Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
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$\begingroup$ Since the coefficient of $u^n$ is a polynomial in $q$ of degree, roughly, $n^2$, there should be no closed form expression that is a rational function. Since the coefficients of $u^n$ satisfy a linear recurrence, the $q$-degrees would have to grow linearly in $n$. Maybe it could have a closed expression that is a rational function in $u$, but the coefficients are not all polynomials in $q$. $\endgroup$– Jason StarrJul 3, 2014 at 12:34
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$\begingroup$ Didn't you rather mean $q^{-n(n+1)/2}$ ? $\endgroup$– LucianJul 3, 2014 at 15:30
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$\begingroup$ @Lucian: Of course I have no idea what the OP meant, but I thought that I would mention that my comment is still valid if the factor is $q^{-n(n+1)/2}$. After a change of variables $s=1/q$, my comment shows that the expression cannot be a rational function in $s$ and $u$. But every rational function in $q$ and $u$ is also a rational function in $s$ and $u$. $\endgroup$– Jason StarrJul 3, 2014 at 15:57
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1$\begingroup$ What does this have to do with lie groups and lie algebras? $\endgroup$– Richard RastJul 3, 2014 at 19:49
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1$\begingroup$ @RichardRast it comes out to be generating function of multiplicities of Demazure modules in a Demazure flag of weyl module. $\endgroup$– Rekha BiswalJul 4, 2014 at 4:56
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