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

Propa Sine Clarification