t 174 ] 



XIV. A Generalized Hyper geometric Function with n para- 

 meters. By Dorothy Wrinch, M'.Sc., Fellow of Girton 

 College, Cambridge, and Member of the Research Staff-, 

 University College, London *. 



'HE behaviour o£ the series 



!• + — ^ - + —^r- - +... 



nn(i+«,.) 2! n(l+«r)(2 + «r) 



r=l r=l 



as | £ | tends to infinity has been investigated in several 

 particular cases. In the case when n = l the series is easily 

 to be derived from the Bessel function, and its behaviour on 

 the circle | ^ | = oo is well known. In a recent paper by the 

 present author, the case when n = 3 was worked out. The 

 asymptotic expansion of the series in the general case has 

 gradually become apparent, and is established in this pap^r. 

 We may obtain the form of the asymptotic expansion of 

 the series 



F„_i(l + ai, l + a s ...L+a n -i m , -) 



= 1+ 2 





when | z | is large after the usual manner. 

 F n _i(^) satisfies the differential equation 



a(&-+*i)...(3 + a»_i)y=j*, 



which becomes 



$i$-r n* 1 )...($+na n _. 1 )y=yt», 



if by analogy with the treatment of Bessel functions we put 



z=(t/n)», 



t/n being any one of the nth roots of z. Putting y = eH x y, 

 the equation becomes 



(d + * — #)($ + i+n«i-*)..-.^ + i + na fl _i-% 1 =^ 1 . 



If x be chosen in such a way that the coefficients of the 

 leading power of t on both sides are equal, the asymptotic 

 expansion for F„_i{U/n)»} will be 



* Communicated by the Author. 



