Zur Theorie der Heineschen Reihe. 
63 
wo : 
(r = 0, 1, 2, . . .), 
zu Grunde gelegt werden. Man hat sodann 1 ): 
(1 — X) ■ <P(X) = 1 + (fr+l—fy) ■ x v + x 
U 
und : 
_ (1 - q n + v ) (1 — q ß + v ) — (1 — g y+v ) (1 ~ g* +v ) f 
Tv ~ (i_ gy +,)(i_^+v) •/>' 
(1 — 3 y+v )(i — q 5+v )' fr cf ’ 
wo : 
Q y = qj -f- <£' 5 — — (g“ -) - qß — q a +P+ v ). 
Nun ist identisch 
0 =(q y +q s -q y+s+t ’) (q a +q ß -q a + ß + v ) - (q°+ q ß - q"-+ß+ r ) (q y +q' ) - qY+ s + v ) 
= (jjjY qt> _ ^/'+'5+>') (cj aJ r' y -\- qß+ r — q a -\-ß+ 2*') 
— (g a + qP— q a +ß+ v ) (qr+ v +q ,s + v - ^/+' 5 +' 2v ). 
Subtrahiert man diese Identität von Q,. und beachtet, dab: 
1 — g°+ v — q^ v -f 5 a +^+ 2v == (1 — g a + v ) (1 — qP +r ) 
1 — q?+ v — q s + v 4- qy+ , ' i + lv = (1 — g ; ’+ v ) (1 — g ,5 +’’), 
so folgt: 
Qy = (q y + g' 5 — g ! ' +,5+v ) (1 — q a+r ) (1 — q ß+v ) 
— (q a + q ß —q a+ P +v ') (1 — §'+”) (1 - 
also : 
fr+i fv ~ = (q*+q s -qr+ s + v ) • f v+ 1 q* - (q n + q ß - q n + ß + v ) ■ f v q v 
2J ’■(/’>■+'-/.■) •^ v+, = (g y_I fy+ i {qx) v + l -(q a +qP)x^fy(qxy 
0 U 0 
- qy + s ~ 2 • 2»' f y+ 1 • (g 2 a;) v +i -q^Px-^vfriq*- x) v 
o 
o 
4 Vgl. a. a. 0., p. 30. 
