The principal numbers of K. Saito for the types AlAl, DlDl and ElEl

The principal number σ(Γ)σ(Γ) for a tree ΓΓ is introduced by K. Saito [Principal ΓΓ-cone for a tree ΓΓ, Adv. Math. (2007), to appear, (Available online 16 February 2007)] as the maximal number of chambers contained in components of the complement of the graphic arrangement attached to ΓΓ (see Section 1).The purpose of the present paper is to determine the principal numbers for the Coxeter–Dynkin graphs1 of types AlAl, DlDl and ElEl. We show that the generating series ∑l=1∞σ(Al)l!xl, ∑l=3∞σ(Dl)l!xl and ∑l=4∞σ(El)l!xl satisfy certain differential equations of the first order. By solving the equations, we obtain the following results:∑l=1∞σ(Al)l!xl=tanx2+π4-1,∑l=3∞σ(Dl)l!xl=2(x-1)tanx2+π4-x2+2,∑l=4∞σ(El)l!xl=12x2-2x+3tanx2+π4-3x3-x-3.