Chain sequence explained

In the analytic theory of continued fractions, a chain sequence is an infinite sequence of non-negative real numbers chained together with another sequence of non-negative real numbers by the equations

a _{1} = ( 1- g _{0)g} _{1} a _{2} = ( 1- g _{1)g} _{2} a _{n} = ( 1- g _{n-1} )g _{n}

where either (a) 0 ≤ g_n < 1, or (b) 0 < g_n ≤ 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^[1] shows that

f(z) = \cfrac{a _{1z}{1} + \cfrac{a _{2z}{1} + \cfrac{a _{3z}{1} + \cfrac{a _{4z}{\ddots}}}}}

converges uniformly on the closed unit disk |z| ≤ 1 if the coefficients are a chain sequence.

An example

The sequence appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g₀ = g₁ = g₂ = ... = , it is clearly a chain sequence. This sequence has two important properties.

• Since f(x) = x - x² is a maximum when x = , this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if  = , and x < , the resulting sequence will be an endless repetition of a real number y that is less than .
• The choice g_n =  is not the only set of generators for this particular chain sequence. Notice that setting

g _{0} = 0 g _{1} = { style (1)/(4) } \quad g_2 = \quad g_3 = \;\dots

generates the same unending sequence .

Notes

1. [Hubert Stanley Wall|Wall]

References

• H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973),