The identity in question was actually first published by Jackson (1928) in slightly disguised form as the fifth equation on page 170 in his paper, though this early appearance of this identity is not well-known. It became widely known as equation 39 (and 83) in the collection of identities due to Slater (1952).

Jackson, F. H. "Examples of a Generalization of Euler's Transformation for Power Series." Messenger Math.57,
169-187, 1928.Leininger, V. E. and Milne, S. C. "Some
New Infinite Families of -Function Identities." Methods Appl. Anal.6,
225-248, 1999.Mc Laughlin, J.; Sills, A. V.; and Zimmer, P. "Dynamic
Survey DS15: Rogers-Ramanujan-Slater Type Identities." Electronic J. Combinatorics,
DS15, 1-59, May 31, 2008. http://www.combinatorics.org/Surveys/ds15.pdf.Slater,
L. J. "Further Identities of the Rogers-Ramanujan Type." Proc.
London Math. Soc. Ser. 254, 147-167, 1952.Sloane, N. J. A.
Sequence A069910 in "The On-Line Encyclopedia
of Integer Sequences."