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q-Trinomial Coefficients and Rogers-Ramanujan Type Identities

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Analytic Number Theory

Part of the book series: Progress in Mathematics ((PM,volume 85))

Abstract

There are many proofs [4], [2, Ch.7] of the celebrated Rogers-Ramanujan identities:

$$1 + \sum\limits_{n = 1}^\infty {\frac{{{q^{{n^2}}}}}{{\left( {1 - q} \right)\left( {1 - {q^2}} \right) \ldots \left( {1 - {q^n}} \right)}}} = \prod\limits_{n = 0}^\infty {\frac{1}{{\left( {1 - {q^{5n + 1}}} \right)\left( {1 - {q^{5n + 4}}} \right)}}}$$
(1.1)
$$1 + \sum\limits_{n = 1}^\infty {\frac{{{q^{{n^2} + n}}}}{{\left( {1 - q} \right)\left( {1 - {q^2}} \right) \ldots \left( {1 - {q^n}} \right)}}} = \prod\limits_{n = 0}^\infty {\frac{1}{{\left( {1 - {q^{5n + 2}}} \right)\left( {1 - {q^{5n + 3}}} \right)}}}.$$
(1.2)

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References

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  1. Dept. of Mathematics, The Pensylvania State University, 16802, University Park, PA, USA

    George E. Andrews

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  1. George E. Andrews
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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 west Green Street, 61801, Urbana, Illinois, USA

    Bruce C. Berndt , Harold G. Diamond , Heini Halberstam  & Adolf Hildebrand , ,  & 

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To my friend, Paul Bateman, on his seventieth birthday

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© 1990 Bikhäuser Boston

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Andrews, G.E. (1990). q-Trinomial Coefficients and Rogers-Ramanujan Type Identities. In: Berndt, B.C., Diamond, H.G., Halberstam, H., Hildebrand, A. (eds) Analytic Number Theory. Progress in Mathematics, vol 85. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3464-7_1

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  • DOI: https://doi.org/10.1007/978-1-4612-3464-7_1

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3481-0

  • Online ISBN: 978-1-4612-3464-7

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