• Title/Summary/Keyword: Euler polynomials

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A NEW FAMILY OF FUBINI TYPE NUMBERS AND POLYNOMIALS ASSOCIATED WITH APOSTOL-BERNOULLI NUMBERS AND POLYNOMIALS

  • Kilar, Neslihan;Simsek, Yilmaz
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1605-1621
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    • 2017
  • The purpose of this paper is to construct a new family of the special numbers which are related to the Fubini type numbers and the other well-known special numbers such as the Apostol-Bernoulli numbers, the Frobenius-Euler numbers and the Stirling numbers. We investigate some fundamental properties of these numbers and polynomials. By using generating functions and their functional equations, we derive various formulas and relations related to these numbers and polynomials. In order to compute the values of these numbers and polynomials, we give their recurrence relations. We give combinatorial sums including the Fubini type numbers and the others. Moreover, we give remarks and observation on these numbers and polynomials.

ON POLY-EULERIAN NUMBERS

  • Son, Jin-Woo;Kim, Min-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.47-61
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    • 1999
  • In this paper we difine poly-Euler numbers which generalize ordinary Euler numbers. We construct a p-adic poly-Euler measure by the poly-Euler polynomials and derive an integral formula.

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A NEW CLASS OF q-HERMITE-BASED APOSTOL TYPE FROBENIUS GENOCCHI POLYNOMIALS

  • Kang, Jung Yoog;Khan, Waseem A.
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.759-771
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    • 2020
  • In this article, a hybrid class of the q-Hermite based Apostol type Frobenius-Genocchi polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating function methods. Furthermore, we consider some relationships for q-Hermite based Apostol type Frobenius-Genocchi polynomials of order α associated with q-Apostol Bernoulli polynomials, q-Apostol Euler polynomials and q-Apostol Genocchi polynomials.

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.

FRACTIONAL EULER'S INTEGRAL OF FIRST AND SECOND KINDS. APPLICATION TO FRACTIONAL HERMITE'S POLYNOMIALS AND TO PROBABILITY DENSITY OF FRACTIONAL ORDER

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • v.28 no.1_2
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    • pp.257-273
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    • 2010
  • One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.

Some Properties of the Generalized Apostol Type Hermite-Based Polynomials

  • KHAN, WASEEM AHMAD
    • Kyungpook Mathematical Journal
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    • v.55 no.3
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    • pp.597-614
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    • 2015
  • In this paper, we study some properties of the generalized Apostol type Hermite-based polynomials. which extend some known results. We also deduce some properties of the generalized Apostol-Bernoulli polynomials, the generalized Apostol-Euler polynomials and the generalized Apostol-Genocchi polynomials of high order. Numerous properties of these polynomials and some relationships between $F_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ and $_HF_n{^{({\alpha})}}(x;{\lambda};{\mu};{\nu};c)$ are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.

SYMMETRIC PROPERTIES OF CARLITZ'S TYPE (p, q)-GENOCCHI POLYNOMIALS

  • KIM, A HYUN
    • Journal of applied mathematics & informatics
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    • v.37 no.3_4
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    • pp.317-328
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    • 2019
  • This paper defines Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, and explains fourteen properties which can be complemented by Carlitz's type (p, q)-Genocchi polynomials and Carlitz's type (h, p, q)-Genocchi polynomials, including distribution relation, symmetric property, and property of complement. Also, it explores alternating powers sums by proving symmetric property related to Carlitz's type (p, q)-Genocchi polynomials.