Generalized Ramanujan master theorem for powers of the cosecant

Let $H(\delta)=\{z\in\mathbb{C}\mid\operatorname{Re}(z)\ge -\delta\}$ for some $0<\delta<1$. Define polynomials by $P_1(x)=1$, $P_2(x)=x$, and

$$P_m(x)=(x^2+(m-2)^2\pi^2)P_{m-2}(x)$$

for $m>2$. Let $g(z)$ be analytic on $H(\delta)$, let $m$ be a positive integer, and let $s$ satisfy $0<\operatorname{Re}(s)<\delta$. Generalized Ramanujan master theorem conjecture. Under some suitable growth conditions on $g$,

$$\int_0^\infty x^{s-1}\sum_{n=0}^\infty (-1)^{mn}\left[P_m\left(\frac{d}{dz}+\log(x)\right)g(z)\right]_{z=n}x^n\,dx=\frac{(-1)^{m-1}(m-1)!\pi^m}{\sin^m(\pi s)}g(-s).$$

This conjecture extends the Ramanujan master theorem to the replacement of $\pi/\sin(\pi s)$ by its $m$-th power, where higher-order poles make the residue computation substantially more complicated. The polynomial sequence appears in work of H. Airault on Fourier transform computations related to hyperbolic measures, while the proposed identity is based on numerical evaluation of the first several cases; its validity under precise growth hypotheses remains open.