m
Hm
DETERMINATIO SERIEI NOSTRAE PER AEQUATIONEM DIFFERENTIALEM ETC.
217
[89] F[a, fi, afi, 1 — x)
= -ng^h^-T) (log«+¥(a-l)+V(6-l)-a V(0)j P(a, <5, 1, *) '
_ H(a + g—l) | A CU6
II(a—l)II(6 —l) i
2.1.2
+ [A+B+C
-f- etc.}
■XX
a.a+l.a+2.6.6+1.6 + 2 3
_____ — t %
. 3 .1.2. 3
ubi
A=l + |-2. B
— U — 2u
a -f-1 g 1 a '
C ' — ¿TT2+rb — *’ etC ’
Ita e. g. pro a — \, fi = £ obtinemus (cfr. form. 52, 71)
[90] F{hh 1. 1 — *)
= logrV^ • T’ * > ^
— ii 2 • ( 2 + i) 524- ( 2 + -H- A)
1.1.3.3.5.5 3
X
4.4.6 . 6
+ (2 + ^+^+^:‘:^xX+ etc.,
= — — {logi, <#)~h\ x “h-rli x?>
i 1 0 2 5 0 1 r> 5 —U 1 3 9 4 2 3 9 I afn I
6 5 5 3 6 0 * 10485 JIO* —J— CUv. j
Denique casum tertium, ubi 7 est integer positivus unitate maior, seorsim
tractare haud necesse est, quum sit
F{ a, fi, a-\-fi-\-\—y, \—x) — —7, 6 + 1 — 7, a+6 + l —7, l—x)
transformatioque seriei _F(« + 1— 7, 6 + 1— 7, a+ 6 + 1— 7, 1 — x) pro 7 + 1
ad casum primum sponte reducatur.
47.
Transimus ad alias transformationes, inter quas primum locum obtineat
Hinc fit dx = —adeoque
y—1 (y—0
substitutio x
dP
— —^(1—yf, differentiando denuo fit
y dP
a dP
d dx —
ddP , /. x4 ddP , V 3dP
— "H 1 — y) T^~ 2 ( 1— y) di
(i—y) 2d d^ + 2 ( 1— ^) dP ’ adeo( l ue
dP
dy“ V } dy
28