We continue the investigation of the product whose argument has been shown, in [2], to be related to the cubic Gauss sum.

Let Ω be a smoothly bounded pseudoconvex domain in ℂ2 and let bΩ be of finite type m. Then we prove the stability of Hölder estimates for under some perturbations of bΩ. As an application, we prove the Mergelyan property with respect to () norms for 0 ≤ α < 1/m.

The content of Part I is nothing else than, the theory of binomial polynomial sequences in infinite variables (u(1), u(2), u(3), …) with weight u(1) = l. However, sometimes we are concerned with specialization therefore, we call the elements in K[u(1), u(2), u(3), …] differential polynomials. As analogies of special polynomials with binomial property, we may construct special differential polynomials with binomial property.

In Part II, we shall be concerned with applications of classical invariant theory, to statistic physics and to theta functions. Main theorem in Chapter 2 is stated as follows:

For a partition function

satisfying γl ≥ 0 (l ≥ 1) and α > 0, the 2n-apolar of ξ(s)

has the expansion

such that βn,1 ≥ 0 (l ≥ 2). This means, for a given partition function ξ(s) with nonnegative relative probabilities, we construct a sequence of partition functions A2n (ξ(s), ξ(s))n≥1 with the same properties, which may be considered a sequence of symbolical higher derivative of ξ(s). The main theorem in Chapter 3 is stated as follows: For given theta functions φ1(z) and φ2(z) of level n1 and n2 respectively, in g variables z = (z1, z2,…, zg), then r = (r1, r2,…, rg-apolar

is a theta function of level n1 + n2, and

The imprimitive unitary reflection group G(m, p, n) acts on the vector space V =Cn naturally. The symmetric group Sk acts on ⊗kV by permuting the tensor product factors. We show that the algebra of all matrices on ⊗kV commuting with G(m, p, n) is generated by Sk and three other elements. This is a generalization of Jones’s results for the symmetric group case [J].

Let Ψ ∈ C2[0,1] be a positive real valued function on (0, 1]. Under certain assumptions on Ψ, the set is a pseudoconvex domain with C2-boundary which may be infinite type. If Ψ has flatness at 0 so that then we can obtain Lp(1 ≤ p ≤ ∞) estimates for on D.

Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hörmander as follows; for any Fourier hyperfunction u, (x0, ξ0) does not belong to the quasianalytic singular spectrum W FM(u) if and only if there exist positive constants C, γ and N, and a neighborhood of x0 and a conic neighborhood Г of ξ0 such that

for all x ∈ U, |ξ| ∈ Γ and |ξ| ≥ N, where M(t) is the associated function of the defining sequence Mp. This result simplifies Hörmander’s definition and unify the singular spectra for the C∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.

A rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).