Hypergeometric Differential Equation

$$x(x-1){d^2y\over dx^2} + [(1+\alpha+\beta)x-\gamma]{dy\over dx} + \alpha\beta y = 0.$$

It has Regular Singular Points at 0, 1, and $\infty$. Every Ordinary Differential Equation of second-order with at most three Regular Singular Points can be transformed into the hypergeometric differential equation.

See also Confluent Hypergeometric Differential Equation, Confluent Hypergeometric Function, Hypergeometric Function

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 542-543, 1953.