On a Family of Sequences Related to Chebyshev Polynomials

We consider the appearance of primes in a family of linear recurrence sequences labelled by a positive integer n. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in n) dilated versions of the so-called Chebyshev polynomials of the fourth kind, also known as airfoil polynomials. We prove that when the value of n is given by a dilated Chebyshev polynomial of the first kind evaluated at a suitable integer, either the sequence contains a single prime, or no term is prime. For all other values of n, we conjecture that the sequence contains infinitely many primes, whose distribution has analogous properties to the distribution of Mersenne primes among the Mersenne numbers. Similar results are obtained for the sequences associated with negative integers n, which correspond to Chebyshev polynomials of the third kind, and to another family of Lehmer numbers

A trick around Fibonacci, Lucas and Chebyshev

In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of this trick to more general sequences. This study leads us to interesting connections between Fibonacci, Lucas sequences and Chebyshev polynomials.Comment: 23 page

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson--Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets