• Title/Summary/Keyword: p-adic L-function

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ON A q-ANALOGUE OF THE p-ADIC GENERALIZED TWISTED L-FUNCTIONS AND p-ADIC q-INTEGRALS

  • Lee, Chae-Jang
    • Journal of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.1-10
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    • 2007
  • The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p-adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted q-Bernoulli numbers. This is the generalization of Kim's h-extension of p-adic q-L-function which was constructed in [5] and is a partial answer for the open question which was remained in [3].

ON BERNOULLI NUMBERS

  • Kim, Min-Soo;Son, Jin-Woo
    • Journal of the Korean Mathematical Society
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    • v.37 no.3
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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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Lq-ESTIMATES OF MAXIMAL OPERATORS ON THE p-ADIC VECTOR SPACE

  • Kim, Yong-Cheol
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.367-379
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    • 2009
  • For a prime number p, let $\mathbb{Q}_p$ denote the p-adic field and let $\mathbb{Q}_p^d$ denote a vector space over $\mathbb{Q}_p$ which consists of all d-tuples of $\mathbb{Q}_p$. For a function f ${\in}L_{loc}^1(\mathbb{Q}_p^d)$, we define the Hardy-Littlewood maximal function of f on $\mathbb{Q}_p^d$ by $$M_pf(x)=sup\frac{1}{\gamma{\in}\mathbb{Z}|B_{\gamma}(x)|H}{\int}_{B\gamma(x)}|f(y)|dy$$, where |E|$_H$ denotes the Haar measure of a measurable subset E of $\mathbb{Q}_p^d$ and $B_\gamma(x)$ denotes the p-adic ball with center x ${\in}\;\mathbb{Q}_p^d$ and radius $p^\gamma$. If 1 < q $\leq\;\infty$, then we prove that $M_p$ is a bounded operator of $L^q(\mathbb{Q}_p^d)$ into $L^q(\mathbb{Q}_p^d)$; moreover, $M_p$ is of weak type (1, 1) on $L^1(\mathbb{Q}_p^d)$, that is to say, |{$x{\in}\mathbb{Q}_p^d:|M_pf(x)|$>$\lambda$}|$_H{\leq}\frac{p^d}{\lambda}||f||_{L^1(\mathbb{Q}_p^d)},\;\lambda$ > 0 for any f ${\in}L^1(\mathbb{Q}_p^d)$.

GENERALIZED EULER POWER SERIES

  • KIM, MIN-SOO
    • Journal of applied mathematics & informatics
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    • v.38 no.5_6
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    • pp.591-600
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    • 2020
  • This work is a continuation of our investigations for p-adic analogue of the alternating form Dirichlet L-functions $$L_E(s,{\chi})={\sum\limits_{n=1}^{\infty}}{\frac{(-1)^n{\chi}(n)}{n^s}},\;Re(s)>0$$. Let Lp,E(s, t; χ) be the p-adic Euler L-function of two variables. In this paper, for any α ∈ ℂp, |α|p ≤ 1, we give a power series expansion of Lp,E(s, t; χ) in terms of the variable t. From this, we derive a power series expansion of the generalized Euler polynomials with negative index, that is, we prove that $$E_{-n,{\chi}}(t)={\sum\limits_{m=0}^{\infty}}\(\array{-n\\m}\)E_{-(m+n),{\chi}^{t^m}},\;n{\in}{\mathbb{N}}$$, where t ∈ ℂp with |t|p < 1. Some further properties for Lp,E(s, t; χ) has also been shown.

AN APPLICATION OF p-ADIC ANALYSIS TO WINDOWED FOURIER TRANSFORM

  • Park, Sook Young;Chung, Phil Ung
    • Korean Journal of Mathematics
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    • v.12 no.2
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    • pp.193-200
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    • 2004
  • We shall introduce the notion of the windowed Fourier transform in $\mathbb{Q}_p$ and show that, for any given function $g{\in}L^2(\mathbb{Q}_p)$ of norm, the windowed Fourier transform of $f$ with respect to $g$ be a function of norms, and moreover be expressible to a summation form. The results obtained in this paper will be usable to the field of research in data compression for signal processing according to the following scheme.

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ON p-ADIC q-BERNOULLl NUMBERS

  • Kim, Tae-Kyun
    • Journal of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.21-30
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    • 2000
  • We give a proof of the distribution relation for q-Bernoulli polynomials $B_{k}$(x : q) by using q-integral and evaluate the values of p-adic q-L-function.n.

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A duplication formula for the double gamma function $Gamma_2$

  • Park, Junesang
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.289-294
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    • 1996
  • The double Gamma function had been defined and studied by Barnes [4], [5], [6] and others in about 1900, not appearing in the tables of the most well-known special functions, cited in the exercise by Whittaker and Waston [25, pp. 264]. Recently this function has been revived according to the study of determinants of Laplacians [8], [11], [15], [16], [19], [20], [22] and [24]. Shintani [21] also uses this function to prove the classical Kronecker limit formula. Its p-adic analytic extension appeared in a formula of Casson Nogues [7] for the p-adic L-functions at the point 0.

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