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.
Best approximation and moduli of smoothness: Computation and equivalence theorems
In this paper we investigate the trigonometric series with the @b-general monotone coefficients. First, we study the uniform convergence criterion. The estimates of best approximations and moduli of smoothness of the series in uniform metrics are obtained in terms of coefficients. These results imply several important relations between moduli of smoothness of different orders (in particular, Marchaud-type inequality) and best approximations.