We study a singular system of ordinary differential equations, Lu ≡ {z^p+1DI_N−A(z)}u = 0, where z ∈ C, p ≥ 0, D = d/dz and A(z) is an N square matrix of holomorphic functions in a neighborhood of z = 0. We call a matrix of operators L ≡ z^p+1DI_N−A(z) := (p, A(z)) a system, for short.
In this paper we introduce a notion of T-expansion of a matrix function A(z), which gives a summation expression of A(z) different from the usual Taylor expansion. The idea comes from the result by L. R. Volevič [Vol], where he studied a general matrix of partial differential operators A(∂_x) and he presented a way of finding out a leading part from the matrix operators which we call Volevič's lemma (cf. Section 3).
By using the T-expansion of A(z), we obtain an algorithm of the reduction procedure of the system L into a decomposition by irreducible subsystems (cf. Theorem A_δ and (4.23) in Subsection 4.3). From this decomposition we can define the Newton polygon N(L) by taking the characteristic polynomial of each irreducible subsystem in Definitions 2.2 and 2.3. The importance of the Newton polygon N(L) will be shown by proving an index formula of the operator L on a formal Gevrey space ${\cal G}$^s (1 ≤ s ≤ ∞) in Theorem C^(∞), which is obtained from the vertical coordinate of an associated vertex of N(L). This is an extension of J.-P. Ramis's results [Ram1,2] for single operators. The index formula is proved by applying the index formula for general matrix of ordinary differential operators obtained in a joint paper with M. Yoshino [M-Y2]. Many other problems concerned with the study of the singular system L = (p, A(z)) are studied. For example, in Subsection 4.4 we give a structure of fundamental matrix solution of Lu = 0 in exact form. In other words, the reduction procedure into Hukuhara-Turrittin's canonical form is exactly shown. The reduction procedure seems to peel one piece of peel of an onion one piece.

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In this paper we offer criteria for property (A) and the oscillation of the third-order nonlinear functional differential equation with mixed arguments [a(t)[x′(t)]^γ]″ + q(t)f(x[τ(t)]) + p(t)h(x[σ(t)]) = 0, where ∫^∞ a^−1/γ(s)ds = ∞. We deduce properties of the studied equations by establishing new comparison theorems so that property (A) and the oscillation are resulted from the oscillation of a set of the first order equations.

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Three hypergeometric series F(a, b, c; x) with the same parameters (a, b, c) up to additive integers are linearly related over rational functions in x. This paper makes this linear relation explicit: the coefficients are given from sums of products of hypergeometric series.

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We consider stationary states of a string with one end fixed in air flows. Describing with a boundary value problem, we study how many stationary states there exist and if they are locally unique and stable. A bifurcation phenomenon of stationary states will be found.

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We obtain Morrey regularity of weak solutions to linear degenerate elliptic Dirichlet problems with A₂ weight on Hörmander's vector fields by using representation formula of weak solutions and properties of the Green function.

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In this paper we prove that, for asymptotically bounded holomorphic functions defined in a polysector in Cⁿ, the existence of a strong asymptotic expansion in Majima's sense following a single multidirection towards the vertex entails (global) asymptotic expansion in the whole polysector. Moreover, we specialize this result for Gevrey strong asymptotic expansions. This is a generalization of a result proved by A. Fruchard and C. Zhang for asymptotic expansions in one variable, but the proof, mainly in the Gevrey case, involves different techniques of a functional-analytic nature.

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