Alpha-Pfaffian, pfaffian point process and shifted Schur measure

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