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Polylogarithms have interesting connections in the Theory of Partitions. This is mainly because a class of generating functions for bivariate partition statistics can be approximated by polylogarithms.

Consider for example the class of generating functions with $s>1,$

$\displaystyle \sum_{n=0}^\infty \sum_{m=0}^\infty p(n,m)x^n u^m =\prod_{k=1}^\infty \frac{1}{(1-xu^k)^{k^{s-2}}}.$

When $s=2,$ $p(n,m)$ is the number of partitions of $m$ with $n$ parts. When $s=3,$ $p(n,m)$ represents the number of plane partitions of $m$ with trace $n.$

For simplicity assume $1>x>0,$ as $u\to ^-1,$

$\displaystyle \prod_{k=1}^\infty \frac{1}{(1-xu^k)^{k^{s-2}}} \thicksim\sqrt[ - \frac{1}{\zeta(2-s)}]{1-x}\exp(\Gamma(s-1)Li_{s}(x)/(\ln u^{-1})^{s})u^{-1})^{s-1}).$

When used in conjunction with other techniques like the Circle Method, this estimate serves as a foundation for many facts about $p(n,m).$

Off the top of my head here are three papers that use this kind of estimate: http://www.math.drexel.edu/~rboyer/papers/partitions_experimental.pdf http://plms.oxfordjournals.org/content/s2-36/1/117.full.pdf http://www.springerlink.com/content/d65348128235x7m1/

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Polylogarithms have interesting connections in the Theory of Partitions. This is mainly because a class of generating functions for bivariate partition statistics can be approximated by polylogarithms.

Consider for example the class of generating functions with $s>1,$

$\displaystyle \sum_{n=0}^\infty \sum_{m=0}^\infty p(n,m)x^n u^m =\prod_{k=1}^\infty \frac{1}{(1-xu^k)^{k^{s-2}}}.$

When $s=2,$ $p(n,m)$ is the number of partitions of $m$ with $n$ parts. When $s=3,$ $p(n,m)$ represents the number of plane partitions of $m$ with trace $n.$

For simplicity assume $1>x>0,$ as $u\to ^-1,$

$\displaystyle \prod_{k=1}^\infty \frac{1}{(1-xu^k)^{k^{s-2}}} \thicksim\sqrt[ - \frac{1}{\zeta(2-s)}]{1-x}\exp(Li_{s}(x)/(\ln frac{1}{\zeta(2-s)}]{1-x}\exp(\Gamma(s-1)Li_{s}(x)/(\ln u^{-1})^{s}).$

When used in conjunction with other techniques like the Circle Method, this estimate serves as a foundation for many facts about $p(n,m).$

Off the top of my head here are three papers that use this kind of estimate: http://www.math.drexel.edu/~rboyer/papers/partitions_experimental.pdf http://plms.oxfordjournals.org/content/s2-36/1/117.full.pdf http://www.springerlink.com/content/d65348128235x7m1/

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Polylogarithms have interesting connections in the Theory of Partitions. This is mainly because a class of generating functions for bivariate partition statistics can be approximated by polylogarithms.

Consider for example the class of generating functions with $s>1,$

$\displaystyle \sum_{n=0}^\infty \sum_{m=0}^\infty p(n,m)x^n u^m =\prod_{k=1}^\infty \frac{1}{(1-xu^k)^{k^{s-2}}}.$

When $s=2,$ $p(n,m)$ is the number of partitions of $m$ with $n$ parts. When $s=3,$ $p(n,m)$ represents the number of plane partitions of $m$ with trace $n.$

For simplicity assume $1>x>0,$ as $u\to ^-1,$

$\displaystyle \prod_{k=1}^\infty \frac{1}{(1-xu^n)^{k^{s-2}}} frac{1}{(1-xu^k)^{k^{s-2}}} \thicksim\sqrt[ - \frac{1}{\zeta(2-s)}]{1-x}\exp(Li_{s}(x)/(\ln u^{-1})^{s}).$

When used in conjunction with other techniques like the Circle Method, this estimate serves as a foundation for many facts about $p(n,m).$

Off the top of my head here are three papers that use this kind of estimate: http://www.math.drexel.edu/~rboyer/papers/partitions_experimental.pdf http://plms.oxfordjournals.org/content/s2-36/1/117.full.pdf http://www.springerlink.com/content/d65348128235x7m1/

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