Abstract
A Göllnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q i(n) denote the number of partitions of n into distinct parts ≢ i (mod 4). By attaching weights which are powers of 2 and imposing certain parity conditions on Göllnitz-Gordon partitions, we show that these are equinumerous with Q i(n) for i = 0, 2. These complement results of Göllnitz on Q i(n) for i = 1, 3, and of Alladi who provided a uniform treatment of all four Q i(n), i = 0, 1, 2, 3, in terms of weighted partitions into parts differing by ≥ 4. Our approach here provides a uniform treatment of all four Q i(n) in terms of certain double series representations. These double series identities are part of a new infinite hierarchy of multiple series identities.
The first author was supported in part by National Science Foundation Grant DMS-0088975
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References
K. Alladi, “On a partition theorem of Göllnitz and quartic transformations” (with an appendix by B. Gordon), J. Num. Th., 69 (1998), 153–180.
G. E. Andrews, “q-series, their development and applications in analysis, number theory, combinatorics, physics, and computer algebra”, CBMS Regional Conf. Series in Math., 66, Amer. Math. Soc., Providence, R.I. (1986).
G. E. Andrews, “An introduction to Ramanujan’s Lost Notebook”, Amer. Math. Monthly, 86 (1979), 89–108.
G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading (1976).
G. E. Andrews and R. J. Baxter, “Lattice gas generalization of the hard hexagon model III: q-trinomial coefficients”, J. Stat. Phys. 47 (1987), 297–330.
H. Göllnitz, “Partitionen mit Differenzenbedingungen”, J. Reine Angew. Math., 225 (1967), 154–190.
B. Gordon, “Some continued fractions of the Rogers-Ramanujan type”, Duke Math. J., 32 (1965), 741–748.
L. J. Slater, “A new proof of Rogers’ transformation of infinite series”, Proc. London Math. Soc, (2), 53 (1951), 460–475.
L. J. Slater, “Further identities of Rogers-Ramanujan type”, Proc. London Math. Soc. (2), 54 (1952), 147–167.
S. O. Warnaar, “The generalized Borwein conjecture II: refined (q-trinomial coefficients”, to appear in Discrete Math, arXiv: math.CO/0110307.
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Alladi, K., Berkovich, A. (2005). Göllnitz-Gordon Partitions with Weights and Parity Conditions. In: Aoki, T., Kanemitsu, S., Nakahara, M., Ohno, Y. (eds) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol 14. Springer, Boston, MA. https://doi.org/10.1007/0-387-24981-8_1
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DOI: https://doi.org/10.1007/0-387-24981-8_1
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