Dimension formula of solution spaces of A-hypergeometric differential-difference systems
Let A = (a ij) be a matrix in M (d, n, Z ≥0). We suppose that the set of the column vectors of A spans Z d. We denote by S i : f (s i) → f (s i − 1) the difference operator with respect to a variable s i. Definition 1. We call the following differential-difference system H A an A-hypergeometric differential-difference system:   n j=1 a ij x j ∂ j − s i   • f = 0, ∂ j − d i=1 S a ij i • f = 0. Definition 2. Let I be a left ideal of the ring of differential-difference operators D = C We define the rank of I by rank(I) = dim C(x,s) D/I, where D/I is a vector space over C(x, s). Theorem 1. The rank of H A agrees with the normalized volume of A. Let a i be the i-th column vector of the matrix A and F (β, x) the integral F (β, x) = C exp n i=1 x i t a i t −β−1 dt, t = (t 1 ,. .. , t d).
HYPERGEOMETRIC FUNCTIONS
Given the hypergeometric function F (a, b; c; z) = ∞ P n=0 (a)n(b)n (c)n(1)n zn, we place conditions on a, b and c to guarante that zpF (a, b; c; z) will be in various subclasses of p-valent starlike and p-valent convex functions. Operators related to the hypergeometric function are also examined.
Structure and orientation control of organopolysilanes and their application to electronic devices
Abstract Molecular structure and packing of various organopolysilanes having symmetrical side-chains of alkyl substituents, such as poly(di-methyl silane), poly(di-ethyl silane), poly(di-propyl silane), poly(di-butyl silane), poly(di-pentyl silane), and poly(di-hexyl silane), are examined, and the method of orientation control of the polysilane thin film is proposed. The application of the evaporated film to the electro-luminescent device is also demonstrated.