Real Tate integrals of Gaussian times polynomial: Γ_ℝ-factorisation

Fix a natural number $n$ (a degree bound) and $\delta \in \mathbb{N}$ with $\delta \le 1$. The assertion is the existence of a single function $E$ of four arguments, a coefficient vector $c \in \mathbb{C}^{n+1}$, two reals $A, B$ and a complex $z$, with the following four properties. First, for every $c$ and all real $A, B$ (no positivity required), $z \mapsto E(c,A,B,z)$ is differentiable on all of $\mathbb{C}$, i.e. entire. Second, for every $c$, all real $A, B$ with $A > 0$, and every $z$ with $\operatorname{Re} z > 0$, the integral over $\rho \in \mathbb{R}$ of $\bigl(\sum_{j=0}^{n} c_j \rho^{j}\bigr) e^{-\pi (A\rho^{2} + 2B\rho)} \operatorname{sign}(\rho)^{\delta} |\rho|^{z-1}$ (the powers of $|\rho|$ being complex powers) equals $\Gamma_{\mathbb R}(z+\delta)\, E(c,A,B,z)$, where $\Gamma_{\mathbb R}(w) = \pi^{-w/2}\Gamma(w/2)$. Third, for every vertical strip, given by reals $\sigma_1, \sigma_2$, there are constants $C, M \in \mathbb{R}$ and $N \in \mathbb{N}$, depending only on the strip (and on $n$, $\delta$), such that for all $c$, all $A > 0$, all real $B$ and all $z$ with $\sigma_1 \le \operatorname{Re} z \le \sigma_2$,

Fourth, for each fixed $z$, the map $(c,A,B) \mapsto E(c,A,B,z)$ is continuous on the set where $A > 0$.

This is the archimedean (real-place) local computation behind Tate-style integrals: the Mellin transform of a polynomial times a Gaussian with a linear term factors as the real Gamma factor $\Gamma_{\mathbb R}(z+\delta)$ times an entire function whose growth in vertical strips and dependence on the Gaussian data are controlled uniformly. It is used in the cubic-induction step of the Langlands–Tunnell argument, where the analogous unfolding integral is shown to be $\Gamma_{\mathbb R}$ times a differentiable function; the estimate rests on the two-sided bounds for $\|\Gamma\|$ in vertical strips.