scholarly journals Continued fraction expansions of the generating functions of Bernoulli and related numbers

2020 ◽  
Vol 31 (4) ◽  
pp. 695-713
Author(s):  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Continued Fraction Expansions

scholarly journals Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions

10.37236/1059 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Continued Fraction ◽  
Perfect Matchings ◽  
Symmetric Distribution ◽  
Set Partitions ◽  
Continued Fraction Expansions

We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes a recent result of Klazar and Noy in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions.

scholarly journals A q-CONTINUED FRACTION

2006 ◽  
Vol 02 (04) ◽  
pp. 523-547 ◽  
Author(s):  
D. BOWMAN ◽  
J. Mc LAUGHLIN ◽  
N. J. WYSHINSKI
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Continued Fraction Expansions ◽  
Ramanujan's Continued Fraction

We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q2; q3)∞/(q; q3)∞and [Formula: see text]. In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

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