Intriguing polynomials coming from a combinatorial physics problem

Some digging through Koekoek and Swarttouw's The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue reveals that $[k,n]_q$ is related to the Al-Salam-Carlitz I polynomials (see page 115 of Koekoek and Swarttouw). More precisely,

$$1-[k,n]_q={}_2\phi_1\left({{q^{-n},q}\atop{0}}; q, q^{k+1}\right)=(-1)^n q^{\frac{n}2(2k-n+3)}U_n^{(q^{-k-1})}\left(\frac1{q};q\right)$$

In particular, letting $S(n,k;q)=1-[k,n]_q$, there is the three-term recurrence

$$S(n+1,k;q)=(1+q^{k+1}-q^{k-n})S(n,k;q)-q^{k+1}(1-q^{-n})S(n-1,k;q)$$

with the initial conditions $S(0,k;q)=1,\quad S(1,k;q)=q^{k+1}-q^k+1$.

From the relation with the Al-Salam-Carlitz II polynomials (see page 116 of Koekoek and Swarttouw), we have another basic hypergeometric expression:

$$1-[k,n]_q={}_2\phi_0\left({{q^n,\frac1{q}}\atop{-}}; \frac1{q}, q^{k-n+1}\right)$$

(Mathematica note: unfortunately Mathematica doesn't have support for ${}_2\phi_0$... yet.)

One can also derive a "reversal" identity by reversing the summation order:

$$\begin{align*}1-[k,n]_q&=q^{n(k+1)}(q^{-n};q)_n\; {}_1 \phi_1\left({{q^{-n}}\atop{q^{-n}}};q,q^{-k}\right)\\&=(-1)^n q^{\frac{n}{2}(2k-n+1)}(q;q)_n\; {}_1 \phi_1\left({{q^{-n}}\atop{q^{-n}}};q,q^{-k}\right)\end{align*}$$

I'll update this post if I manage to dig up more information...

Hope this does not goes against the rules here, but I wanted to post a more permanent summary of the brainstorming that took place in the comments to the question.

• J.M. suggested writing

$$[k,n]_q = -\sum_{m=1}^n \prod_{\ell=0}^{m-1} \left(q^{k+1}-q^{k+\ell-n+1}\right)$$

and

$$[k,n]_q = 1-{}_2\phi_1\left({{q^{-n},q}\atop{0}}; q, q^{k+1}\right)$$

• anon noticed that point (4.) implies point (1.)

Not a full answer, but there's a recurrence $$[k,1]_q = q^{k}-q^{k+1}$$ $$[k,n+1]_q = \left(q^{k+1}-q^{k-n}\right) \left([k,n]_q -1\right)$$

One derivation is $$[k,n+1]_q = -\sum_{m=1}^{n+1} \prod_{\ell=0}^{m-1} \left(q^{k+1}-q^{k+\ell-n}\right)$$ $$ = -\left(q^{k+1}-q^{k-n}\right) \sum_{m=1}^{n+1} \prod_{\ell=1}^{m-1} \left(q^{k+1}-q^{k+\ell-n}\right)$$ Subst. $\ell^\prime = \ell - 1, \; m^\prime = m-1$ $$ = -\left(q^{k+1}-q^{k-n}\right) \sum_{m^\prime=0}^{n} \prod_{\ell^\prime=0}^{m^\prime-1} \left(q^{k+1}-q^{k+\ell^\prime-n+1}\right)$$ $$ = \left(q^{k+1}-q^{k-n}\right) \left(-1-\sum_{m^\prime=1}^{n} \prod_{\ell^\prime=0}^{m^\prime-1} \left(q^{k+1}-q^{k+\ell^\prime-n+1}\right)\right)$$ $$ = \left(q^{k+1}-q^{k-n}\right) \left([k,n]_q -1\right) $$