A semi-recursion for the number of involutions in special orthogonal groups over finite fields
Finite Fields and Their Applications
Let I(n) be the number of involutions in a special orthogonal group SO(n,F-q) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers 1(n) form a semi-recursion, in that for m > 1 we have I(2m + 3) = (q(2m+2) + 1)I(2m + 1) + q(2m)(q(2m) - 1)I(2m - 2). We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over F-q. (C) 2011 Elsevier Inc. All rights reserved.
Vinroot, C. Ryan and Jiang, Feiqi, A semi-recursion for the number of involutions in special orthogonal groups over finite fields (2011). Finite Fields and Their Applications, 17(6), 532-551.