The distributional products of particular distributions

Abstract Let f be a C∞ function on R and P be a quadratic form defined by P ( x ) = P ( x 1 , x 2 , … , x m ) = x 1 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 ) · δ ( k ) ( P ) = ∑ i = 0 k k i f ( i ) ( 0 ) δ ( k - i ) ( P ) , where δ(k)(P) is given by ( δ ( k ) ( P ) , ϕ ) = ( - 1 ) k ∫ 0 ∞ ∂ 2 r ∂ r k r p - 2 ψ ( r , s ) 2 r = s s q - 1 d s . In particular, we have P n · δ ( k ) ( P ) = n ! k n δ ( k - n ) ( P ) if k ⩾ n , 0 if k n , which solves a problem raised by Li in 2004.