I want to find a function $f(s,x)$ such that
$f(s,x)$ is analytic
for any $s \in Z^+ $,
$f(s,x)=B_s(x)$, where $B_s(x)$ are the Bernoulli polynomials$f(a, x)$ is elementary against $x$ at any constant $a$
If possible, $f(s,b)$ is elementary against $s$ at any constant $b$
The last condition is not mandatory.