# Tagged Questions

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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### q-Shifted factorials

The q-shifted factorial is defined as $(a;q)_n := (1-a)(1-aq)\ldots(1-aq^{n-1})$. It is supposed to be an analog of the Pochhammer symbol, or falling factorial: $x(x-1)\ldots(x-n+1)$. But the formulas ...
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### $q$-digamma function evaluation

What is the value of $\psi_2^{(0)}(1)$, where $\psi_q^{(0)}(z)$ is the $q$-digamma function? My attempt: \begin{align*} \psi_2^{(0)}(z) &=\frac{1}{\Gamma_2(z)}\frac{d\Gamma_2(z)}{dz} \\&=\...
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### Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
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### 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 ...
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### Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?

In this question, user begins shows that, for each $k\in \mathbb{N}$, there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Primitive roots as roots of a $q$-multinomial.

If $n$ is divisible by $m$, why is it the case that the $m$th primitive roots of unity are also roots of $\binom{n}{k}_q$ if and only if $m$ does not divide $k$? I'm viewing $\binom{n}{k}_q$ as a ...
### Deriving Cauchy's identity from the $q$-binomial theorem?
Cauchy's identity states that $$\prod_{i\geq 0}\frac{1-axq^i}{1-xq^i}=\sum_{n\geq 0}\frac{(1-a)(1-aq)\cdots(1-aq^{n-1})}{(1-q)(1-q^2)\cdots(1-q^n)}x^n.$$ Is it possible to somehow derive this ...