Some Equalities for Continued Fractions of Generalized Rogers-Ramanujan Type

Abstract

Abstract. In this paper, we first discuss the convergence of the continued fractions of generalized Rogers-Ramanujan type in the modified sense. Then we prove several equalities concerning these continued fractions. The proofs of our main results are mainly based on the Bauer-Muir transformation. Preliminary materialIf the sequence {S n (0)} of the approximants of the continued fraction b 0 + K(a n /b n ) converges to a point f in the extended complex plane C = C ∪ {∞}, then we call that the continued fraction b 0 + K(a n /b n ) converges to f in the classical sense, and write .).As in [5] or [8], we introduce the following definition.

Some $$q$$ q -continued fractions of Ramanujan, their explicit values, and equalities

“…Li and Wang [9] also proved some equalities on generalized Rogers-Ramanujan type continued fractions. Lee and Sohn [8] studied certain equalities of the continued fractions (1.9) and (1.10).…”

“…To prove (ii), replacing n by 1/n in the definition of b n in (i) and simplifying using the result J 1/n = 1/J n from [1, p. 9, Theorem 6.1], we find that 5,6,7,8,9,10,11,13,15,16,17,18,19,23,25,31,36, and 49. Yi [21] also evaluated J n for n = 1, 2, 3, 4, 5, 8, and 9.…”