Bailey Mod 9 Identities

The Bailey mod 9 identities are a set of three Rogers-Ramanujan-like identities appearing as equations (1.6), (1.8), and (1.7) on p. 422 of Bailey (1947) given by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

(OEIS A104467, A104468, and A104469).

Unfortunately, Bailey used non-standard (and essentially unreadable) notation in the paper where these identities first appeared. All three of these identities appear in the list of Slater (1952) as equations (42), (41), and (40) in that order. However, all three contain misprints.

In one sense, these identities are the next logical step in the following sequence:

1. The two Rogers-Ramanujan identities (triple product on mod 5 over ).

2. The three Rogers-Selberg identities (triple product on mod 7 over ).

3. The (sort of) four Bailey mod 9 identities (triple product on mod 9 over ).

Here, "sort of" refers to the fact that between  and , there is an "identity" in which the product side contains , so the identity reduces to  and therefore is not listed.

REFERENCES:

Bailey, W. N. "Some Identities in Combinatory Analysis." Proc. London Math. Soc. 49, 421-435, 1947.

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. 2 54, 147-167, 1952.

Sloane, N. J. A. Sequences A104467, A104468, and A104469 in "The On-Line Encyclopedia of Integer Sequences."