In this paper Euler considers the properties of the pentagonal numbers, those
numbers of the form \$3n^2 n2\$. He recalls that the infinite
product \$(1-x)(1-x^2)(1-x^3)...\$ expands into an infinite series with exponents
the pentagonal numbers, and tries substituting the roots of this infinite
product into this infinite series. I am not sure what he is doing in some
parts: in particular, he does some complicated calculations about the roots of
unity and sums of them, their squares, reciprocals, etc., and also sums some
divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any
suggestions or corrections about these parts.
%0 Generic
%1 citeulike:3036290
%A Euler, Leonhard
%D 2005
%K Vor1800 available-in-tex-format mathematics number-theory pre1800
%T On the remarkable properties of the pentagonal numbers
%U http://arxiv.org/abs/math/0505373
%X In this paper Euler considers the properties of the pentagonal numbers, those
numbers of the form \$3n^2 n2\$. He recalls that the infinite
product \$(1-x)(1-x^2)(1-x^3)...\$ expands into an infinite series with exponents
the pentagonal numbers, and tries substituting the roots of this infinite
product into this infinite series. I am not sure what he is doing in some
parts: in particular, he does some complicated calculations about the roots of
unity and sums of them, their squares, reciprocals, etc., and also sums some
divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any
suggestions or corrections about these parts.
@misc{citeulike:3036290,
abstract = {In this paper Euler considers the properties of the pentagonal numbers, those
numbers of the form \$\frac{3n^2 \pm n}{2}\$. He recalls that the infinite
product \$(1-x)(1-x^2)(1-x^3)...\$ expands into an infinite series with exponents
the pentagonal numbers, and tries substituting the roots of this infinite
product into this infinite series. I am not sure what he is doing in some
parts: in particular, he does some complicated calculations about the roots of
unity and sums of them, their squares, reciprocals, etc., and also sums some
divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any
suggestions or corrections about these parts.},
added-at = {2009-08-02T17:14:35.000+0200},
archiveprefix = {arXiv},
author = {Euler, Leonhard},
biburl = {https://www.bibsonomy.org/bibtex/2cf8a9ebebdd40db54b94cfa7db7e350c/rwst},
citeulike-article-id = {3036290},
citeulike-linkout-0 = {http://arxiv.org/abs/math/0505373},
citeulike-linkout-1 = {http://arxiv.org/pdf/math/0505373},
description = {my bookmarks from citeulike},
eprint = {math/0505373},
interhash = {ffec598eced9609d2c15b872c8a010e6},
intrahash = {cf8a9ebebdd40db54b94cfa7db7e350c},
keywords = {Vor1800 available-in-tex-format mathematics number-theory pre1800},
month = May,
posted-at = {2008-07-23 08:50:20},
priority = {2},
timestamp = {2009-08-06T10:36:18.000+0200},
title = {On the remarkable properties of the pentagonal numbers},
url = {http://arxiv.org/abs/math/0505373},
year = 2005
}