What are some existing notions of positivity (or nonnegativity) for $\mathbf{R}[q]$ such that all $q$-integers $[n]_q = 1+q+...+q^{n-1}$ are positive, as well as all polynomials that can be expressed in the form $[m_1]_q [m_2]_q \dots / [n_1]_q [n_2]_q \dots$, such as the algebraically natural $q$-Catalan numbers $[2n]_q / [n]_q [n+1]_q$?

Of course, one such property is that the polynomial takes on only nonnegative values when $q>0$, but I suspect that there are others.

Note however that the coefficients need not be nonnegative; e.g., the cyclotomic polynomial $1-q+q^2$ can be written as $[6]_q / [2]_q [3]_q$.

What are some existing notions of positivity (or nonnegativity) for $\mathbf{R}[q]$ such that all $q$-integers $[n]_q = 1+q+...+q^{n-1}$ are positive, as well as all polynomials that can be expressed in the form $[m_1]_q [m_2]_q \dots / [n_1]_q [n_2]_q \dots$, such as the algebraically natural $q$-Catalan numbers $[2n]_q / [n]_q [n+1]_q$?

Of course, one such property is that the polynomial takes on only nonnegative values when $q \geq 0$, but I suspect that there are others.

Note however that the coefficients need not be nonnegative; e.g., the cyclotomic polynomial $1-q+q^2$ can be written as $[6]_q / [2]_q [3]_q$.

What are some existing notions of positivity (or nonnegativity) for $\mathbf{R}[q]$ such that all $q$-integers $[n]_q = 1+q+...+q^{n-1}$ are positive, as well as all polynomials that can be expressed in the form $[m_1]_q [m_2]_q \dots / [n_1]_q [n_2]_q \dots$, such as the algebraically natural $q$-Catalan numbers $[2n]_q / [n]_q [n+1]_q$?

Of course, one such property is that the polynomial takes on only nonnegative values when $q>0$, but I suspect that there are others.

What are some existing notions of positivity (or nonnegativity) for $\mathbf{R}[q]$ such that all $q$-integers $[n]_q = 1+q+...+q^{n-1}$ are positive, as well as all polynomials that can be expressed in the form $[m_1]_q [m_2]_q \dots / [n_1]_q [n_2]_q \dots$, such as the algebraically natural $q$-Catalan numbers $[2n]_q / [n]_q [n+1]_q$?

Of course, one such property is that the polynomial takes on only nonnegative values when $q>0$, but I suspect that there are others.

Note however that the coefficients need not be nonnegative; e.g., the cyclotomic polynomial $1-q+q^2$ can be written as $[6]_q / [2]_q [3]_q$.