Two asymptotic distributions related to Rényi-type continued fraction expansions
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DOI: 10.1007/s10998-021-00444-4
Abstract: We attempt to investigate a two-dimensional Gauss–Kuzmin theorem for Rényi-type continued fraction expansions. More precisely, our focus is to obtain specific lower and upper bounds for the error term considered which imply the convergence rate… read more here.
Keywords: nyi type; continued fraction; two asymptotic; fraction expansions ... See more keywords
New theta function identities for a continued fraction of Ramanujan and their applications
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DOI: 10.1007/s13370-020-00823-z
Abstract: We prove three new theta function identities for the continued fraction H ( q ) defined by $$\begin{aligned} H(q):=q^{1/8}-\frac{q^{7/8}}{1-q}_{+}\frac{q^2}{1+q^2}_{-}\frac{q^3}{1-q^3}_{+}\frac{q^4}{1+q^4}_{- \cdots }, \vert q\vert read more here.
Keywords: function identities; theta function; continued fraction; new theta ... See more keywords
Some quicker continued fraction approximations and inequalities towards Euler's constant
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DOI: 10.1016/j.jnt.2016.11.024
Abstract: Abstract In this paper, we define some new continued fraction sequences towards Euler's constant and two related inequalities. We also present some numerical simulations to demonstrate the superiority of the optimal new sequences over new… read more here.
Keywords: continued fraction; approximations inequalities; towards euler; euler constant ... See more keywords
Asymptotic expansions of the prime counting function
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DOI: 10.1016/j.jnt.2021.07.032
Abstract: We provide several asymptotic expansions of the prime counting function $\pi(x)$ and related functions. We define an {\it asymptotic continued fraction expansion} of a complex-valued function of a real or complex variable to be a… read more here.
Keywords: asymptotic expansions; continued fraction; prime counting; expansions prime ... See more keywords
Polynomial continued fractions for exp(π)
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DOI: 10.1080/10236198.2023.2204979
Abstract: We present two (inequivalent) polynomial continued fraction representations of the number eπ with all their elements in Q; no such representation was seemingly known before. More generally, a similar result for erπ is obtained for… read more here.
Keywords: continued fraction; fractions exp; polynomial continued; continued fractions ... See more keywords
ON THE LARGEST PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS
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DOI: 10.1142/s0218348x21500997
Abstract: Let [Formula: see text] be the continued fraction expansion of an irrational [Formula: see text]. For any [Formula: see text], write [Formula: see text]. This paper is concerned with the Hausdorff dimension of the set… read more here.
Keywords: continued fraction; largest partial; see text; formula see ... See more keywords
A Continued Fraction of Ramanujan and Some Ramanujan-Weber Class Invariants
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DOI: 10.2298/fil1713975a
Abstract: On Page 36 of his ``lost" notebook, Ramanujan recorded four $q$-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two $q$-series representations of Ramanujan's continued fraction found in his ``lost" notebook.… read more here.
Keywords: weber class; class invariants; continued fraction; ramanujan weber ... See more keywords
Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions
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DOI: 10.3390/axioms10040310
Abstract: The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and… read more here.
Keywords: function; continued fraction; branched continued; generalized hypergeometric ... See more keywords
A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms
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DOI: 10.3390/math9030255
Abstract: 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… read more here.
Keywords: continued fractions; type approach; lochs type; continued fraction ... See more keywords
Continued Fraction Expansions of Stable Discrete-Time Systems of Difference Equations
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DOI: 10.3390/sym14061226
Abstract: We provide a systematic procedure for generating the coefficients of the continued fraction expansion of the test function associated with the characteristic polynomial of a stable system of difference equations. We illustrate the feasibility of… read more here.
Keywords: stable discrete; difference equations; fraction expansions; continued fraction ... See more keywords