108 THE REV. F. H. JACKSON ON 
The series within the large brackets, although simple in form, offers considerable 
ditticulty in summation. The sum is 
2+ Diet +I) «GP A41)- GH DpteT) =. Gr +E as) 
The sum in general for all values of 7 is 
ofA]. (2) Dlr + 1) Py((r +1) | 
Pe Pees ee ic) 
The reason that the simple series (p= 1) 
Qr 
2r+2 
2r(21r — 2) 
Q@r+ Darth 
8 
aS 
4 
is easily summed as 
r! r! 
i 
or! 
while the general series offers difficulty, is that the functions I, are present, both to 
the base p* and the base p” in the series (12) and in expression (14). 
Herne has shown in his “ Kugelfunctionen” that 
${a, b, e pal=1+ pote (1 =a)(1 = ap)(1 = (1 = bp), 9 
n= C27) (epee e(l=c) 
a ~ bap" )(1 — aD) Boe 
ae es (1 — ap")(1 — ep”) |e | (15) 
Consider now the series 
7) (el eS) eal ee 0) feet a [r] [r-1].. plSeee ll) 
Ti getl ieee ek ee Sea peer al. aaa eo) 
Since 
Zita. 
(il > ete ane 
4 4—] 
Bi pete 
we write series (16) as the sum of two Hetnr’s series 
S toe eel 1)(p"* — 1) 
dy +P] S, {i+ Fe eda ees Asal 
; 7] 1 s(s+3 al) eee all 
+P ta] { i > off feuse sy te 0 i. =3 ee = 1 cease \ » (ls) 
These series we transform by means of (15). 
First, for the series 8, we put 
