242 BELL SYSTEM TECHNICAL JOURNAL 



Thus a recurrence relation expressing A„,n in terms of A,n-\.n-\ and x 

 Am-2, n-2 evidently requires a relation between Fa+y+, F, and Fa-y^. 

 Referring to Gauss' tables/' we find 



(y- a- \)F + aF^+ - (y - 1)/^.- - 0, 

 7(1 - z)F- yF„_ + (7 - ^>F„+ = 0. 



From the second of these two equations we form two more equations by 

 substituting a + 1 for a in one case and 7 — 1 for 7 in the other, 

 giving 



7(1 - s)F„+ - yF+ (y- ^)zF^+y+ = 



(7 - 1)(1 - z)Fy^ - (7 - 1)F„_,_ + (7 - 1 - l3)zF = 0. 



Now eliminating Fa+ and F^_ from the first, third, and fourth of the 

 equations, we obtain 



a(/3 - y)zF„+y+ + 7[7 - 1 + (a - (3)z:\F 



which is the relation desired. Substituting the value of Fa+y+ in 

 terms of Amn, F in terms of A,„-i, „_i, and Fa-y- in terms of A,n-2. n-2 

 gives the recurrence formula of Equation (30). 



In using (30) we may find, as for instance in calculating Amo, Ami, 

 Aon, Ain, that the right hand member involves coefficients with nega- 

 tive subscripts. A simple rule for treating such cases may be demon- 

 strated as follows. We first note that if we replace m by —ni in 

 (28) the value of the right hand member is unchanged.^ Hence since 

 (30) is derivable directly from (28), we conclude that correct results 

 are obtained from (30) if we adopt the convention, 



A = A 



The case of n negative is a little more difficult because if n is a 



negative integer in (28), an indeterminate form results. However, 



making use of the result just obtained on the interchangeability of sign 



of the subscripts, w, m — 1, m — 2 in (30), we can demonstrate a 



^ Gauss, Werke, Bd. Ill, page 130. The equations used here are numbered (5) 

 and (8) by Gauss. 



' If we express {-)'^'^T ( ^-^ j / T ( j-^ — j in terms of (-)-'«'2 



( —m -\- n — \\ / r^ { -m — n -\- ?,\ . 



Y { ^ ) / ^ I 9 / y successive apphcations of the recurrence 



formula for the gamma function, we find the two quantities are equivalent. Chang- 

 ing the sign of m in the hypergeometric function merely interchanges a and /3, and 

 hence does not change the value of the function. 



