Pentagonal Number Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that prod_{n=1}^infty (1-x^n)=sum_{k=-infty}^infty(-1)^kx^{k(3k-1)/2}. In other words, (1-x)(1-x^2)(1-x^3) cdots = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} + cdots. A striking feature of this expansion is the amount of cancellation in the product. The indices 1, 2, 5, 7, 12, ... appearing on the…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that prod_{n=1}^infty (1-x^n)=sum_{k=-infty}^infty(-1)^kx^{k(3k-1)/2}. In other words, (1-x)(1-x^2)(1-x^3) cdots = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} + cdots. A striking feature of this expansion is the amount of cancellation in the product. The indices 1, 2, 5, 7, 12, ... appearing on the right hand side are called pentagonal numbers (or more accurately, generalized pentagonal numbers).