Alpha-Pfaffian, pfaffian point process and shifted Schur measure

For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the $\alpha$-determinant. It gives the pfaffian at $\alpha=-1$. We give some formulas for $\alpha$-pfaffians and study the positivity. Further we define point processes determined by the $\alpha$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure.