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Special Functions

  • Mayer Humi
  • William Miller
Part of the Universitext book series (UTX)

Abstract

In the year 1812 Johann Frederick Carl Gauss published a comprehensive work studying the hypergeometric series which has the form
$$1+\frac{\alpha \beta}{\gamma}\ {x}+\frac{\alpha(1+\alpha)\beta(1+\beta)}{2!\gamma(1+\gamma)}{x}^{2}\ +\frac{\alpha(1+\alpha)(2+\alpha)\beta (1+\beta)(2+\beta)}{3!\gamma(1+\gamma)(2+\gamma)}{x}^{3}\ +\ ...$$
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
$$1\ +{x}\ +\ {x}^{2}\ +\ {x}^{3}\ +\ ...$$
which is the well-known geometric series.

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.

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Mayer Humi
    • 1
  • William Miller
    • 1
  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

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