Hypergeometric series

Gauss series

A series of the form

$$ F ( \alpha , \beta ; \gamma ; z) = $$

$$ = \ 1 + \sum _ {n = 1 } ^ \infty \frac{\alpha \dots ( \alpha + n - 1) \beta \dots ( \beta + n - 1) }{n! \gamma \dots ( \gamma + n - 1) } z ^ {n} . $$

Such a series is meaningful if $ \gamma $ is not equal to zero or to a negative integer; it converges for $ | z | < 1 $. If, in addition, $ \mathop{\rm Re} ( \gamma - \alpha - \beta ) > 0 $, it also converges for $ z = 1 $. In such a case the Gauss formula

$$ F ( \alpha , \beta ; \gamma ; 1) = \ \frac{\Gamma ( \gamma ) \Gamma ( \gamma - \alpha - \beta ) }{\Gamma ( \gamma - \alpha ) \Gamma ( \gamma - \beta ) } , $$

where $ \Gamma ( z) $ is the gamma-function, holds. An analytic function defined with the aid of a hypergeometric series is said to be a hypergeometric function.

A generalized hypergeometric series is a series of the form

$$ {} _ {p} F _ {q} ( \alpha _ {1} \dots \alpha _ {p} ; \ \gamma _ {1} \dots \gamma _ {q} ; z) = $$

$$ = \ \sum _ {n = 0 } ^ \infty { \frac{1}{n!} } \frac{( \alpha _ {1} ) _ {n} \dots ( \alpha _ {p} ) _ {n} }{( \gamma _ {1} ) _ {n} \dots ( \gamma _ {q} ) _ {n} } z ^ {n} , $$

where $ ( x) _ {n} \equiv x( x + 1) \dots ( x + n - 1) $. If this notation is used, the series

is written as $ {} _ {2} F _ {1} ( \alpha , \beta ; \gamma ; z) $.

Comments

Generalized hypergeometric series can be characterized as power series $ \sum _ {n = 0 } ^ \infty A _ {n} z ^ {n} $ such that $ A _ {n + 1 } /A _ {n} $ is a rational function of $ n $. An analogous characterization for series in two variables was given by J. Horn. This yields a class of power series in two variables which includes the various Appell's hypergeometric series, cf. [a1].

Basic hypergeometric series can be characterized as power series $ \sum _ {n = 0 } ^ \infty A _ {n} z ^ {n} $ such that $ A _ {n + 1 } /A _ {n} $ is a rational function of $ q ^ {n} $, where $ q $ is a fixed complex number not equal to 0 or 1. Such series have the form

$$ {} _ {r} \phi _ {s} ( a _ {1} \dots a _ {r} ; \ b _ {1} \dots b _ {s} ; \ q, z) = $$

$$ = \ \sum _ {n = 0 } ^ \infty \frac{[ ( - 1 ) ^ {n} q ^ {n ( n - 1)/2 } ] ^ {s - r + 1 } }{( q; q ) _ {n} } \frac{( a _ {1} ; q ) _ {n} \dots ( a _ {r} ; q ) _ {n} }{ ( b _ {1} ; q ) _ {n} \dots ( b _ {s} ; q ) _ {n} } z ^ {n} , $$

where $ ( x; q) _ {n} \equiv ( 1 - x) ( 1 - qx) \dots ( 1 - q ^ {n - 1 } x) $. See [a2].

Hypergeometric functions of matrix argument have also been studied, cf. [a3].

References

[a1]	P. Appell, M.J. Kampé de Fériet, "Fonctions hypergéométriques et hypersphériques: Polynômes d'Hermite" , Gauthier-Villars (1926)
[a2]	G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press (1989)
[a3]	K. Gross, D. Richards, "Special functions of matrix argument I" Trans. Amer. Math. Soc. , 301 (1987) pp. 781–811