A recurrence formula for leaping convergents of non-regular continued fractions

Abstract

Given a continued fraction [ a 0 ; a 1 , a 2 , … ] , p n / q n = [ a 0 ; a 1 , … , a n ] is called the n-th convergent for n = 0 , 1 , 2 , … . Leaping convergents are those of every r-th convergent p rn + i / q rn + i ( n = 0 , 1 , 2 , … ) for fixed integers r and i with r ⩾ 2 and i = 0 , 1 , … , r - 1 . This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p 3 n + 1 / q 3 n + 1 for the continued fraction of e = [ 2 ; 1 , 2 k , 1 ¯ ] k = 1 ∞ and obtained some arithmetic properties. Komatsu studied those p 3 n / q 3 n for e 1 / s = [ 1 ; s ( 2 k - 1 ) - 1 , 1 , 1 ¯ ] k = 1 ∞ ( s ⩾ 2 ). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ ( 3 ) is shown.