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Ramanujan's Lost Notebook pp 181–204Cite as

Theorems about the Partition Function on Pages 189 and 182

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Abstract

Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences p(5n+4)≡0 (mod 5) and p(7n+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity

$$\sum_{n=0}^{\infty}p(5n+4)q^n = 5 \frac{(q^5;q^5)_{\i}^5}{(q;q)_{\infty}^6}, \qquad |q|<1, $$

which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence p(7n+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function p r (n) defined by

$$ \frac{1}{(q;q)_{\infty}^r}=\sum_{n=0}^{\infty}p_r(n)q^n, \qquad |q|<1. $$

In particular, he states new congruences for p r (n).

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    George E. Andrews

  2. Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, USA

    Bruce C. Berndt

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  1. George E. Andrews
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  2. Bruce C. Berndt
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Correspondence to George E. Andrews .

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Andrews, G.E., Berndt, B.C. (2012). Theorems about the Partition Function on Pages 189 and 182. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_6

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