From time to time, I run into the finite product $\prod_{j=1}^n(1+q^j)$. And, the more it happens, the more fascinated I've become. So, herein, I wish to get help in collecting such results. To give some perspective into what I look for, check out the below examples. First, some nomenclature: $(q)_k=(1-q)(1-q^2)\cdots(1-q^k)$ and $\binom{n}k_q=\frac{(q)_n}{(q)_k(q)_{n-k}}$.
(0) It's almost silly, but the sum of elementary functions of the specialization $\pmb{q}=(q,q^2,\dots,q^n)$:
$$e_0(\pmb{q})+e_1(\pmb{q})+\cdots+e_n(\pmb{q})=\prod_{j=1}^n(1+q^j).$$
(1) The classical $q$-binomial theorem, which results from counting restricted distinct partitions or number of weighted tilings:
$$\sum_{k=0}^nq^{\binom{k+1}2}\binom{n}k_q=\prod_{j=1}^n(1+q^j).$$
(2) I can't remember where I saw this (do you?) but
$$\sum_{k=0}^nq^k\binom{n}k_{q^2}=\prod_{j=1}^n(1+q^j).$$
(3) The $H$-polynomial of a symplectic monoid $MSp_n$ (see this paper, page 13):
$$\sum_{k=0}^n(-1)^kq^{k^2}\binom{n}k_{q^2}^2\prod_{i=1}^k(1-q^{2i})\prod_{j=1}^{n-k}(1+q^j)^2=\prod_{j=1}^{2n}(1+q^j),$$
although the authors did not seem to be aware of the RHS.
QUESTION. Can you provide such formulas (in any field) with the same RHS (always a finite product) as in above, together with resources or references?
Thank you.
Best Answer
Up to scaling, $\prod_{j=1}^n(1+q^j)$ is the character of the principal specialization of the spinor representation of $\mathfrak{so}(2n+1)$. This was first explicitly stated by J. W. B. Hughes, Lie algebraic proofs of some theorems on partitions, in Number Theory and Algebra, Academic Press, 1977, pp. 135--155.