• Title/Summary/Keyword: transcendental entire solution

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ENTIRE SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATION AND FERMAT TYPE q-DIFFERENCE DIFFERENTIAL EQUATIONS

  • CHEN, MIN FENG;GAO, ZONG SHENG
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.447-456
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    • 2015
  • In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

History of Transcendental numbers and Open Problems (초월수의 역사와 미해결 문제)

  • Park, Choon-Sung;Ahn, Soo-Yeop
    • Journal for History of Mathematics
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    • v.23 no.3
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    • pp.57-73
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    • 2010
  • Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.

VALUE DISTRIBUTION OF SOME q-DIFFERENCE POLYNOMIALS

  • Xu, Na;Zhong, Chun-Ping
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.29-38
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    • 2016
  • For a transcendental entire function f(z) with zero order, the purpose of this article is to study the value distributions of q-difference polynomial $f(qz)-a(f(z))^n$ and $f(q_1z)f(q_2z){\cdots}f(q_mz)-a(f(z))^n$. The property of entire solution of a certain q-difference equation is also considered.

On the Growth of Transcendental Meromorphic Solutions of Certain algebraic Difference Equations

  • Xinjun Yao;Yong Liu;Chaofeng Gao
    • Kyungpook Mathematical Journal
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    • v.64 no.1
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    • pp.185-196
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    • 2024
  • In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), bi(z) for i = 0, 1, 2 and dj (z) for j = 0, ..., 4 are given functions, △cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the bi(z) and the dj(z) are polynomials, and d4(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d0(z) + 1, or, deg a(z) = deg d0(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

Non-homogeneous Linear Differential Equations with Solutions of Finite Order

  • Belaidi, Benharrat
    • Kyungpook Mathematical Journal
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    • v.45 no.1
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    • pp.105-114
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    • 2005
  • In this paper we investigate the growth of finite order solutions of the differential equation $f^{(k)}\;+\;A_{k-1}(Z)f^{(k-l)}\;+\;{\cdots}\;+\;A_1(z)f^{\prime}\;+\;A_0(z)f\;=\;F(z)$, where $A_0(z),\;{\cdots}\;,\;A_{k-1}(Z)\;and\;F(z)\;{\neq}\;0$ are entire functions. We find conditions on the coefficients which will guarantees the existence of an asymptotic value for a transcendental entire solution of finite order and its derivatives. We also estimate the lower bounds of order of solutions if one of the coefficient is dominant in the sense that has larger order than any other coefficients.

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