View source for Hypergeometric series

''Gauss series''

A series of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484601.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484602.png" /></td> </tr></table>

Such a series is meaningful if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484603.png" /> is not equal to zero or to a negative integer; it converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484604.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484605.png" />, it also converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484606.png" />. In such a case the Gauss formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484607.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484608.png" /> is the gamma-function, holds. An analytic function defined with the aid of a hypergeometric series is said to be a [[Hypergeometric function|hypergeometric function]].

A generalized hypergeometric series is a series of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h0484609.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846011.png" />. If this notation is used, the series

is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846012.png" />.



====Comments====
Generalized hypergeometric series can be characterized as power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846014.png" /> is a rational function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846015.png" />. 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. [[#References|[a1]]].

Basic hypergeometric series can be characterized as power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846017.png" /> is a rational function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846019.png" /> is a fixed complex number not equal to 0 or 1. Such series have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846020.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846021.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048460/h04846022.png" />. See [[#References|[a2]]].

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

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