Determinants through the Looking Glass

Using a recurrence derived from Dodgson's Condensation Method, we provide numerous explicit evaluations of determinants. They were all conjectured, and then rigorously proved, by computer-assisted methods, that should be amenable to full automation. We also mention a rst step towards that goal, our Maple package, DODGSON, that automates the special case of Hankel and Toeplitz hypergeometric determinants. AMS Subject Classi cation: Primary 05A, 15A This article is motivated by the computation in [1] that was inspired by the short proof [6] of MacMahon's determinant evaluation [4], using a determinantal identity of Charles Dodgson [2]. Many special cases of the sampled determinants given below were independently discovered by M. Petkov sek [5]. For an excellent and detailed survey of existing methods of proofs of determinant identities, see [3]. For any n by n matrix A, let Ar(i; j) denote the r by r minor consisting of r contiguous rows and columns of A, starting with row i and column j. In particular, An(1; 1) = detA. Then, according to Dodgson [2], (Lewis) An(1; 1)An 2(2; 2) = An 1(1; 1)An 1(2; 2) An 1(2; 1)An 1(1; 2): For many cases, An(i; j) turn out, conjectured at rst, to have an explicit expression, involving single and double products. Whenever this is the case the proof of the conjectured evaluation is completely routine, by induction on n, by checking that (Lewis) is satis ed by that conjectured expression, and by checking the trivial initial conditions for n = 0 and n = 1. Finally, to get an explicit expression for the original determinant, all one has to do is plug in i = 1 and j = 1.