A theorem on Pfaffians

Abstract We prove the Pfaffian analog of the well-known diagonal expansion theorem for determinants. If P k is the sum of the Pfaffians of all the k -square principal triangular subarrays of a given N -square triangular array A then P (λ)=Σ 1 N P k λ k =Pfaffian [ A (λ)] with a ij (λ)= a ij −(−1) j−i λ 2 . Our proof is an application of Wick's theorem.