A SURVEY ON SOME OLD AND NEW IDENTITIES ASSOCIATED WITH LAPLACE DISTRIBUTION AND BERNOULLI NUMBERS
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (29)
- M. Abramowitz, "Handbook of Mathematical Functions with For-mulas, Graphs and Mathematical Tables" (1970), Dover Publication
- T. S. Chihara, "An Introduction to Orthogonal Polynomials" (1978), Gordon and Breach Sci-ence Publishers
- J. B. Conway, "Functions of One Complex Variable" (1978), Springer
- M. H. DeGroot, "Probability and Statistics", pp.893~ (2012), Pearson Addison-Wesley
- R. Dere, "Hermite base Bernoulli type polynomials on the umbral algebra", Russ. J. Math. Phys, 22(1), pp.1~5 (2015)
- W. Feller, "An Introduction to Probability Theory and Its Applications" (1966), John Wiley & Sons
- C. C. Grosjean, "The weight functions, generating functions and miscellaneous prop-erties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials", J. Comput. Appl. Math, 16(3), pp.259~307 (1986)
- M. E. H. Ismail, "Determinants with orthogonal polynomial entries", J. Comput. Appl. Math, 178, pp.255~266 (2005)
- N. L. Johnson, "Continuous Univariate Distributions, Volume 2" (1995), John Wiley & Sons, Inc
- T. Kim, "Euler numbers and polynomials associated with zeta functions", Abstr. Appl. Anal (2008)[DOI]
- D. S. Kim, "Identities of symmetry for type 2 Bernoulli and Euler Polynomials", Symmetry, 11(5), pp.613~ (2019)
- S. Kotz, "The Laplace Distribution and Generaliza-tion a Revisit with New Applications" (2001), Reno and Indianapolis
- I. Kucukoglu, "An approach to negative hypergeometric distribution by generating function for special numbers and polynomials", Turk. J. Math, 43, pp.2337~2353 (2019)
- I. Kucukoglu, "Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson-Charlier Polynomials and Probability Distribution Function", Axioms, 8(4), pp.112~ (2019)
- N. N. Lebedev, "Special functions and their applications" (1965), Prentice-Hall, Inc
- E. Lukacs, "Characteristic function" (1970), Charles griffin & Company Limited
- P. Natalini, "A generalization of the Bernoulli polynomials", Journal of Applied Mathematics, 3, pp.155~163 (2003)[DOI]
- H. Rademacher, "Topics in Analytic Number Theory" (1973), Springer-Verlag
- S. Roman, "The Umbral Calculus", pp.193~ (1984), Academic Press, Inc
- P. N. Sadjang, "Moments of classical orthogonal polynomials", 박사, pp.2013
- Y. Simsek, "Special functions related to Dedekind-type DC-sums and their applications", Russ. J. Math. Phys, 17(4), pp.495~508 (2010)
- Y. Simsek, "New families of special numbers for computing negtive order euler numbers and related numbers and polynomials", Appl. Anal. Discrete Math, 12, pp.1~35 (2018)
- B. Simsek, "The computation of expected values and moments of spe-cial polynomials via characteristic and generating functions", AIP Conf. Proc, 1863, pp.300012-1~300012-5 (2017)
- H. M. Srivastava, "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments", J. Math. Anal. Appl, 246, pp.331~351 (2000)
- H. M. Srivastava, "Series Associated with the Zeta and Related Functions" (2001), Kluwer Academic Publishers
- H.M. Srivastava, "Zeta and q-Zeta Functions and Associated Series and Integrals" (2012), Elsevier Science Publishers
- Some Properties of the Log-Laplace Distrubition, "V. R. R. Uppuluri", Mathemat-ics and Statistics Research Department Computer Sciences Division Union Carbide Corporation, ~
- C.-F Wei, "Several closed expressions for the Euler numbers", J. Inequal Appl, 219 (2015)[DOI]
- R. Zhang, "Matrix inversion using orthogonal polynomials", Arab. J. Math. Sci, 17, pp.11~30 (2011)