Chain sequence (continued fraction)
Encyclopedia
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 the equations
where either (a) 0 ≤ gn < 1, or (b) 0 < gn ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite
continued fractions.
The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem shows that
converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {an} are a chain sequence.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
of continued fractions
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 denominators can assume arbitrary real or complex values....
, 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 the equations
where either (a) 0 ≤ gn < 1, or (b) 0 < gn ≤ 1. Chain sequences arise in the study of the convergence problem – both in connection with the parabola theorem, and also as part of the theory of positive definite
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
continued fractions.
The infinite continued fraction of Worpitzky's theorem contains a chain sequence. A closely related theorem shows that
converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients {an} are a chain sequence.
An example
The sequence {¼, ¼, ¼, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = ½, it is clearly a chain sequence. This sequence has two important properties.- Since f(x) = x − x2 is a maximum when x = ½, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {gn} = {x}, and x < ½, the resulting sequence {an} will be an endless repetition of a real number y that is less than ¼.
- The choice gn = ½ is not the only set of generators for this particular chain sequence. Notice that setting
-
- generates the same unending sequence {¼, ¼, ¼, ...}.