• 제목/요약/키워드: Bernoulli number

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SOME REMARKS ON A q-ANALOGUE OF BERNOULLI NUMBERS

  • Kim, Min-Soo;Son, Jin-Woo
    • 대한수학회지
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    • 제39권2호
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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].

OME PROPERTIES OF THE BERNOULLI NUMBERS OF THE SECOND KIND AND THEIR GENERATING FUNCTION

  • Qi, Feng;Zhao, Jiao-Lian
    • 대한수학회보
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    • 제55권6호
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    • pp.1909-1920
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    • 2018
  • In the paper, the authors find a common solution to three series of differential equations related to the generating function of the Bernoulli numbers of the second kind and present a recurrence relation, an explicit formula in terms of the Stirling numbers of the first kind, and a determinantal expression for the Bernoulli numbers of the second kind.

ON BERNOULLI NUMBERS

  • Kim, Min-Soo;Son, Jin-Woo
    • 대한수학회지
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    • 제37권3호
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    • pp.391-410
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    • 2000
  • In the complex case, we construct a q-analogue of the Riemann zeta function q(s) and a q-analogue of the Dirichlet L-function L(s,X), which interpolate the 1-analogue Bernoulli numbers. Using the properties of p-adic integrals and measures, we show that Kummer type congruences for the q-analogue Bernoulli numbers are the generalizations of the usual Kummer congruences for the ordinary Bernoulli numbers. We also construct a q0analogue of the p-adic L-function Lp(s, X;q) which interpolates the q-analogue Bernoulli numbers at non positive integers.

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APPLICATIONS OF CLASS NUMBERS AND BERNOULLI NUMBERS TO HARMONIC TYPE SUMS

  • Goral, Haydar;Sertbas, Doga Can
    • 대한수학회보
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    • 제58권6호
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    • pp.1463-1481
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    • 2021
  • Divisibility properties of harmonic numbers by a prime number p have been a recurrent topic. However, finding the exact p-adic orders of them is not easy. Using class numbers of number fields and Bernoulli numbers, we compute the exact p-adic orders of harmonic type sums. Moreover, we obtain an asymptotic formula for generalized harmonic numbers whose p-adic orders are exactly one.

Bernoulli and Euler Polynomials in Two Variables

  • Claudio Pita-Ruiz
    • Kyungpook Mathematical Journal
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    • 제64권1호
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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.

AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND

  • Kim, Y.H.;Jung, H.Y.;Ryoo, C.S.
    • Journal of applied mathematics & informatics
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    • 제32권3_4호
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    • pp.465-474
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    • 2014
  • Many mathematicians have studied various relations beween Euler number $E_n$, Bernoulli number $B_n$ and Genocchi number $G_n$ (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

Natural Frequencies of Euler-Bernoulli Beam with Open Cracks on Elastic Foundations

  • Shin Young-Jae;Yun Jong-Hak;Seong Kyeong-Youn;Kim Jae-Ho;Kang Sung-Hwang
    • Journal of Mechanical Science and Technology
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    • 제20권4호
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    • pp.467-472
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    • 2006
  • A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Pasternak foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.