The distributional products of particular distributions

Let f be a C∞ function on R and P be a quadratic form defined by P(x)=P(x1,x2,…,xm)=x12+⋯+xp2-xp+12-⋯-xp+q2 with p + q = m. In this paper, we mainly show thatf(P)·δ(k)(P)=∑i=0kkif(i)(0)δ(k-i)(P),where δ(k)(P) is given by(δ(k)(P),ϕ)=(-1)k∫0∞∂2r∂rkrp-2ψ(r,s)2r=ssq-1ds.In particular, we havePn·δ(k)(P)=n!knδ(k-n)(P)ifk⩾n,0ifk