The distributional products of particular distributions

Let f be a C-infinity function on R and P be a quadratic form defined by P(x) = P(x(1), x(2),..., x(m)) = x(2) + ... + x(p)(2) - x(p+1)(2) - (. . .) - x(p+q)(2) with p + q = m. In this paper, we mainly show that f(P) .delta((k))(P) = [GRAPHICS] f((i))(0)delta((k-i))(P), where delta((k))(P) is given by (delta((k))(P), phi) = (-1)(k) integral(infinity)(0) [(partial derivative/2r partial derivative r)(k) {r(p-2) psi(r,s)/2}](r=s) s(q-1) ds. In particular, we have P-n . delta((k)) (P) = { [GRAPHICS] if k >= n, {0 if k < n, which solves a problem raised by Li in 2004. (c) 2006 Elsevier Inc. All rights reserved.