A semi-recursion for the number of involutions in special orthogonal groups over finite fields

Let I(n)I(n) be the number of involutions in a special orthogonal group SO(n,Fq)SO(n,Fq) defined over a finite field with q elements, where q   is the power of an odd prime. Then the numbers I(n)I(n) form a semi-recursion, in that for m>1m>1 we haveI(2m+3)=(q2m+2+1)I(2m+1)+q2m(q2m−1)I(2m−2).I(2m+3)=(q2m+2+1)I(2m+1)+q2m(q2m−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 FqFq.