6 THE REV. F. H. JACKSON ON 
Taking (30) from (29) we obtain by (23) that 
1l—n d por d yn 
ay - Ny (4p) = Lene ae) a \° ae OS nal’ mn) b = Te | NF ni 2X) t 
The expression on the right is equal to 
1 [x] 1 i d aa l 
y Tinga) ar sa re Tins) } 
1 [-x] reat ial th ad x ’ a 
_ ay x J in(wr) Tes reat J iny(@ ) . s | ( ) 
Now 
n =) 
ay. Le (0d) 7 x a! J png(#A) 
is equal to 
[20] 
dp” =u in(#X) 
and therefore we have finally 
ce {J tn(@A) — Deny(a , Ap) | a?” da? Taled)- Pe AG ete) .. (2p 
a very See equation. 
From (23) it follows that 
Tin-n(@?d) = Gr { [2n]I (arp) +p 2+ 4]Tnyol@Ap) toes ee L (38) 
6. 
The Function P,,,(xA). 
If P,,,(z) denote the series 
ny _ [nr] [nr oe fee 2 Oe neg SS 
Aja ees Bey? ine aT oa eee $4 
Then * P,,,(a) satisfies the differential equation 
rd ly ee dy 
al da”) a dz™ #41 =e ls We 1} | att aP [~][- i ly=P; [n—- ne) pas aC ) (35) 
Introducing a constant parameter \ we have more generally the series 
es [» 1 RP arenes n 1] [w- 2] [nw - 3], re aes : 
Af a - Ooe ie Bi Boe ae =F ee 
Satisfying 
path — 5a gan + {1-(el-[-e- 1) fo! + fn) [ny = Pha) -Ph_(o"r) (87) 
Give to the arbitrary constant A the value 
[Qn]! A” 
(2),{7]! [nr]! 
* Proc. Edin. Math. Soc., vol. xxi. 
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