A simple proof of Watson's partition congruences for powers of 7
A simple proof of Watson's partition congruences for powers of 7
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DOI：
10.1017/s1446788700025386
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发表时间：
1984-06
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影响因子：
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通讯作者：
F. Garvan
中科院分区：
文献类型：
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作者：
F. Garvan
Ramanujan猜想，如果n是一个特定的形式，那么n的无限制分割数p(n)可以被7的高次幂整除。g·n·沃森证明了拉马努金猜想的一个修正版本。本文建立了合适的生成公式，从这个公式可以很容易地推导出沃森的结果。我们的证明比沃森的更直接。它们是初等的，只依赖于欧拉和雅可比的经典恒等式。沃森的证明依赖于七阶模方程。我们也需要模方程但是我们用O. Kolberg的基本技巧推导它。
Abstract Ramanujan conjectured that if n is of a specific form then p(n), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson. In this paper we establish appropriate generating formulae, from which Watson's results follow easily. Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.