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In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…

We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…

We give an asymptotic formula for the total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.

In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of…

We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…

The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of…

We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle…

We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.

Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.

A palintiple is a natural number which is an integer multiple of its digit reversal. A previous paper partitions all palintiples into three distinct classes according to patterns in the carries and then determines all palintiples belonging…

We describe a database of 1779 continued fractions with polynomial coefficients, of which more than 1500 are new, both for interesting constants and for transcendental functions, and provide the database inside the \TeX\ source of the…

This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely…

It is well known that the arithmetic nature of Mills' prime-representing constant is uncertain: we do not know if Mills' constant is a rational or irrational number. In the case of other prime-representing constants, irrationality can be…

In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.

What is the higher-dimensional analog of a permutation? If we think of a permutation as given by a permutation matrix, then the following definition suggests itself: A d-dimensional permutation of order n is an [n]^(d+1) array of zeros and…

A permutation $\sigma$ of a multiset is called Stirling permutation if $\sigma(s)\ge \sigma(i)$ as soon as $\sigma(i)=\sigma(j)$ and $i<s<j.$ In our paper we study Stirling polynomials that arise in the generating function for descent…

A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of…

The coefficient of x^{-1} of a formal Laurent series f(x) is called the formal residue of f(x). Many combinatorial numbers can be represented by the formal residues of hypergeometric terms. With these representations and the extended…

A natural number N is said to be palindromic if its binary representation reads the same forwards and backwards. In this paper we study the quotients of two palindromic numbers and answer some basic questions about the resulting sets of…