Symmetric identity for polynomial sequences satisfying $A_{n+1}^\prime (x)=(n+1)A_n(x)$

Farid Bencherif; Rachid Boumahdi; Tarek Garici

Displaying similar documents to “Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ ”

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers $(p, n)$ , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree $8$ polynomials. ...

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae

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Let $P_{k}$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_{k}$ in $f = \sum_{k} a_{k} P_{k}$ . A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_{k}$ results in obtaining a low order of the recurrence.

A generalisation of Amitsur's A-polynomials

Adam Owen, Susanne Pumplün (2021)

Communications in Mathematics

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We find examples of polynomials $f \in D [t; σ, δ]$ whose eigenring $ℰ (f)$ is a central simple algebra over the field $F = C \cap Fix (σ) \cap Const (δ)$ .

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ in the special case where $f (x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f (x)$ is monic and linear.

Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

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We analyse the roots of the polynomial $x^{n} - p x^{n - 1} - q x - 1$ for $p ⩾ q ⩾ 1$ . This is the characteristic polynomial of the recurrence relation $F_{k, p, q} (n) = p F_{k, p, q} (n - 1) + q F_{k, p, q} (n - k + 1) + F_{k, p, q} (n - k)$ for $n ⩾ k$ , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

Weak polynomial identities and their applications

Vesselin Drensky (2021)

Communications in Mathematics

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Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$ . The polynomial $f (x_{1}, ..., x_{n})$ of the free associative algebra $K 〈 x_{1}, x_{2}, ... 〉$ is a weak polynomial identity for the pair $(R, V)$ if it vanishes in $R$ when evaluated on $V$ . We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of...

Approximation by weighted polynomials in $ℝ^{k}$

Maritza M. Branker (2005)

Annales Polonici Mathematici

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We apply pluripotential theory to establish results in $ℝ^{k}$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $Σ \subset ℝ^{k}$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

Jumps of ternary cyclotomic coefficients

Bartłomiej Bzdęga (2014)

Acta Arithmetica

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It is known that two consecutive coefficients of a ternary cyclotomic polynomial $Φ_{p q r} (x) = \sum_{k} a_{p q r} (k) x^{k}$ differ by at most one. We characterize all k such that $| a_{p q r} (k) - a_{p q r} (k - 1) | = 1$ . We use this to prove that the number of nonzero coefficients of the nth ternary cyclotomic polynomial is greater than $n^{1 / 3}$ .

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $∥ \cdot ∥$ be the uniform norm in the unit disk. We study the quantities $M_{n} (α) : = inf (∥ z P (z) + α ∥ - α)$ where the infimum is taken over all polynomials $P$ of degree $n - 1$ with $∥ P (z) ∥ = 1$ and $α > 0$ . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{α > 0} M_{n} (α) = 1 / n$ . We find the exact values of $M_{n} (α)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of $ℂ^{N}$ . We define a pluricomplex Green’s function $V_{K, θ}$ for θ-incomplete polynomials. We establish properties of $V_{K, θ}$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of $ℝ^{N}$ , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K ∖ s u p p {(d d^{c} V_{K, θ})}^{N}$ . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $s u p p {(d d^{c} V_{K, θ})}^{N}$ when K is a compact...

Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang (2014)

Acta Arithmetica

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We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f (x), g (x) \in_{p} [x]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1 / E (1 + κ / (1 - κ)} > M > p^{ε}$ , where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $| f ([0, M]) \cap g ([0, M]) | ≲ M^{1 - ε}$ .

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_{m}$ be the mth coefficient of the square f(x)² of...

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

On the Gauss-Lucas'lemma in positive characteristic

Umberto Bartocci, Maria Cristina Vipera (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$ , then $f(x)$ has at least one root a with the following property: if $\mu\leq k\leq n$ , where $\mu$ is the multiplicity of $\alpha$ , then $f^{(k)}(\alpha)\neq 0$ (such a root is said to be a "free" root of $f(x)$ ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$ ) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it...

On the distribution of the roots of polynomial $z^{k} - z^{k - 1} - \dots - z - 1$

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

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We consider the polynomial $f_{k} (z) = z^{k} - z^{k - 1} - \dots - z - 1$ for $k \geq 2$ which arises as the characteristic polynomial of the $k$ -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_{k} (z)$ which lie inside the unit disk.

On the lattice of polynomials with integer coefficients: the covering radius in $L_{p} (0, 1)$

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

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The paper deals with the approximation by polynomials with integer coefficients in $L_{p} (0, 1)$ , 1 ≤ p ≤ ∞. Let $P_{n, r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^{r} {(1 - x)}^{r}$ , r ≥ 0, and let $P_{n, r}^{ℤ} \subset P_{n, r}$ be the set of polynomials with integer coefficients. Let $μ (P_{n, r}^{ℤ}; L_{p})$ be the maximal distance of elements of $P_{n, r}$ from $P_{n, r}^{ℤ}$ in $L_{p} (0, 1)$ . We give rather precise quantitative estimates of $μ (P_{n, r}^{ℤ}; L ₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ (P_{n, r}^{ℤ}; L_{p})$ for p ≠ 2. It follows that $μ (P_{n, r}^{ℤ}; L_{p}) ≍ n^{- 2 r - 2 / p}$ as n → ∞. The results...

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Uniqueness results for differential polynomials sharing a set

Soniya Sultana, Pulak Sahoo (2025)

Mathematica Bohemica

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We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${(Q (f))}^{(k)}$ and ${(Q (g))}^{(k)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{'}$ . The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).

Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. II

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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Some hypergeometric polynomials associated with the Lauricella function $F_{D}$ of several variables. I

L. Carlitz, H. M. Srivastava (1976)

Matematički Vesnik

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On sets of polynomials whose difference set contains no squares

Thái Hoàng Lê, Yu-Ru Liu (2013)

Acta Arithmetica

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Let $_{q} [t]$ be the polynomial ring over the finite field $_{q}$ , and let $_{N}$ be the subset of $_{q} [t]$ containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set $A \subseteq_{N}$ for which A-A contains no squares of polynomials. By combining the polynomial Hardy-Littlewood circle method with the density increment technology developed by Pintz, Steiger and Szemerédi, we prove that $D (N) ≪ q^{N} {(l o g N)}^{7} / N$ .

On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$ , $0 \leq p \leq \infty$

Laiyi Zhu, Xingjun Zhao (2022)

Czechoslovak Mathematical Journal

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Let $𝕋_{n}$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $∥ (1 / n) T_{n}^{'} ∥_{p} \leq {∥ T_{n} ∥}_{p}$ for $0 \leq p \leq \infty$ and $T_{n} \in 𝕋_{n}$ . We derive the multivariate version of the result of Golitschek and Lorentz ${∥{|T_{n} cos α + \frac{1}{n} \nabla T_{n} sin α|}_{l_{\infty}^{(m)}}∥}_{p} \leq {∥ T_{n} ∥}_{p}, 0 \leq p \leq \infty$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$ .

On the Gauss-Lucas'lemma in positive characteristic

Umberto Bartocci, Maria Cristina Vipera (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Displaying similar documents to “Symmetric identity for polynomial sequences satisfying A n + 1 ' ( x ) = ( n + 1 ) A n ( x ) ”

Displaying similar documents to “Symmetric identity for polynomial sequences satisfying $A_{n + 1}^{'} (x) = (n + 1) A_{n} (x)$ ”