MathOverflow

The $q$-binomial theorem states$$\frac{(az;q)_\infty}{(z;q)_\infty}=\sum_{n\geq 0}\frac{(a;q)_\infty}{(q;q)_\infty}z^n,$$where $(a;q)_n=\prod_{r=0}^{n-1}(1-aq^r)$ is the $q$-Pochhammer symbol.
Is there a closed-form equation for its reciprocal? That is, a formula for the coefficients $c_n$ in$$\frac{(z;q)_\infty}{(az;q)_\infty}=\sum_{n\geq 0}c_nz^n?$$...

Let $d\ge 2$ and let $\mathcal U=\{U_1,\dots,U_K\}\subset U(d)$ be a finite set of unitaries. For an integer $t\ge 1$ consider the $t$-th moment operator$$M_t := \frac{1}{K}\sum_{a=1}^K U_a^{\otimes t}\otimes \overline{U_a}^{\otimes t}\in \mathrm{End...

How does the Galois groups of Taylor polynomials of entire functions relate to the Weierstrass factorization?
Setup: Let $f$ be a non-polynomial entire function with Weierstrass factorization:$$f(z)=z^m e^{g(z)}\prod_{n=1}^\infty E_{p_n}\!\left(\frac{z}{a_n}\right)$$where $$E_k(u)=(1-u)\exp\!\left(\sum_{j=1}^k u^j/j\right).$$ Define $\hat f(z)=\frac{f(z)}{e...

It is well known that all nonzero commutative rings satisfy the strong rank condition (and in fact also the potentially stronger Orzech property; whether they are equivalent for rings is the subject of another question). Recently, Thomas Browning verified...