A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequence

Using variable exponents, we build a new class of rearrangement-invariant Banach function spaces, independent of the variable Lebesgue spaces, whose function norm is ρ(f ) = ess sup_{x∈(0,1)} ρ_{p(x)} (δ(x)f (·)), where ρ_{p(x)} denotes the norm of the Lebesgue space of exponent p(x) (assumed measurable and possibly infinite) and δ is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the grand Lebesgue spaces, and the EXP_α spaces (α > 0). We analyze the function norm and we prove a boundedness result for the Hardy–Littlewood maximal operator, via a Hardy type inequality

Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval (0 , 1) ⊂ R, which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the EXPα spaces (α> 0). In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let (Formula presented.), where w is a weight, 0 < δ(·) ≤ 1 ≤ p(·) < ∞, we show that if p+= ‖ p‖ ∞< + ∞, the inequality ǁMfǁp[·], δ[·], w ≤ cǁfǁp[·], δ[·], w holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class Ap+

Let X_1,…,X_k be quasinormed spaces with quasinorms | ⋅ |_j, j = 1,…,k, respectively. For any f = (f_1,⋯,f_k) ∈ X_1 ×⋯× X_k let ρ(f) be the unique non-negative root of the Cauchy polynomial pf(x)=x^k-\sum_{j=1}^k!f_j|_j^jx^{k-j} . We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X_1 ×⋯× X_k, 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 X_1,…,X_k 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.\ud \ud \ud Read More: http://www.worldscientific.com/doi/10.1142/S0129167X1550083

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