• Title/Summary/Keyword: Bernoulli polynomial

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SOME REMARKS ON A q-ANALOGUE OF BERNOULLI NUMBERS

  • Kim, Min-Soo;Son, Jin-Woo
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.221-236
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    • 2002
  • Using the p-adic q-integral due to T. Kim[4], we define a number B*$_{n}$(q) and a polynomial B*$_{n}$(q) which are p-adic q-analogue of the ordinary Bernoulli number and Bernoulli polynomial, respectively. We investigate some properties of these. Also, we give slightly different construction of Tsumura's p-adic function $\ell$$_{p}$(u, s, $\chi$) [14] using the p-adic q-integral in [4].n [4].

A NOTE OF THE MODIFIED BERNOULLI POLYNOMIALS AND IT'S THE LOCATION OF THE ROOTS

  • LEE, Hui Young
    • Journal of applied mathematics & informatics
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    • v.38 no.3_4
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    • pp.291-300
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    • 2020
  • This type of polynomial is a generating function that substitutes eλt for et in the denominator of the generating function for the Bernoulli polynomial, but polynomials by using this generating function has interesting properties involving the location of the roots. We define these generation functions and observe the properties of the generation functions.

Bernoulli and Euler Polynomials in Two Variables

  • Claudio Pita-Ruiz
    • Kyungpook Mathematical Journal
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    • v.64 no.1
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    • pp.133-159
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    • 2024
  • In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2)a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ≠ 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1p2 (x1, x2). By using results for S(1,x2,p2)1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

TRIPLE AND FIFTH PRODUCT OF DIVISOR FUNCTIONS AND TREE MODEL

  • KIM, DAEYEOUL;CHEONG, CHEOLJO;PARK, HWASIN
    • Journal of applied mathematics & informatics
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    • v.34 no.1_2
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    • pp.145-156
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    • 2016
  • It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. In this article, we consider relationship between fifth-order combinatoric convolution sums of divisor functions and Bernoulli polynomials. As applications of these identities, we give a concrete interpretation in terms of the procedural modeling method.

SOME SYMMETRY IDENTITIES FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY UNRAMIFIED ROOTS OF UNITY

  • Kim, Dae San
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.603-618
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    • 2015
  • We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the p-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

  • Sarkar, Korak;Ganguli, Ranjan
    • Structural Engineering and Mechanics
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    • v.51 no.5
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    • pp.773-792
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    • 2014
  • In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Transverse Vibration of a Uniform Euler-Bernoulli Beam Under Varying Axial Force Using Differential Transformation Method

  • Shin Young-Jae;Yun Jong-Hak
    • Journal of Mechanical Science and Technology
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    • v.20 no.2
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    • pp.191-196
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    • 2006
  • This paper presents the application of techniques of differential transformation method (DTM) to analyze the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force. The governing differential equation of the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force is derived and verified. The varying axial force was extended to the more general case which was high polynomial consisted of many terms. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The accuracy and the convergence in solving the problem by DTM are discussed.

Lucas-Euler Relations Using Balancing and Lucas-Balancing Polynomials

  • Frontczak, Robert;Goy, Taras
    • Kyungpook Mathematical Journal
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    • v.61 no.3
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    • pp.473-486
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    • 2021
  • We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.