## Abstract

In the year 1812 Johann Frederick Carl Gauss published a comprehensive work studying the where α, β, γ, x are real numbers. The name hypergeometric comes from the fact that this series is a generalization of the geometric series. In fact, if we set α = 1 and β = γ in (1.1) above we have which is the well-known geometric series.

*hypergeometric series*which has the form$$1+\frac{\alpha \beta}{\gamma}\ {x}+\frac{\alpha(1+\alpha)\beta(1+\beta)}{2!\gamma(1+\gamma)}{x}^{2}\ +\frac{\alpha(1+\alpha)(2+\alpha)\beta (1+\beta)(2+\beta)}{3!\gamma(1+\gamma)(2+\gamma)}{x}^{3}\ +\ ...$$

$$1\ +{x}\ +\ {x}^{2}\ +\ {x}^{3}\ +\ ...$$

## Keywords

Bessel Function Recurrence Relation Gamma Function Hypergeometric Function Legendre Polynomial
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.

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## Copyright information

© Springer-Verlag New York Inc. 1988