A q-continued fraction

Let a, b, c, d be complex numbers with d 6= 0 and |q| \u3c 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