Related papers: Continuants and convergence of certain continued f…

This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and…

In this note, we describe an interpretation of the (continuous) Fourier transform from the perspective of the Chinese Remainder Theorem. Some related issues, including a new derivation of Poisson summation formula, are discussed.

This is a brief historical note about famous Legendre's criterium for convergent of continued fraction expansion. The paper is written in Russian.

We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we…

We give an elementary geometric proof using Ford circles that the convergents of the continued fraction expansion of a real number $\alpha$ coincide with the rationals that are best approximations of the second kind of $\alpha$.

There are infinite processes (matrix products, continued fractions, $(r,s)$-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a…

The Hankel transform of an integer sequence is a much studied and much applied mathematical operation. In this note, we extend the notion in a natural way to sequences of $d$ integer sequences. We explore links to generalized continued…

We present a general introduction to continued fractions, with special consideration to the function fields case. These notes were prepared for a summer class given this year in Beijing at Beihang university.

Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of…

We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…

We discuss the classical, and forgotten, notion of perpetuants. We give a proof of the Theorem of Stroh computing their dimensions, and exhibit a basis of perpetuants, thus closing an old line of investigation.

The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate…

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for…

We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…

We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial…

Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then…

We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…

The direct calculation of the Generalized operator entropy proves difficult by the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient…

For $\alpha_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction and $\sigma$ a linear fractional transformation, we study the continued fraction expansion of $\sigma(\alpha_0)$ and its convergents. We provide the continued…

Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than…