170 071 Expansions in Tetraehoric Fmictions 
As in the case of the Incomplete F-function multiply each side by e^^ and inte- 
grate. The limits of the integral on the left-hand side will be x - 0 and a; = 1, as 
we take the value of the integral outside these limits to be zero. 
The ^ limits will therefore be —p/(p + q) for x — O and q/ip + q) for 
Then 
J -Pip+q ^yP><l) J -'X, v27ro- 
I.e. ^-^dx=\ e^^"^^ (b{- I)) ~==du, 
Jo ^(P,q) J— ^ V27r 
and e-^p/y^+s — ^ —L — dx= 6 {0(t) \ ^,_^ ~dy 
= 0(^cr) ' — dy 
, J - co v Ztt 
Now 
= <j>{da)ei'''^' (xxi). 
1 xP-" (1 - a;)'^-! , 
^ ^l + 0x + ~- + , +... + ——-{- ...\ dx 
But 
B(p, ?) 
"/o B(p,9) ■ ' 2! ■ 8! ,s'! 
_B{p,q) B{p + l,q) lM£+2^ , , e^ B{p + S,q) 
^{p,q) ^{p,q) 2! B{p,q) ^-^^s! B (p, 
B{p + s,q)_ p{p + l) ...jp + s-l) 
B(/^, g) (^ + <?)(;;>^-f 1)... {p + q + s-iy 
therefore ^.-^ — -\ dx^l + 0 1- ^ , — — 
Jo ^(P,q) P + q 2\ (p + q){p + q + l) 
+ + p(p + l)...(p + s-l) _^ 
si {p + q)(p + q + l) ...{p + q + s -1) 
From equation (xxi) 
1 + 2 ! (p-|-g)(p-|-g + l) 
+ p(p + l)...{p + s-l) I 
^■•■^s! + ... + g + j ^ ^" 
Let ^ (- i)) = a„ - ttiD + a.,D' - ...{-If asD" + 
4> (da) = a„ + tti (^0-) + {day + a, {Oaf + a, {Oay + . . . 
= c„ -t- c^e + c.^Q'' -f- . . . -H c,^-^- -t- . . . , 
where Cs = «.,cr''. 
By equating coefficients of powers of 6 in equation (xxii) the coefficients in 
<\> (da-) can be obtained in terms of p and q, for 
