General Solution to Hypergeometric Differential Equation

Theorem

Consider the hypergeometric differential equation:

$(1): \quad x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \dfrac {\d y} {\d x} - a b y = 0$


The general solution to $(1)$ is:

$y = A \map F {a, b; c; x} + B x^{1 - c} \map F {a - c + 1, b - c + 1; 2 - c; x}$

where $\map F {a, b; c; x}$ denotes the Gaussian hypergeometric function of $x$:

$\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}$


Proof


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Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: General Solution of The Hypergeometric Differential Equation: $31.13$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 32$: Hypergeometric Functions: General Solution of The Hypergeometric Differential Equation: $32.13.$
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