The hypergeometric differential equation

$$\frac {d^2 f} {d z^2}+ \left( {\frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} + \frac {1 - \gamma - \gamma'} {z - c} } \right) \frac {d f} {d z} +\left(\frac{\alpha \alpha^{\prime}(a-b)(a-c)}{z-a}+\frac{\beta \beta^{\prime}(b-c)(b-a)}{z-b}+\frac{\gamma \gamma^{\prime}(c-a)(c-b)}{z-c}\right) \frac{f}{(z-a)(z-b)(z-c)} = 0$$

contains the constants $\alpha, \alpha', \beta, \beta', \gamma, \gamma'$.

I have often heard said constants referred to as "exponents" of the hypergeometric differential equation.

What exactly does this mean? I think I have gathered that near the point $z=a$, as the equation is a second-order differential equation, the two solutions should behave like $e^{\alpha}$, $e^{\alpha'}$, similarly at $b, c$, but I may be misunderstanding this.

What is the correct interpretation of the term "exponent", here to describe these quantities? How can we show these exponents arise?

  • $\begingroup$ What you call "hypergeometric differential equation" is in fact Riemann's differential equation. You will find the answer to your question in the link. $\endgroup$
    – Gonçalo
    May 15 at 15:00


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