Weak and strong $q$-analogs of the Laguerre--P\'olya class

Abstract

For $0<q<1$ we compare two natural $q$-analogs of the Laguerre--P\'olya class. The first one is a coefficient-side class, defined as the inverse image of the classical Laguerre--P\'olya class under the normalized $q$-Borel transform \[ \Bq\left(\sum_{k\ge 0}a_k\frac{z^k}{k!}\right) =\sum_{k\ge 0}a_k\frac{q^{k(k-1)/2}(1-q)^k}{(q;q)_k}z^k . \] 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--P\'olya 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--P\'olya 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 \[ \qLPs\subsetneq \LP\subsetneq \qLPw, \qquad \qLPIs\subsetneq \LPI\subsetneq \qLPIw . \]