Weak and strong $q$-analogs of the Laguerre--Pólya class

For $0<q<1$ we compare two natural $q$-analogs of the Laguerre–Pólya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre–Pólya class under the normalized $q$-Borel transform $$\text{Bq}\left(\sum _{k\ge 0}{a}_k\frac{z^k}{k!}\right)=\sum _{k\ge 0}{a}_k\frac{q^{k(k-1)\mathrm{/}2}(1-q{)}^k}{(q;q{)}_k}{z}^k\mathrm{.}$$ The second one is a zero-side class, defined as the locally uniform closure of real polynomials whose nonzero zeros are logarithmically $q$-separated on each side of the origin. We prove that the normalized $q$-Borel transform maps the classical Laguerre–Pólya class, and its type-I subclass, into themselves. This yields a $q$-Jensen-polynomial criterion and shows that the coefficient-side class strictly contains the classical Laguerre–Pólya class. On the zero side, we prove a genus-zero product representation. The logarithmic separation condition prevents zeros escaping to infinity from producing a residual exponential factor; consequently no nonconstant exponential factor can occur. For every $q\in (0,1)$ we obtain the strict chains $$\text{qLPs}⊊\text{LP}⊊\text{qLPw},\text{qLPIs}⊊\text{LPI}⊊\text{qLPIw}\mathrm{.}$$