106 BELL SYSTEM TECHNICAL JOURNAL 



that the average value of i?" is, when —2<re («) < — |, 



2-'r(i+'i).„ 



R" = / , -' / »•-"-' /o(Pr)/„(Or)e"*'"'" dr 



r'-" 



(\ 00 00 



^ / *:=0 m=0 



-'i) (-ti;)^(-3')'" (3.10-25) 



2/k+m 



k\k\m\m\ 



It appears very probable that this result could be extended, by analytic 

 continuation, to positive integer values of ;/. W'e have used the notation 



(«)o = 1, {a)k = a{a+\) --• {a + k -\) 



p2 01 (3.10-26) 



X = — , -v = — 



2\{/o ' ' 2\po 



and have denoted the Legendre polynomial by Pkis). The series converge 

 for all values of P, Q, and xpo and terminate when n is an even positive integer. 

 WTien X or y, or both, are large in comparison with unity we ma}' use the 

 integral for R"^ to obtain the asymptotic expansion, assuming Q < P so 

 that y < X, 



m- 



n 



21k T^ /, n , 11 . y 



^^^len 11 is an even positive integer this series terminates and gives the same 

 expression as (3.10-25). When n is an odd integer the 2^1 may be expressed 

 :n terms of the complete elliptic functions R and K of modulus y "x~ ": 



\ Xl -K TT \ X/ 



(3.10-28) 



iF.ihh^-A =-K 



\ X/ -K 



The higher terms may be computed from 



a{\ -3)%Fi(a+l,a+l;l;s) = (2a- 1)(1 + s)oFi(a, a; l;z) 



+ (1 - a^F^ia- l,a- l;l;s) (3.10-29) 



