Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture

In this paper we pursue the refined global Gross-Prasad conjecture for Bessel periods formulated by Yifeng Liu in the case of special Bessel periods for $\mathrm{SO}\left(2n+1\right)\times\mathrm{SO}\left(2\right)$. Recall that a Bessel period for $\mathrm{SO}\left(2n+1\right)\times\mathrm{SO}\left(2\right)$ is called special when the representation of $\mathrm{SO}\left(2\right)$ is trivial. Let $\pi$ be an irreducible cuspidal tempered automorphic representation of a special orthogonal group of an odd dimensional quadratic space over a totally real number field $F$ whose local component $\pi_v$ at any archimedean place $v$ of $F$ is a discrete series representation. Let $E$ be a quadratic extension of $F$ and suppose that the special Bessel period corresponding to $E$ does not vanish identically on $\pi$. Then we prove the Ichino-Ikeda type explicit formula conjectured by Liu for the central value $L\left(1/2,\pi\right)L\left(1/2,\pi\times\chi_E\right)$, where $\chi_E$ denotes the quadratic character corresponding to $E$. Our result yields a proof of Boecherer's conecture on holomorphic Siegel cusp forms of degree two which are Hecke eigenforms.