Shifted genus expanded ${\mathcal{W}}_{\mathcal{\infty }}$ algebra and shifted Hurwtiz numbers

We construct the shifted genus expanded ${\mathcal{W}}_{\mathcal{\infty }}$ algebra, which is isomorphic to the central subalgebra ${\mathcal{A}}_{\mathcal{\infty }}$ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra ${\mathrm{\Lambda }}^{\ast }$ defined by A. Y. Okounkov and G. I. Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers, thus we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.