Partner with ConvertIt.com
 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 15. Hypergeometric Functions Mathematical Properties 15.1. Gauss Series, Special Elementary Cases, Special Values of the Argument Gauss Series The circle of convergence of the Gauss hyper- geometric series 15.1.1 F(a, b; c; 2) =2F1(a, b; c; 2) - r(4 5 r(a+n)r(b+n) - Z* r (a) r (b) r (c+n) n! - is the unit circle 1z1=1. series on its circle of convergence is: The behavior of this (a) Divergence when 5%' (c--a-b)I -1. (b) Absolute convergence when 5%' (c-a-b)>O. (c) Conditional convergence when - 1 <9 (c-a - b ) l O ; the point z=1 is excluded. The Gauss series reduces to a polynomial of degree n in z when a or b is equal to -n, (n=O, 1, 2, . . . ). (For these cases see also 15.4.) The series 15.1.1 is not defined when c is equal to -m, (m=O, 1, 2, . . .), provided a or b is not a negative integer n with nO) *See page n. 556 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.