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Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j}$$ and the Lucas polynomials $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \... 0answers 425 views A q-analogue of Foulkes' character related to alternating permutations My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is ... 0answers 154 views Some q-analogues of  \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}. Let {\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} ) and let  {{n}\brack{k}}_q denote a q-binomial coefficient. I am interested in q-analogues of the identity  \sum\... 0answers 131 views What is known about the q-analogue of the simplex? I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ... 0answers 216 views What is the q-analog of \Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}? I would expect the q-Gamma function to have the property which would be the q-analog of the Euler reflection formula from my question title. More concretely: \Gamma(z) has simple poles at ... 0answers 55 views How to prove that \sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}=\frac{(ab;q)_n}{(q;q)_n}? By Cauchy identity,$${}_1\phi_0(a;—;q,z)=\sum_{n\geq0}\frac{(a;q)_n}{(q;q)_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_\infty},\quad|z|<1,|q|<1,$$we can obtain the q-analogue of (1-z)^{-a}(1-z)^{-b}=... 0answers 23 views How to prove {}_2\phi_1(q^{-n},b;c;q,q)=\frac{(c/b;q)_n}{(c;q)_n}b^n. Firstly, we have already known the one of q-analogues of Vandermonde's formula, which is$${}_2\phi_1(q^{-n},b;c;q,cq^n/b)=\frac{(c/b;q)_n}{(c;q)_n}. And there is a hint, when we change the order ...
It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over \$...