Denote by $ {n\brack {k}}$ a $q-$binomial coefficient.
Let ${D_{n,k}}(t,q) = \sum\limits_{j = 0}^{n - k} {{q^{{j^2} + kj}}}{n\brack {j}}{n\brack {k+j}}t^j $
and
${R_n}(x,t,q) = \sum\limits_{k = 0}^n {{{( - 1)}^k}{q^{\binom{k}{2}}{n\brack {k}}c(n,k,t)x^ {n-2k}}}$
with
$c(n,k,t)= \sum\limits_{j = 0}^k{k\brack {j}}q^{(n+1-k)j}{{n+j-k-1}\brack{j}}/{{n-1}\brack{j}} t^j$
for $k<n$ and
$c(n,n,t)=1+q^n t^n $.
Computations suggest that
$$ \sum\limits_{k = 0}^n {{D_{n,k}}(t,q){R_k}(x,t,q)} = {x^n}.$$
This implies that the linear functional $L$ defined by $L(R_n)=0$ for $n>0$ has the moments
$L(x^n)= \sum\limits_{j = 0}^{n } {{q^{j^2} }}{n\brack {j}}^2t^j $
which are $q-$Narayana polynomials of type B.
Note that $D_{n,k}(1,1)=\binom{2n}{n+k}$ and $R_n(x,1,1)=L_{2n}(\sqrt{x})$ where $L_n(x)$ is a Lucas polynomial. There are proofs for $q=1$ and for $t=1.$ But these proofs use properties which are lacking in the general case.
I tried to prove this by comparing coefficients or by matrix inversion, but did only succeed in special cases. Are there $q-$ hypergeometric identities for double sums or other methods which can be applied?
Edit In the mean-time I could prove this identity by reducing it to a corresponding identity for $q-$Narayana polynomials of type A.
But I think there must be also a direct computational proof by comparing coefficients. Could anyone provide such a proof?
