Generalized Hyperbolic Functions
TL;DR: In this paper, generalized hyperbolic functions are used to define generalized generalized Hyperbolic Functions (GHF) and generalized generalized homophily functions (GHF) are defined.
Abstract: (1982). Generalized Hyperbolic Functions. The American Mathematical Monthly: Vol. 89, No. 9, pp. 688-691.
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23 Dec 2010
TL;DR: This book provides algorithms and ideas for computationalists on low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields.
Abstract: This book provides algorithms and ideas for computationalists. Subjects treated include low-level algorithms, bit wizardry, combinatorial generation, fast transforms like the Fourier transform, and fast arithmetic for both real numbers and finite fields. Various optimization techniques are described and the actual performance of many given implementations is examined. The focus is on material that does not usually appear in textbooks on algorithms. The implementations are done in C++ and the GP language, written for POSIX-compliant platforms such as the Linux and BSD operating systems.
70 citations
Cites background from "Generalized Hyperbolic Functions"
...which is a three power series analogue of the relation cosh(2)− sinh(2) = 1, see [336]....
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TL;DR: In this article, a probabilistic approach to define a Brownian motion of order n (n≳2) is presented, which is different from Hochberg's and Mandelbrot's.
Abstract: In this paper, a Brownian motion of order n (n≳2) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent R+1/n‐valued random variables (rv) (where R+1/n={z∈C; zn∈R+}). Many properties of the real standard Brownian motion are generalized at order n, but in the case n≳2, it is interesting to describe the Brownian motion of order n on the σ algebra ⊗[B(R+1/n)] R+ [where B(R+1/n) is the σ algebra generated by sets of type A(0,h)={z∈C; zn∈[0;hn[},(heR+*)]. This σ algebra is totally different from ⊗[B(R)] R+. Thus this study shows the fractal nature of the Brownian motion of order n, and given invariance scale (self‐similarity) properties. Then, a stochastic integral and an Ito–Taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average. All these results can be obtained via nonstandard analysis methods (infinitesimal time discretization). Finally, one rem...
21 citations
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TL;DR: The higher-order circular and hyperbolic functions deserve to be better known as discussed by the authors, in order to make them more accessible to teachers and students in calculus, linear algebra and differential equations courses.
Abstract: The higher-order circular and hyperbolic functions deserve to be better known. Here we give their main properties in order to make them more accessible to teachers and students in calculus, linear algebra and differential equations courses. The study of these functions can be related to such diverse topics as the binomial theorem and the fast Fourier transform. Here, for each positive integer n, we define r functions FJ,(x), r = 0,1 .I., n-1. The cases a = 1 and a = -1 correspond, respectively, to what are usually known as generalized hyperbolic functions and generalized circular or trigonometric functions. We find it useful to retain the parameter a; the case a = 0 also gives something interesting. The functions considered here are elemnentary and can be a rich source for student projects and investigations.
20 citations
Cites background from "Generalized Hyperbolic Functions"
..., in 1969 by Battioni [2], in 1978 by Ricci [26], and in 1982 by one of the present authors [35]....
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...The second-named author published a note [35] on these functions in 1982 before some of the references given here were drawn to his attention by the first-named author....
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TL;DR: The class of functions with integral difference ratios (i.e., functions with constant integral difference ratio) was introduced in this paper. But their representation is not a Newton series, and they are not uniformly close to any function having integral different ratios.
Abstract: Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a) - f(b) ≡ 0 (mod (a - b)) for all a > b. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance, all functions x ↦ ⌊e1/a ax x!⌋, with a ∈ ℤ\{0, 1}, and a function equal to ⌊e x!⌋ except on 0. Finally, to study the complement class, we look at functions ℕ → ℝ which are not uniformly close to any function having integral difference ratios.
15 citations
Posted Content•
TL;DR: This class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a) - f(b) ≡ 0 (mod (a - b)) for all a > b is characterized via their representations as Newton series.
Abstract: Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$.
We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto\lfloor e^{1/a}\;a^x\;x!\rfloor$, with $a\in\Z\setminus\{0,1\}$, and a function equal to $\lfloor e\;x!\rfloor$ except on 0. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.
13 citations