4 edition of **Zeros of Bernoulli, generalized Bernoulli, and Euler polynomials** found in the catalog.

Zeros of Bernoulli, generalized Bernoulli, and Euler polynomials

Karl Dilcher

- 112 Want to read
- 21 Currently reading

Published
**1988** by American Mathematical Society in Providence, R.I., USA .

Written in English

- Bernoulli polynomials.,
- Euler polynomials.

**Edition Notes**

Statement | Karl Dilcher. |

Series | Memoirs of the American Mathematical Society,, no. 386 |

Classifications | |
---|---|

LC Classifications | QA3 .A57 no. 386, QA246.5 .A57 no. 386 |

The Physical Object | |

Pagination | iv, 94 p. : |

Number of Pages | 94 |

ID Numbers | |

Open Library | OL2030820M |

ISBN 10 | 082182449X |

LC Control Number | 88006356 |

OCLC/WorldCa | 17733004 |

In mathematics, the Bernoulli numbers B n are a sequence of rational numbers with deep connections to number are closely related to the values of the Riemann zeta function at negative integers.. There are several conventions for Bernoulli numbers. The most common has B n = 0 for all odd n other than 1, and B 1 = −1/2, but some authors use B 1 = +1/2 and some write B n for B 2n. Euler numbers and Bernoulli numbers, we can deﬁne Poly-Euler num-bers and polynomials as follows which also [11], deﬁned it by following method in same times. Deﬁnition 1 (Poly-Euler polynomials):The Poly-Euler polynomials may be deﬁned by using the following generating function, (14) 2Li k(1−e−t) 1+et e xt = P∞ n=0 E(k) n. Polynomial variable, specified as a symbolic variable, expression, function, vector, or matrix. If x is a vector or matrix, bernoulli returns Bernoulli numbers or polynomials for each element of you use the bernoulli function to find Bernoulli polynomials, at least one argument must be a scalar or both arguments must be vectors or matrices of the same size.

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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials About this Title. Karl Dilcher. Publication: Memoirs of the American Mathematical SocietyCited by: Get this from a library.

Zeros of Bernoulli, generalized Bernoulli, and Euler polynomials. [Karl Dilcher] -- This study deals with three classes of polynomials which have important applications in the theory of finite differences, in number theory and in classical analysis. These polynomials are closely.

Title (HTML): Zeros of Bernoulli, Generalized Bernoulli, and Euler Polynomials Author(s) (Product display): Karl Dilcher Book Series Name: Memoirs of. Genre/Form: Electronic books: Additional Physical Format: Print version: Dilcher, Karl, Zeros of Bernoulli, generalized Bernoulli, and Euler polynomials /.

real zeros of Euler polynomials. In the same way, Corollaries 1 and 2 lead to the corresponding statements about the zeros of Bernoulli and generalized Bernoulli polynomials.

This may serve as an explanation for the nature of the results mentioned in the Introduction. POLYNOMIALS.

cerned with the Generalized Bernoulli and Euler polynomials. Furthermore, the properties obtained in the second chapter are also examined for the generalized Bernoulli and Euler polynomials in this part of the thesis.

The Complemen-tary Argument Theorem, the. the calculus of finite difference by Euler [7], a researcher working with the Ber-noulli brothers at Zurich.

As informed by Raugh [8], Euler had followed de Moivre in given B k the denomination Bernoulli numbers introduced the Be, r-noulli polynomials in via the generating function () 0 1 e e1. zt m t m m t B zt m ∞ = = − ∑ (). Expressing this with the generalized Bernoulli polynomial, Σ k=0 n-1 kp = p+1 1 Bp+1()n-Bp+1()0 (') Approximate expression Formula holds only as a concept, and regrettably, we cannot use this for actual calculation.

If it is why, obtaining Σ p at non-integer p is difficult and the generalized Bernoulli polynomial on the domain wider than. Generalized Bernoulli polynomials -- definition and some applications. Generalized Bernoulli but more exciting these are the numerators of the lost Bernoulli numbers as I called them in my blog post on the Euler-Bernoulli diamond offset, suppressed zeros and other OEIS-typical peculiarities) in A, A, A, A (see []).Recently, many authors have studied in the different several areas related to -theory (see [1–13]).In this paper, we present a systemic study of some families of multiple Carlitz's type -Bernoulli numbers and polynomials by using the integral equations of -adic -integralswe derive some interesting equations of -adic -integrals on.

