Category: Continued fractions

Śleszyński–Pringsheim theorem
In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.
Khinchin's constant
In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of
Markov constant
In number theory, specifically in Diophantine approximation theory, the Markov constant of an irrational number is the factor for which Dirichlet's approximation theorem can be improved for .
Stieltjes transformation
In mathematics, the Stieltjes transformation Sρ(z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside I by the formula Under certain conditions we
Convergence problem
In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the
Lochs's theorem
In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by . The theorem states th
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing t
Complete quotient
In the metrical theory of regular continued fractions, the kth complete quotient ζ k is obtained by ignoring the first k partial denominators ai. For example, if a regular continued fraction is given
Solving quadratic equations with continued fractions
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is where a ≠ 0. The quadratic equation on a number can be solved using the well-known quadratic for
Periodic continued fraction
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,
List of mathematical constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it
Minkowski's question-mark function
In mathematics, the Minkowski question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rationa
Rogers–Ramanujan continued fraction
The Rogers–Ramanujan continued fraction is a continued fraction discovered by and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated expl
Engel expansion
The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that For instance, Euler's constant e has the Engel expansion 1, 1, 2, 3, 4, 5, 6, 7, 8,
Hermite's problem
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventuall
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denomin
Padé table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants Rm, n to a given complex formal power series. Certain sequences of approximants lying with
Gauss–Kuzmin–Wirsing operator
In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss ma
Continuant (mathematics)
In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
Gauss–Kuzmin distribution
In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a rand
Chain sequence
In the analytic theory of continued fractions, a chain sequence is an infinite sequence {an} of non-negative real numbers chained together with another sequence {gn} of non-negative real numbers by th
Incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective name
Stern–Brocot tree
In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the
Lévy's constant
In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions.In 193
Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees wi
Pell's equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian
Euler's continued fraction formula
In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published
Gauss's continued fraction
In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathem
Restricted partial quotients
In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction x is said to be restricted, or composed of restricted partial quotie