# The Various Forms of the Differential Equation for the Confluent Hypergeometric Function and the Definitions of their Solutions

• Herbert Buchholz
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 15)

## Abstract

The usual Gauss or hypergeometric differential equation
$$z\left( {z - 1} \right) \cdot \frac{{d^2 y}} {{dz^2 }} + \left\{ {\left( {\alpha _1 + \alpha _2 + 1} \right)z - \beta } \right\} \cdot \frac{{dy}} {{dz}} + \alpha _1 \alpha _2 \cdot y = 0$$
(1)
has the following independent solutions at the regular or weak singularity z = 0 for arbitrary real or complex values of the three parameters α1, α2 and β:
$$\begin{array}{*{20}c} \hfill {y_1 } & \hfill { = \,_2 F_1 \left( {\alpha _1 ,\alpha _2 ;\beta ;z} \right) = \sum\limits_{\lambda = 0}^\infty {\frac{{\left( {\alpha _1 } \right)_\lambda \cdot \left( {\alpha _2 } \right)_\lambda }} {{\left( \beta \right)_\lambda \cdot \lambda !}} \cdot z^\lambda } } \\ \hfill {} & \hfill { = \frac{{\Gamma \left( \beta \right)}} {{\Gamma \left( {\alpha _1 } \right)\Gamma \left( {\alpha _2 } \right)}} \cdot \sum\limits_{\lambda = 0}^\infty {\frac{{\Gamma \left( {\alpha _1 + \lambda } \right)\Gamma \left( {\alpha _2 + \lambda } \right)}} {{\Gamma \left( {\beta + \lambda } \right) \cdot \lambda !}}} \cdot z^\lambda ,} \\ \end{array}$$
(1a)
$$\begin{array}{*{20}c} \hfill {y_2 = z^{1 - \beta } \,_2 F_1 \left( {1 + \alpha _1 - \beta ,\,1 + \alpha _2 - \beta ;\,2 - \beta ;z} \right)} & \hfill {\left( {\left| z \right| < 1} \right)} \\ \end{array}$$
(1b)

## Keywords

Differential Equation Laguerre Polynomial Whittaker Function Cylinder Function Confluent Hypergeometric Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.