Value distribution and uniqueness of difference polynomials and entire solutions of difference equations

Xiaoguang Qi

Displaying similar documents to “Value distribution and uniqueness of difference polynomials and entire solutions of difference equations”

The hyper-order of solutions of certain linear complex differential equations

Guowei Zhang, Ang Chen (2010)

Annales Polonici Mathematici

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We prove some theorems on the hyper-order of solutions of the equation $f^{(k)} - e^{Q} f = a (1 - e^{Q})$ , where Q is an entire function, which is a polynomial or not, and a is an entire function whose order can be larger than 1. We improve some results by J. Wang and X. M. Li.

Complex Oscillation Theory of Differential Polynomials

Abdallah El Farissi, Benharrat Belaïdi (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper, we investigate the relationship between small functions and differential polynomials $g_{f} (z) = d_{2} f^{''} + d_{1} f^{'} + d_{0} f$ , where $d_{0} (z)$ , $d_{1} (z)$ , $d_{2} (z)$ are entire functions that are not all equal to zero with $ρ (d_{j}) < 1$ $(j = 0, 1, 2)$ generated by solutions of the differential equation $f^{''} + A_{1} (z) e^{a z} f^{'} + A_{0} (z) e^{b z} f = F$ , where $a, b$ are complex numbers that satisfy $a b (a - b) \neq 0$ and $A_{j} (z) \neg \equiv 0$ ( $j = 0, 1$ ), $F (z) \neg \equiv 0$ are entire functions such that $max \{ρ (A_{j}), j = 0, 1, ρ (F)\} < 1 .$

Growth of solutions of a class of complex differential equations

Ting-Bin Cao (2009)

Annales Polonici Mathematici

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The main purpose of this paper is to partly answer a question of L. Z. Yang [Israel J. Math. 147 (2005), 359-370] by proving that every entire solution f of the differential equation $f^{'} - e^{P (z)} f = 1$ has infinite order and its hyperorder is a positive integer or infinity, where P is a nonconstant entire function of order less than 1/2. As an application, we obtain a uniqueness theorem for entire functions related to a conjecture of Brück [Results Math. 30 (1996), 21-24].

Entire function sharing two polynomials with its $k$ th derivative

Sujoy Majumder, Nabadwip Sarkar (2024)

Mathematica Bohemica

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We investigate the uniqueness problem of entire functions that share two polynomials with their $k$ th derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible.

Entire solutions of q-difference equations and value distribution of q-difference polynomials

Jilong Zhang, Lianzhong Yang (2013)

Annales Polonici Mathematici

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We investigate the existence and uniqueness of entire solutions of order zero of the nonlinear q-difference equation of the form fⁿ(z) + L(z) = p(z), where p(z) is a polynomial and L(z) is a linear differential-q-difference polynomial of f with small growth coefficients. We also study the zeros distribution of some special type of q-difference polynomials.

Uniqueness of entire functions concerning difference polynomials

Chao Meng (2014)

Mathematica Bohemica

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In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f (z)$ and $g (z)$ be two transcendental entire functions of finite order, and $α (z)$ a small function with respect to both $f (z)$ and $g (z)$ . Suppose that $c$ is a non-zero complex constant and $n \geq 7$ (or $n \geq 10$ ) is an integer. If $f^{n} (z) (f (z) - 1) f (z + c)$ and $g^{n} (z) (g (z) - 1) g (z + c)$ share “ $(α (z), 2)$ ” (or ${(α (z), 2)}^{*}$ ), then $f (z) \equiv g (z)$ . Our results extend and generalize some well known previous...

On deviations from rational functions of entire functions of finite lower order

E. Ciechanowicz, I. I. Marchenko (2007)

Annales Polonici Mathematici

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Let f be a transcendental entire function of finite lower order, and let $q_{ν}$ be rational functions. For 0 < γ < ∞ let B(γ):= πγ/sinπγ if γ ≤ 0.5, B(γ):= πγ if γ > 0.5. We estimate the upper and lower logarithmic density of the set $r : \sum_{1 \leq ν \leq k} l o g ⁺ m a x_{| | z | | = r} {| f (z) - q_{ν} (z) |}^{- 1} < B (γ) T (r, f)$ .

Uniqueness theorems for entire functions whose difference polynomials share a meromorphic function of a smaller order

Xiao-Min Li, Wen-Li Li, Hong-Xun Yi, Zhi-Tao Wen (2011)

Annales Polonici Mathematici

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We deal with uniqueness of entire functions whose difference polynomials share a nonzero polynomial CM, which corresponds to Theorem 2 of I. Laine and C. C. Yang [Proc. Japan Acad. Ser. A 83 (2007), 148-151] and Theorem 1.2 of K. Liu and L. Z. Yang [Arch. Math. 92 (2009), 270-278]. We also deal with uniqueness of entire functions whose difference polynomials share a meromorphic function of a smaller order, improving Theorem 5 of J. L. Zhang [J. Math. Anal. Appl. 367 (2010), 401-408],...

