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Chain sequence

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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

a 1 = ( 1 − g 0 ) g 1 a 2 = ( 1 − g 1 ) g 2 a n = ( 1 − g n − 1 ) g n {\displaystyle a_{1}=(1-g_{0})g_{1}\quad a_{2}=(1-g_{1})g_{2}\quad a_{n}=(1-g_{n-1})g_{n}}

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

f ( z ) = a 1 z 1 + a 2 z 1 + a 3 z 1 + a 4 z ⋱ {\displaystyle f(z)={\cfrac {a_{1}z}{1+{\cfrac {a_{2}z}{1+{\cfrac {a_{3}z}{1+{\cfrac {a_{4}z}{\ddots }}}}}}}}\,}

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

An example

The sequence {⁠1/4⁠, ⁠1/4⁠, ⁠1/4⁠, ...} appears as a limiting case in the statement of Worpitzky's theorem. Since this sequence is generated by setting g0 = g1 = g2 = ... = ⁠1/2⁠, it is clearly a chain sequence. This sequence has two important properties.

  • Since f(x) = x − x is a maximum when x = ⁠1/2⁠, this example is the "biggest" chain sequence that can be generated with a single generating element; or, more precisely, if {gn} = {x}, and x < ⁠1/2⁠, the resulting sequence {an} will be an endless repetition of a real number y that is less than ⁠1/4⁠.
  • The choice gn = ⁠1/2⁠ is not the only set of generators for this particular chain sequence. Notice that setting
g 0 = 0 g 1 = 1 4 g 2 = 1 3 g 3 = 3 8 … {\displaystyle g_{0}=0\quad g_{1}={\textstyle {\frac {1}{4}}}\quad g_{2}={\textstyle {\frac {1}{3}}}\quad g_{3}={\textstyle {\frac {3}{8}}}\;\dots }
generates the same unending sequence {⁠1/4⁠, ⁠1/4⁠, ⁠1/4⁠, ...}.

Notes

  1. Wall traces this result back to Oskar Perron (Wall, 1948, p. 48).

References

  • H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted by Chelsea Publishing Company, (1973), ISBN 0-8284-0207-8
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