160 research outputs found

    New Finite Rogers-Ramanujan Identities

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    We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.Comment: 19 pages. to appear in Ramanujan

    Rogers-Ramanujan Computer Searches

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    We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers–Ramanujan type and identities of false theta functions

    Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem

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    Let r≥1r\geq 1 be a positive integer, AA a real positive semi-definite symmetric r×rr\times r rational matrix, BB a rational vector of length rr, and CC a rational scalar. Nahm's problem is to find all triples (A,B,C)(A,B,C) such that the rr-fold qq-hypergeometric series fA,B,C(q):=∑n=(n1,…,nr)T∈(Z≥0)rq12nTAn+nTB+C(q;q)n1⋯(q;q)nrf_{A,B,C}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T}\in (\mathbb{Z}_{\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}} becomes a modular form, and we call such (A,B,C)(A,B,C) a modular triple. When the rank r=3r=3, after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers-Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier.Comment: 37 pages. We made some changes after the first version. Comments are welcom
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