Bernoulli Polynomials Bernoulli Numbers The “generating function” for the Bernoulli numbers is x ex −1 = X∞ n=0 B n n. () That is, we are to expand the left-hand side of this equation in powers of x, i.e., a Taylor series about x = 0. The coeﬃcient of xn in this expansion is B n/n!. This is consistent with and supplements the results of Howard [2] on the real zeros of Euler polynomials.

In the same way, Corollaries 1 and 2 lead to the corresponding statements about the zeros of Bernoulli and generalized Bernoulli polynomials. This may serve as an explanation for the nature of the results mentioned in the Introduction. Properties of Bernoulli polynomials 3 3.

Fourier series and Bernoulli polynomials 9 4. Bernoulli polynomials on the unit interval [0,1] 12 5.

Asymptotic behavior of Bernoulli polynomials 16 6. The generating function of Bernoulli polynomials 18 7. The von Staudt-Clausen theorem 23 8. The Euler-Maclaurin’s formula 25 9. Zeros of Bernoulli, Generalized Bernoulli, and Euler Polynomials Memoirs of the AMS, 94 pp., Softcover [Purchasing Information] Bernoulli Numbers Bibliography ().

Queen's Papers in Pure and Applied Mathematics, No. 87,pp., Softcover [Purchasing Information] [On-line Version] Number Theory. In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials.

From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities. MSCA15, 05A40, 11B68, 11B75, 33E20, 65Q We recall, for example, the generalized Bernoulli polynomials Bnα(x) recalled in the book of Gatteschi [ 6 ] defined by the generating function tαext et − 1 α = ∞ n=0 Bnα(x) by means of which, Tricomi and Erdélyi [16] gave an asymptotic expansion of the ratio of two gamma functions.

In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of.

relationships between generalized poly-Bernoulli polynomials with parameters and generalized Euler polynomials with parameters.

Let us briefly recall poly-Bernoulli numbers and polynomials. For an integer, put which is the -th polylogarithm if, and a rational function if. Many mathematicians have recently studied various matrices and analogs of these matrices. Especially, these matrices are the Bernoulli, Pascal and Euler matrices [1,2,3,4,5,6,7,8,9,10,11].These matrices and their analogs are obtained using numbers and polynomials such as the Bernoulli, Euler, q-Bernoulli, and q-Euler expressions [5, 12,13,14,15,16,17,18].

In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer.

Bernoulli Theorems and Applications The energy equation and the Bernoulli theorem There is a second class of conservation theorems, closely related to the conservation of energy discussed in Chapter 6. These conservation theorems are collectively called Bernoulli Theorems since the scientist who first contributed in a fundamental way to the.

Fix a fixed m ∈ N, the generalized Bernoulli polynomials of level m are defined by means of the following generating function [1] z The q-analogue of the classical Bernoulli numbers and. and so generally applicable as the Bernoulli numbers." | David Eugene Smith, A Source Book in Mathematics, The Bernoulli numbers are the terms of a sequence of rational numbers discovered independently by the Swiss mathematician Jakob Bernoulli and Japanese mathematician Seki Takakazu [6].

Both encountered. INTRODUCTION The generalized Bernoulli polynomials B(~) (x) of order ~ and the generalized Euler polynomials E(n~) (x) of order (~, each of degree n in x as well as in ~, are defined by means of the following generating functions (see, for details, [3, p. et seq.; 4, Section ; 5, Section ]): ext = B(~a)(x (It[.

A large literature scatters widely in books and journals on Bernoulli numbers Bn, and Bernoulli polynomials Bn (x) and Euler numbers En and Euler polynomials En (x). They can be studied by means of the binomial expressions connecting them, n X n Bk xn−k.

In this chapter, a certain variation of Bernoulli and Euler numbers and polynomials is introduced by means of polylogarithm, particularly, the poly-Bernoulli and Euler numbers and polynomials. Furthermore, a certain generalization of poly-Bernoulli and poly-Euler numbers and polynomials is defined by means of multiple polylogarithm.

