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 quotients, if the sequence of denominators of its partial quotients is bounded; that is x = [ a 0 ; a 1 , a 2 , … ] = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 + ⋱ = a 0 + K ∞ i = 1 1 a i , {\displaystyle x=[a_{0};a_{1},a_{2},\dots ]=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{a_{4}+\ddots }}}}}}}}=a_{0}+{\underset {i=1}{\overset {\infty }{K}}}{\frac {1}{a_{i}}},\,} and there is some positive integer M such that all the (integral) partial denominators ai are less than or equal to M. == Periodic continued fractions == A regular periodic continued fraction consists of a finite initial block of partial denominators followed by a repeating block; if ζ = [ a 0 ; a 1 , a 2 , … , a k , a k + 1 , a k + 2 , … , a k + m ¯ ] , {\displaystyle \zeta =[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}],\,} then ζ is a quadratic irrational number, and its representation as a regular continued fraction is periodic. Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of a0 through ak+m.

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