120 BELL SYSTEM TECHNICAL JOURNAL 



where we have written /„ for Imiy, v) and assumed v > 0. Then, using 

 (5-11), 



n=3,5.... 



-2 z «o/-ir^(n^ + cV] 



n=»2.4.-" 



When we put 

 br.{v) = [an-x{v) - a„+iW]/2, n > 1 (7-8) 



bn{v) = 2/3[(;j - l)-ig-("-i)|^'l - (n -\- iyh-^-+'^\% n = 2, 4, 6, • • • 



bn{v) = 0(i32), w = 1, 3, 5, • • • 

 in (5-11) and use the results of Appendix II we obtain 

 R'^' = i?y^ - iA-y E 5;^[0^ - l)~V(;z - 1, w - 1,5„, 0,0) 



71=2,4,6- •• 



+ (w + ly'Jin -f 1, w + 1, c, 5„ , 0, 0) . (7-9) 



-2(n - ly' J{n - 1, w + 1, c, 5„, 0, 0)] 



The values of the first two J's, obtained by setting m = n ± 1 in (A2-7), 

 may be simplified by using 



c2 + (n ± 1 4- 5)2 = 2(ii ± l)in + 5) 



where we have dropped the subcript n from 5„ . In order to eUminate 5 

 from the denominator we multiply both numerator and denominator by 

 n — 8 and use 



(n - 8) {8+ In ± 2) = (». ± 1)^ -f c^ - 8{n ± 2) 

 «2 _ 52 = 1 + ^2 = ^2 



Setting in the value, given by (A2-9), of the last / and separating the 

 terms (into those which contain the first power of 8 and those which do not) 

 enable us to write the term within the square brackets in (7-9) as 



\n^ _ 8 J ( » - 1) - 1 ( m + 1) + 1 



K\n^ - 1)2 k2 [(^ _ 1)2{,2 + (w - 1)2} "^ (/Z + 1)2{C2 + (W + 1)M 



, 2n{2(>z^ + c^) -/cV-l)}] .710) 



"^ k2(«2 - 1)2(«2 + c2) J 



It is found that when (7-10) is put in (7-9), the contribution of the first 

 two terms within the square bracket of (7-10) exactly cancels the summation 