Uniqueness of entire functions and fixed points

Xiao-Guang Qi, Lian-Zhong Yang (2010)

Annales Polonici Mathematici

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Let f and g be entire functions, n, k and m be positive integers, and λ, μ be complex numbers with |λ| + |μ| ≠ 0. We prove that ${(f ⁿ (z) (λ f^{m} (z) + μ))}^{(} k)$ must have infinitely many fixed points if n ≥ k + 2; furthermore, if ${(f ⁿ (z) (λ f^{m} (z) + μ))}^{(} k)$ and ${(g ⁿ (z) (λ g^{m} (z) + μ))}^{(} k)$ have the same fixed points with the same multiplicities, then either f ≡ cg for a constant c, or f and g assume certain forms provided that n > 2k + m* + 4, where m* is an integer that depends only on λ.

Car-Pólya and Gel’fond’s theorems for $_{q} [T]$

David Adam (2004)

Acta Arithmetica

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Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

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We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{(k)} + a_{k - 1} (z) f^{(k - 1)} + \dots + a ₁ (z) f^{'} + D (z) f = 0$ , (1) where $D (z) = Q ₁ (z) e^{P ₁ (z)} + Q ₂ (z) e^{P ₂ (z)} + Q ₃ (z) e^{P ₃ (z)}$ , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), $a_{j} (z)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

Deformed Heisenberg algebra with reflection and $d$ -orthogonal polynomials

Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani (2017)

Czechoslovak Mathematical Journal

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This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of $d$ -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when $d = 1$ . The underlying algebraic framework allowed a systematic...

A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera, José A. Díaz-García, Graciela González-Farías (2007)

Applicationes Mathematicae

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This work solves the partial differential equation for Jack polynomials $C_{κ}^{α}$ of the second order. When the parameter α of the solution takes the values 1/2, 1 and 2 we get explicit formulas for the quaternionic, complex and real zonal polynomials of the second order, respectively.

Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

Antonio Garijo, Xavier Jarque, Mónica Moreno Rocha (2011)

Fundamenta Mathematicae

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We consider the family of transcendental entire maps given by $f_{a} (z) = a (z - (1 - a)) e x p (z + a)$ where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of $f_{a}$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of...

Distribution of zeros and shared values of difference operators

Jilong Zhang, Zongsheng Gao, Sheng Li (2011)

Annales Polonici Mathematici

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We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that $Δ_{c}^{k} f \neq 0$ for n ≥ 2, and a is a small function with respect to f, then $f ⁿ Δ_{c}^{k} f$ equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

Upper bounds for the $L_{q}$ norm of Fekete polynomials on subarcs

(2012)

Acta Arithmetica

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A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Michal Johanis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that a $C^{k}$ -smooth mapping on an open subset of $ℝ^{n}$ , $k \in ℕ \cup {0, \infty}$ , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

The zero distribution and uniqueness of difference-differential polynomials

Kai Liu, Xin-Ling Liu, Lian-Zhong Yang (2013)

Annales Polonici Mathematici

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We consider the zero distribution of difference-differential polynomials of meromorphic functions and present some results which can be seen as the discrete analogues of the Hayman conjecture. In addition, we also investigate the uniqueness of difference-differential polynomials of entire functions sharing one common value. Our theorems improve some results of Luo and Lin [J. Math. Anal. Appl. 377 (2011), 441-449] and Liu, Liu and Cao [Appl. Math. J. Chinese Univ. 27 (2012), 94-104]. ...

On the approximation of entire functions over Carathéodory domains

Devendra Kumar, Harvir S. Kasana (1994)

Commentationes Mathematicae Universitatis Carolinae

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Let $D$ be a Carathéodory domain. For $1 \leq p \leq \infty$ , let $L^{p} (D)$ be the class of all functions $f$ holomorphic in $D$ such that ${∥ f ∥}_{D, p} = [\frac{1}{A} \int \int_{D}^{} {| f (z) |}^{p} {d x d y]}^{1 / p} < \infty$ , where $A$ is the area of $D$ . For $f \in L^{p} (D)$ , set $E_{n}^{p} (f) = inf_{t \in π_{n}} {∥ f - t ∥}_{D, p};$ $π_{n}$ consists of all polynomials of degree at most $n$ . In this paper we study the growth of an entire function in terms of approximation error in $L^{p}$ -norm on $D$ .