Car-Pólya and Gel’fond’s theorems for $_q[T]$

David Adam

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Displaying similar documents to “Car-Pólya and Gel’fond’s theorems for $_{q} [T]$ ”

Convergence of Taylor series in Fock spaces

Haiying Li (2014)

Studia Mathematica

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It is well known that the Taylor series of every function in the Fock space $F_{α}^{p}$ converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in $F_{α}^{p}$ do not necessarily converge “in norm”.

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.

Stević-Sharma type operators on Fock spaces in several variables

Lijun Ma, Zicong Yang (2024)

Czechoslovak Mathematical Journal

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Let $ϕ$ be an entire self-map of $ℂ^{N}$ , $u_{0}$ be an entire function on $ℂ^{N}$ and $𝐮 = (u_{1}, \dots, u_{N})$ be a vector-valued entire function on $ℂ^{N}$ . We extend the Stević-Sharma type operator to the classcial Fock spaces, by defining an operator $T_{u_{0}, 𝐮, ϕ}$ as follows: $- . 4 p t T_{u_{0}, 𝐮, ϕ} f = u_{0} \cdot f \circ ϕ + \sum_{i = 1}^{N} u_{i} \cdot \frac{\partial f}{\partial z_{i}} \circ ϕ .$ We investigate the boundedness and compactness of $T_{u_{0}, 𝐮, ϕ}$ on Fock spaces. The complex symmetry and self-adjointness of $T_{u_{0}, 𝐮, ϕ}$ are also characterized.

A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Diego Marques, Elaine Silva (2017)

Communications in Mathematics

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In this note, we prove that there is no transcendental entire function $f (z) \in ℚ [[z]]$ such that $f (ℚ) \subseteq ℚ$ and $den f (p / q) = F (q)$ , for all sufficiently large $q$ , where $F (z) \in ℤ [z]$ .

Entire functions of exponential type not vanishing in the half-plane $ℑ z > k$ , where $k > 0$

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $P (z)$ be a polynomial of degree $n$ having no zeros in $| z | < k$ , $k \leq 1$ , and let $Q (z) : = z^{n} \bar{P (1 / \bar{z})}$ . It was shown by Govil that if ${max}_{| z | = 1} | P^{'} (z) |$ and ${max}_{| z | = 1} | Q^{'} (z) |$ are attained at the same point of the unit circle $| z | = 1$ , then $max_{| z | = 1} | P^{'} (z) | \leq \frac{n}{1 + k^{n}} max_{| z | = 1} | P (z) | .$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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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.