fractional linear transformations
The Pfunction possesses a simple symmetry under the action of fractional linear transformations known as Möbius transformations (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group GL(2, C). Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

u=\frac{Az+B}{Cz+D} \quad \text{ and } \quad \eta=\frac{Aa+B}{Ca+D}
and

\zeta=\frac{Ab+B}{Cb+D} \quad \text{ and } \quad \theta=\frac{Ac+B}{Cc+D}
then one has the simple relation

P \left\{ \begin{matrix} a & b & c & \; \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\} =P \left\{ \begin{matrix} \eta & \zeta & \theta & \; \\ \alpha & \beta & \gamma & u \\ \alpha' & \beta' & \gamma' & \; \end{matrix} \right\}
expressing the symmetry.
See also
Notes

^ Siklos, Stephen. "The Papperitz equation". Retrieved 21 April 2014.
References

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)

Chapter 15 Hypergeometric Functions

Section 15.6 Riemann's Differential Equation
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