292 MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
And when a > 2, we have asymptotically for all values of |arg x\ 
V 
exp {x v °e a } v e c n T (1-/3) 
R 'a '2t7uj.R o. ™nla F2mtxnla tw / i O 
Ea (* ; 0’P] - a ^_ p L H to x n/a e 2m,in/a f (1 - y8 - n)J ’ 
wherein p takes all integral values, positive, negative, or zero, such that 
27t/x + arg x < |-a7r 
Part IX. 
The Function jF] {a; p;x\. 
§ 50. Generalised hypergeometric functions form a wide class of integral functions 
whose asymptotic expansions are closely connected with the theory of linear 
differential equations. 
The general type of such series is 
1 + a i • • • a p x + g*i(«i + l)--.« P (^+ 1 ) x 2 +... 
1 . Pl ... pg 1.2 .pi (pi + 1 ) Pq {pg + I ) 
r( p 1 )...r(p g ) | r(* l +n)...r(* p +n) ^ 
T ( ai ) ...V (a p ) n=oT (n+l)T ( Pl + n) ...r (p q + n) 
wherein p < q. 
This series we shall denote by pF^jaj, ..., a p ; p u ..., p q ; x} or briefly by pF g {cc}. 
The series satisfies the differential equation 
| (3 + ai)...($ + u p )-^-p+p,-l)'-‘{3 + pg~l)}y = 0 • • • l 1 )’ 
where 3 = x —. The equation is of order (q + 1), and the other q solutions are 
dx 
given by 
X l ~ px pFq { Ctl — Pl + 1, Up — Pl+1 ; 2 — Pl, p 2 — Pl+ G •••) P? —P1+ 1 > X }’ 
and (q— 1) similar functions. 
I give here some of the results which I have obtained by applying integrals of the 
types previously used to the theory of these series. I or detailed mvestigations I 
refer to a forthcoming paper.* 
I. The series ^ {a ; p ; x} satisfies the equation 
x 
d 2 y , \dy 
—2- — (x—p) -2--a.il = 
dx 2 
dx 
ay 
0. 
* ‘ Cambridge Philosophical Transactions,’ vol. ‘20, 
