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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 (from any field) with the same RHS (always a finite product) as in above, together with resources or references?

Thank you.

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.

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 (any help?) 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 (from any field) with the same RHS (always finite product) as in above, together with resources or references?

Thank you.

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 (from any field) with the same RHS (always a finite product) as in above, together with resources or references?

Thank you.

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 $\pmb{q}=(q_1,\dots,q_n)$: $$e_0(\pmb{q})+e_1(\pmb{q})+\cdots+e_n(\pmb{q})=\prod_{j=1}^n(1+q^j).$$

(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 (any help?) 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 (from any field) with the same RHS (always finite product) as in above, together with resources or references?

Thank you.

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 (any help?) 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 (from any field) with the same RHS (always finite product) as in above, together with resources or references?

Thank you.