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Our interest is in the generating function  E N,β (I;ξ;w β )  for the probabilities  E N,β (n;I;w β )  that in a matrix ensemble with unitary ( β  = 2) or orthogonal ( β  = 1) symmetry, characterized by the weight  w β  (λ) and having  N  eigenvalues, the interval  I  contains exactly  n  eigenvalues. Using a determinant formula for  E   N ,2 , a general quadratic identity is obtained which relates  E   N ,2  in the case  I  and  w  2 ( x ) even to a product of generating functions  E   N ,2  with different  I ,  w  2 (λ) and  N , and for which the eigenvalues are positive. Also, generalizing some earlier calculations, the sum  E   N ,1  (2 n  − 1;  I ;  w  1 ) +  E   N ,1  (2 n ;  I ;  w  1 ) for  N  even,  I  = (− t ,  t ) and  w  1  an even classical weight is shown to equal  E   N /2,2 ( n ; (0,  t  2 );  w  2 ) for  w  2  related to  w  1 . Implications of these identities are discussed.

Evenness symmetry and inter-relationships between gap probabilities in random matrix theory