A MATHEMATICAL THEORY OF RANDOM MIGRATION 

 Therefore by (xvi) : 



K=jr^f(P- 1 - s )(p- 2 - s )---(- s )(- 1 ) s ~ 1 j'y' s ~ 2eWd ^ 

 = { _ 1)p s(s-y. .£-p+i) jy e - xdx> i£x= _ m 



47 



Thus 



E s (r) = 2vo> |a> - sco, + v (2!) / o, 4 i ^ >- co s +.. 



.(lxxxiv) 



= 27rcr 2 C/ s (r)* J say. 



Now consider : 



/; 



^^(-l)^)*- 1 S A . ("wlr 



d (cr 2 )' 



/; 



-if'la* 



.(lxxxv). 



= a» 2s -S0) M _ 2 



We can now express I a (r) in terms of tu-functions. 



We have : 



I s (r)=j"s\E s (r) r ^ 



,„ , T 00 f «(«-!) s(s-l)(s-2) 1 rdr 



= s!27rcr 2 l J6> -so> 2 + ',„ ' at,--* — T^i / o»,+ ...|-^ r 



, / ,, f sot, s(s — 1) s(s— l)(s — 2) , I 



=<l2 ^(' +1 )|^-rrii + -2!3i ^~ Bui ^ + -j 



= s!27ro- a (s+l)F,(r). 

 Thus /„( C ) = iy-&4^V£(( S +l)^(c)F s (a)) (lxxxvi), 



where : 



T7- / \ s s(s— 1) s(s — l)(s — 2) 



t / *(^) = »o-/J]y"i+ / 2 !) 2 *"« (WW w 6+"-' 



T r / \ s s(s— 1) S(s— l)(s — 2) 



F A r ) = w »-IT2"! & ' a + ~2T3T^ 314! £ * 6+ -' 



a result which allows of fairly rapid determination from tables of o^co^. 



There is, perhaps, less difficulty in this form in allowing for the first term 

 or two of the operator Q t , for U s (r) and V s (r) can be at once differentiated with 

 regard to o- 2 , but even then the final result has considerable complexity. 



* This result involves the expression of any power of r 2 in x-f unc ti on3 - 



