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Generalized golden ratios of ternary alphabets

Vilmos Komornik, Anna Chiara Lai, Marco Pedicini (2011)

Journal of the European Mathematical Society

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence...

Generalized Schröder matrices arising from enumeration of lattice paths

Lin Yang, Sheng-Liang Yang, Tian-Xiao He (2020)

Czechoslovak Mathematical Journal

We introduce a new family of generalized Schröder matrices from the Riordan arrays which are obtained by counting of the weighted lattice paths with steps E = ( 1 , 0 ) , D = ( 1 , 1 ) , N = ( 0 , 1 ) , and D ' = ( 1 , 2 ) and not going above the line y = x . We also consider the half of the generalized Delannoy matrix which is derived from the enumeration of these lattice paths with no restrictions. Correlations between these matrices are considered. By way of illustration, we give several examples of Riordan arrays of combinatorial interest. In addition,...

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let A m be the mth coefficient of the square f(x)² of a unimodular...

Identities arising from higher-order Daehee polynomial bases

Dae San Kim, Taekyun Kim (2015)

Open Mathematics

Here we will derive formulas for expressing any polynomial as linear combinations of two kinds of higherorder Daehee polynomial basis. Then we will apply these formulas to certain polynomials in order to get new and interesting identities involving higher-order Daehee polynomials of the first kind and of the second kind.

Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Luca Q. Zamboni (2000)

Annales de l'institut Fourier

In a 1982 paper Rauzy showed that the subshift ( X , T ) generated by the morphism 1 ↦ 12 , 2 ↦ 13 and 3 ↦ 1 is a natural coding of a rotation on the two-dimensional torus 𝕋 2 , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in ℝ 2 , each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity 2 n + 1 satisfying a combinatorial criterion known as the ☆ condition of Arnoux and Rauzy codes the orbit of a point...

Integrals of logarithmic and hypergeometric functions

Anthony Sofo (2016)

Communications in Mathematics

Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.

Jumping sequences.

Butler, Steve, Graham, Ron, Zang, Nan (2008)

Journal of Integer Sequences [electronic only]

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