Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and C
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Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers Damla Gun1
· Yilmaz Simsek1
Received: 10 April 2020 / Accepted: 2 July 2020 © The Royal Academy of Sciences, Madrid 2020
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. Keywords Generating function · Special functions · Bernoulli numbers and polynomials · Eulerian numbers · Stirling numbers · Catalan numbers Mathematics Subject Classification 11S80 · 11B68 · 05A15 · 05A19 · 11M35 · 30C15 · 26C05 · 12D10 · 33C45
1 Introduction It has been seen in recent years that not only generating functions, but also the numbers and polynomials produced by them are used in many different disciplines, especially mathematics. In this paper, we investigate some families of generating functions including the Bernoulli numbers and polynomials of higher order, the Stirling numbers, and the Catalan numbers. By applying the Laplace transform to the generating function of the Bernoulli polynomials of higher order, we give interesting identity involving these polynomials. Using generating
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Yilmaz Simsek [email protected] Damla Gun [email protected]
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Department of Mathematics, Faculty of Science University of Akdeniz, 07058 Antalya, Turkey 0123456789().: V,-vol
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D. Gun, Y. Simsek
functions with their functional equations, gamma and beta functions, we give some identities and relations including the Bernoulli numbers of higher order, the Stirling numbers, and the Catalan numbers. We give integral representations for these numbers. We also give some inequalities including binomial coefficients, the Bernoulli numbers of negative higher order, the Stirling numbers, and the Catalan numbers. In this paper we use the following definitions, relations, and notations: Let N, Z, Q, R and C denote the set of natural numbers, the set of integers, the set of rational numbers, the set of real numbers and the set of complex numbers, respectively. N0 = N∪ {0}. The Bernoulli numbers and polynomials of higher order are defined by means of the following generating functions, respectively: n ∞ k t (n) t FB (t, n) = = B (1) k et − 1 k! k=0
and
∞
G B (t, x, n) = FB (t, n)e
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