Lüroth-type alternating series representations for real numbers

Metric properties of alternating Oppenheim expansions

“…Note also that for the alternating Lüroth expansions, H n (j) = 1 for all n and j, so V n = D n in this case. Therefore, the subsequent statements on the sequence V n reestablish the results of Kalpazidou et al [6] on the Lüroth case.…”

“…The fundamental inequality (9) is a necessary and sufficient condition for obtaining the digits D j at (7) by the expansion (5) and (6). In particular, there are numbers x for which equality holds at (9) infinitely many times.…”

“…In fact, only the special cases a n (j) = 1 and b n (j) = j(j +1) for all n ≥ 1 and j ≥ 1 (the Lüroth case; see Kalpazidou et al [6] and Indlekofer et al [5]), and a n (j) = 1 and b n (j) = j (alternating Engel series or Pierce expansions; see Shallit [9]). We now fill in this gap.…”

On some properties of the Lüroth‐type alternating series representations for real numbers

“…Similar further results were derived later by Kalpazidou, Knopfmacher, and Knopfmacher (see [4,5]), for the alternating Lüroth-type series.…”

Properties of distributions of random variables with independent differences of consecutive elements of the Ostrogradskii series

“…A generalization of the Ostrogradskiȋ algorithm for approximations in Banach spaces is also proposed in [9]. In the paper [10], an algorithm for sign alternating series expansion of a number is introduced leading, under a certain choice of the parameters, to the Lüroth series, Engel series, and Ostrogradskiȋ series (though the latter series are not studied in that paper). We also note that the algorithm of theQ ∞ -representation (see [11] or [12]) can give, under a certain choice of the setQ ∞ , the sign alternating series expansions of the Lüroth type.…”