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A SURVEY ON SOME OLD AND NEW IDENTITIES ASSOCIATED WITH LAPLACE DISTRIBUTION AND BERNOULLI NUMBERS
ZEHRA SELIN ASKAN,IREM KUCUKOGLU,Yilmaz Simsek 장전수학회 2020 Advanced Studies in Contemporary Mathematics Vol.30 No.4
The purpose of this paper is to give some survey on old and new identities related to the characteristic function, the Laplace distribution and special numbers and polynomials with comparative results and observations. Additionally, we give some computation formulas for the higher-order moments of some kinds of random variables with the Laplace distribution in terms of the Bernoulli numbers of the first kind, the Euler numbers of the second kind and Riemann zeta function by using the techniques of generating functions and characteristic function of the aforementioned random variables. Finally, with the aid of the Hankel determinants formed by the moments corresponding to the weight function that reveals the orthogonality feature of the orthogonal polynomials, we give futher remarks and observations on not only orthogonality properties of some orthogonal polynomials such as the Hermite polynomials, but also construction methods of the three-term recurrence relations for the orthogonal polynomials.