Use this tag for questions pretaining to q-analogs of functions, for example q-Binomials, $q$-derivatives, the q-theta function, the q-Pochammer symbol, etc.

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4
votes
2answers
279 views

Recurrence for $q$-analog for the Stirling numbers?

I read in some papers that the Stirling numbers (of the second kind) have a natural $q$-analog $S_q(n,k)$, which satisfy the recurrence $$ S_q(n,k)=(k)_qS_q(n-1,k)+q^{k-1}S_q(n-1,k-1) $$ with the ...
13
votes
4answers
494 views

Intriguing polynomials coming from a combinatorial physics problem

For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ ...
4
votes
1answer
212 views

Reciprocity Law of the Gaussian (or $q$-Binomial) Coefficient

It is a standard exercise in combinatorics to show that the binomial coefficient satisfies the reciprocity law $\binom{-n}{k} = (-1)^k \binom{n+k-1}{k}$ for $n, k \geqslant 0$, which is the multiset ...
1
vote
2answers
338 views

Representing the $q$-binomial coefficient as a polynomial with coefficients in $\mathbb{Q}(q)$?

Trying a bit of combinatorics this winter break, and I don't understand a certain claim. The claim is that for each $k$ there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are ...
7
votes
2answers
255 views

q-Analogue of the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$.

The stirling numbers of the second kind satisfy the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$, where $(x)_k$ is the falling factorial. Consider the $q$-analog recursive definition of the ...
6
votes
1answer
253 views

Intermediate step in deducing Jacobi's triple product identity.

An intermediate step deduces Jacobi's triple product identity by taking the $q$-binomial theorem $$ \prod_{i=1}^{m-1}(1+xq^i)=\sum_{j=0}^m\binom{m}{j}_q q^{\binom{j}{2}}x^j $$ and deducing $$ ...
1
vote
1answer
124 views

Interpolation of a sequence of polynomials (viewed in terms of q-analogue powers)

I'm trying to find closed-form expressions for a sequence of coefficients, such that the index of the coefficient occurs as number such that I can later interpolate to fractional indexes as well. ...