Euler polynomial, generalized Bernoulli number, generalized Bernoulli polinomial, generalized Euler number, generalized Euler polynomial.

The authors were supported in part by NNSF (#) of China, SF for the Prominent Youth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province. Agoh, Recurrences for Bernoulli and Euler polynomials and numbers, Exp. Math. 18(), – zbMATH MathSciNet Google Scholar [8] T.

Agoh, On Fermat and Wilson quotients, Exp. Math. 14 (), – zbMATH MathSciNet Google Scholar. Further, the numbers related to the q-Bernoulli–Euler polynomials are considered, and the graph of the q-Bernoulli–Euler polynomials is also drawn for index n = 3 and \(q = 1/2 \).

View Show. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation + + (−) = for an arbitrary complex number α, the order of the Bessel function.

Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values. Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials, J. Approx. Theory 49 (), MR 88g 7.

Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters, J. Number Theory 25 (), MR 88a 8.

Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been the subject of extensive study in recent years and much progress have been made both mathematically and computationally. Using computer, a realistic study for q-Euler polynomials [n,q](x) is very interesting.

In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in number Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for.

On the Symmetries of the q -Bernoulli Polynomials Kim, Taekyun, Abstract and Applied Analysis, ; A Note on the Multiple Twisted Carlitz's Type q -Bernoulli Polynomials Jang, Lee-Chae and Ryoo, Cheon-Seoung, Abstract and Applied Analysis, ; On the Modified q -Bernoulli Numbers of Higher Order with Weight Kim, T., Choi, J., Kim, Y.-H., and Rim, S.-H., Abstract and Applied Analysis, Dilcher, K.

Zeros of Bernoulli, generalized Bernoulli, and Euler polynomials, CIP galley (closely related to corresponding sequences of numbers, namely the Bernoulli numbers) found: Math.

subj. classif.: (Number theory, XX; Sequences and sets, 11Bxx; Bernoulli and Euler numbers and polynomials, 11B68). The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials.

The investigation includes derivations of generating functions, series definitions, and several important properties and identities of the hybrid q-special polynomials. BERNOULLI_NUMBER2 evaluates the Bernoulli numbers.

BERNOULLI_NUMBER3 computes the value of the Bernoulli number B(N). BERNOULLI_NUMBER_VALUES returns some values of the Bernoulli numbers. BERNOULLI_POLY evaluates the Bernoulli polynomial of order N at X. BERNOULLI_POLY2 evaluates the N-th Bernoulli polynomial at X.

In this paper, we perform a further investigation for the Frobenius-Euler polynomials. Some new formulae of products of the Frobenius-Euler polynomials are established by applying the generating function methods and some summation transform techniques.

It turns out that some corresponding known results are obtained as special cases. MSCB68, 05A The origin of r- or 1-generalized Fibonacci sequences goes back to Euler.

In [7, Chapter XVII] he discussed Daniel Bernoulli’s method of using linear recurrences to approximate zeros of (mainly polynomial) functions.

In this paper, we rst study the relationship between a given polynomial. Generalization of Bernoulli polynomials of classical Bernoulli polynomials B0 n (x).In (2), we take x = 0, a generalized Bernoulli numbers are tm et − m n−1 h=0 th h.

n=0 B[m−1] t n n. (3) Since G[m−1](x,t)=A(t)ext, the generalized Bernoulli polynomials belong to the class of Appell polynomials. involving the Bernoulli and Euler polynomials. In this paper, a uni- ed approach by means of P[x] is presented in the study of Bernoulli polynomials and numbers and Euler polynomials and numbers.

For any square matrix A, the exponential of Ais de ned as the following matrix in a series form: eA= I+ A+ 1 2! A2 + 1 3! A3 + = X k=0 1 k! Ak.C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol.

In particular, six approaches to the theory of Bernoulli polynomials are known, these are associated with the names of J. Bernoulli, L. Euler, P. E. Appel, A. Hurwitz, E.

Lucas and D. H. Lehmer. Also Apostol and Qiu-Ming Luo defined new generalizations of Bernoulli polynomials that we have used in this paper. [sections]1.