DETERMINATIO SERIEI NOSTRAE PER AEQUATIONEM DIFFERENTIALEM ETC. 
211 
aequatio 83 iit 
F{a, fi, a+6+l — 7, 1—0?) 
= f{a,fi,y)F{a,fi,y,at) 
-\-g[a, fi, y)(1—o?) Y-a-6 o? 1_Y .F(l — a, l — fi, 2—7, 0?) 
Eidem aequationi adiumento formulae 82 hanc quoque formam tribuere licet 
7,^+1—7, a+€+l—7, l — w) = 
f{a, fi, 7)( 1 — o?) Y_a_6 F{7—a, 7—fi, 7, oc)-\-g[a, fi, 7)o? 1_Y -F(a-f-1 —7, fi-]-1 —7,2—7, a?) 
sive dividendo per o? 1_Y , mutandoque resp. a, fi, 7 in a-j-i — 7, *> + l—7, 2 — 7 
F(a, fi, a-{-fi-\-\—7, 1—0?) 
= ^(a+1 — 7, @+l —7, 2—'7)jP(a, @, 7, 0?) 
4-/(a4-l—■7, @4-1 — 7, 2 — 7)(1 — oo x ~~'iF[\—a, i—fi, 2—7,0?) 
Quae quum identica esse debeat cum formula praecedenti, habemus 
g{a, fi, 7) — 7, @4-1—7, 2—7) 
itaque 
[84] F{a, fi, a4-@4-l — 7, 1 — 0?) 
= /(a, @, 7).F(a, @, 7, 0?) 
4-/(a-f-l —7, @4-1 —7, 2 — 7)(1 — o?) Y ~ a_6 o? 1_Y .F(l — a, 1 —@, 2—7,0?) 
43. 
lam ut indolem functionis f[a, fi, 7) eruamus, statuamus 0? — 0. Tunc 
patet, esse F{a, @, 7, 0?) = 1, o? 1_Y == 0, quoties quidem 1—7 fuerit quantitas 
positiva. Sed per aequationem 48 habemus 
F{a, fi, a-f-@4-l—7, 1) = n(a—y)n\s—y)) 
Quare sub eadem restrictione demonstratum est, fieri 
[ 85 ] — n(a — y)II(6 — y) 
Hanc vero formulam generalem esse, ita demonstramus. 
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