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Abstract
We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.
Highlights
The proof of a q-series identity, whether a series-to-series identity such as the second iterate of Heine’s transformation (see (4.1) below), a basic hypergeometric summation formula such as the q-Binomial Theorem (see (2.1)) or one of the Rogers-Ramanujan identities (see (S14) below), generally falls into one of two broad camps.In the one camp, there are a variety of analytic methods
The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and use analytic methods to prove the full version
We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan
Summary
The proof of a q-series identity, whether a series-to-series identity such as the second iterate of Heine’s transformation (see (4.1) below), a basic hypergeometric summation formula such as the q-Binomial Theorem (see (2.1)) or one of the Rogers-Ramanujan identities (see (S14) below), generally falls into one of two broad camps. There are a variety of analytic methods These include (but are certainly not limited to) elementary q-series manipulations (as in the proof of the BaileyDaum summation formula on page 18 of [15]), the use of difference operators (as in the electronic journal of combinatorics 18 (2011), #P60. In the present paper we use a “hybrid” method to prove a number of basic hypergeometric identities. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. We use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities
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