A note on the rank parity function

Abstract

a b s t r a c tWe provide some further theorems on the partitions generated by the rank parity function. New Bailey pairs are established, which are of independent interest.

On Bailey pairs and certain q-series related to quadratic and ternary quadratic forms

“…A particularly nice example that was established in [4], and bears a close resemblance to Euler's Pentagonal Number Theorem [10], is given by (see [10]); for convenience we also put (a) n = (a; q) n . The q-series in (1.1) is, in fact, of the Rogers-Ramanujan type, and has the equivalent form ∞ n=0 q 2n 2 (q) 2n = 1 (q 2 , q 3 , q 4 , q 5 , q 11 , q 12 , q 13 , q 14 ; q 16 ) ∞ , which follows from the limiting case ρ 1 , ρ 2 → ∞ of Bailey's lemma (see The motivation of this study is two-fold. We establish another approach to proving identities like (1.1), and go a step further by finding identities that are related to certain ternary quadratic forms.…”

Remarks on a Bailey Pair with One Free Parameter

“…Corollary 3.7 has appeared in [3, pg.233, eq. (9.4.4)], and was noted in [12] due to its relevance to the q-series n≥0 q n(2n+1) (−q) 2n , which was found to be lacunary and related to σ(q) in [12], by using the x → 0 instance of Theorem 2.1.…”

“…Further notes on σ * (q) can be found in [11], and more examples related to real quadratic forms are given in [5,7,8,11,12]. We consider a similar function…”

“…(45)] also offered a proof of the pair with the β n given in (1.2) with a = q using a recurrence approach, and subsequently establishing new partition theorems using different limiting cases of Bailey's lemma than used in [12].…”

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AbstractWe offer a more general Bailey pair than one that was proved in two different papers by two different methods [5,12].

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