Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers


GÜN D., ŞİMŞEK Y.

REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, vol.114, no.4, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 114 Issue: 4
  • Publication Date: 2020
  • Doi Number: 10.1007/s13398-020-00899-z
  • Journal Name: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, MathSciNet, zbMATH, DIALNET, Civil Engineering Abstracts
  • Keywords: Generating function, Special functions, Bernoulli numbers and polynomials, Eulerian numbers, Stirling numbers, Catalan numbers, INTEGRAL PRESENTATIONS, BINOMIAL COEFFICIENTS, COMBINATORIAL SUMS
  • Akdeniz University Affiliated: Yes

Abstract

The aim of this paper is to give some new relations, identities, and inequalities for the Bernoulli polynomials and numbers of higher order, the Stirling numbers of the second kind, the Eulerian numbers, and the Catalan numbers. By applying the Laplace transformation to the generating function of the Bernoulli polynomials of higher order, a novel formula for these polynomials is obtained. Integral and series representations for these polynomials and numbers are given. Moreover, the upper bound and the lower bound for the Bernoulli numbers of negative order are given. Some inequalities including the Bernoulli numbers of negative order and the Stirling numbers of the second kind are also given. Finally, appropriate ligaments of the definitions and results introduced here with those in earlier as well as oncoming investigations will be designated.