Confluent Hyper-geometric Function. 

 M(«, 7, x) 



_ I\ 7 ) , ,. A _ a f, "(«"7-H).l 



133 



r( 7 - 



b-^)--! 1 -^ 



a(« + l)(a-7+l)(a- 7 + 2) 1 



1.2 



J-,} 



+ ? M.,.,-, {l+ i-H^).i 



Both these series diverge for all values of x y but they have 

 the property that the error involved in taking the sum to n 

 terms to be the value of the series, is less than the nth term. 



V - £.M.(«,- V ,*)-J.M(« + 1,7+1,«). 



(10) 



(l-a).fM(a, 7 , •).*.« (l- 7 ). M(«-l, 7 -l, «) + (y-l). 



. . . (11) 



2=0 



Hence the function can easily be differentiated or in- 

 tegrated. 



VI. The following difference relations would be useful for 

 extending the tables : — 



^.M(a+l,7 + l,^) = M( a +l,7,^)~MK7, t f), 



a . M(« + l,7+l, x) = (cc-y) . M(*,7+l, x) +7 • M(«, 7, a?), 



(a + x) . M(a+1,7+1,^) = («-7).M(«,7+1,^) 



+ 7.M(a + l,7,.r), 



«7.M(«+l,7,ar)=7(« + «).M(a,7,d;) U12) 



-^(7-a).M(«,7 + l,a?) J 



* . M(« + 1, y, *) = + 2a— 7) . M(a, 7, #) 



+ (7-a).M(«-i,7,^) 3 



^ lflt .^.M(«,7 + l,^) = Gr + 7-l).M(a,7,^) ■ 



+ (l- 7 ).M(a,7-l,.r).J 



