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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.

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