Banach function norms via Cauchy polynomials and applications

Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial [Formula: see text]. We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.

Note on a -Contour Integral Formula of Gasper-Rahman

We use the -Chu-Vandermonde formula and transformation technique to derive a more general -integral equation given by Gasper and Rahman, which involves the Cauchy polynomial. In addition, some applications of the general formula are presented in this paper.