  Given a split classical group of symplectic type and a split general linear
group over a local field $F$, we use Langlands-Shahidi method to construct
their Rankin-Selberg local $\gamma$-factors and prove the corresponding
analytic stability for generic representations. The idea generalizes the work
of J. Cogdell, F. Shahidi, T.-L. Tsai in 2017 and D. She in 2023 in the study
of asymptotic behaviors of partial Bessel functions. Different from the known
cases, suppose $P=MN$ is the maximal parabolic subgroup with Levi component
$M\simeq \mathrm{GL}_r\times\mathrm{Sp}_{2m}$ that defines the local factors,
the action of the maximal unipotent subgroup of $M$ on $N$ have non-trivial
stabilizers, and the space of integration for the corresponding local
coefficient is no longer isomorphic to a torus. We will separate its toric part
out in our cases and show that it plays the same role as the torus over which
the integral representing the local coefficient is taken in the known cases.
This is a new phenomenon with sufficient generality and we believe that it may
provide us with a possible direction towards a uniform proof of stability of
Langlands-Shahidi $\gamma$-factors in our future work.
