The study of probability distributions of the hyperbolic cosine type

Probability distribution, obtained earlier by the author in a result of the characterization of distributions by the property of the constant regression of quadratic statistics on a linear form and called the hyperbolic cosine type distribution, is considered. Along with the classic normal distribution, gamma distribution, negative binomial and some other distributions, the given three-parameter distribution is related to the class of probability distributions, united by a common characterization condition. The obtained distribution is a generalization of the famous two-parameter Meixner distribution.
In the article, a proof that a function appeared in the result of given characterization is indeed characteristic is given. The structure of this function, in connection with gamma distribution and the corresponding distribution conjugated with the gamma distribution, along with the constant distribution, is presented. Infinite decomposability of distribution is proved.
Based on characteristic function, the probability density function, expressed through the beta function of complex conjugate arguments is recovered in general terms. Along with a unified formula, correlations in elementary functions are deduced for the distribution density functions at integer values of parameter m. Density formulas at odd and even values of the parameter are similar on the structure of cofactors: exponent function, hyperbolic secant or cosecant correspondingly, and polynomial factors.
The distribution under study has multiple applications, not only in probability problems. Moments of distribution at specified parameters change as polynomials of some class with corresponding coefficients. These coefficients can be constructed as number sets (number triangles and number sequences), both known and new with setting of a row of functional relationships.