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Hyperbolic measures, moments and coefficients. Algebra on hyperbolic functions. (English) Zbl 1164.33002

The paper under review mainly studies integrals and expansions concerning hyperbolic functions \(\sinh\), \(\cosh\) and \(\tanh\). In the section 3, the author calculates the densities of some given probabilities containing above mentioned hyperbolic functions, and moreover, various integrals like \(J_p=\int_{-\infty}^{\infty}\frac{x^p}{\cosh(x)\cosh(x+t)}\,dx\) and \(K_p=\int_{-\infty}^{\infty}\frac{x^p}{\sinh(x)\cosh(x+t)}\,dx\), and also integrals containing the square of the hyperbolic cosine. In the section 4, the author considers a random walk problem with two distinct probability densities concerning hyperbolic cosine function, and in the next section, Fourier transforms of some hyperbolic functions.
More precisely, the author shows how the calculations of the Fourier transforms of \(\frac{1}{\cosh(x)^n}\), \((\frac{x}{\sinh(x)})^n\) and of \(\frac{x}{\sinh(x)}(\frac{x}{\tanh(x)})^n\) produce the same kind of polynomials. Section 6 contains expansions of \(\cosh(u)^t\), \((\frac{u}{\sinh(u)})^t\) and \((\frac{u}{\tanh(u)})^t\), and the last section gives more details of the coefficients of \(\sinh(u)^k\) when \(k\) is a positive integer.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C70 Other hypergeometric functions and integrals in several variables
60E10 Characteristic functions; other transforms
60G50 Sums of independent random variables; random walks
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