Displaying similar documents to “Zeros of a certain class of Gauss hypergeometric polynomials”

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence { P i } i = 1 generated by a three-term recurrence relation P i ( x ) + Q 1 ( x ) P i - 1 ( x ) + Q 2 ( x ) P i - 2 ( x ) = 0 with the standard initial conditions P 0 ( x ) = 1 , P - 1 ( x ) = 0 , where Q 1 ( x ) and Q 2 ( x ) are arbitrary real polynomials.

Zero points of quadratic matrix polynomials

Opfer, Gerhard, Janovská, Drahoslava

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Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial 𝐩 , i.e., the terms have the form 𝐀 j 𝐗 j 𝐁 j , where all quantities 𝐗 , 𝐀 j , 𝐁 j , j = 0 , 1 , ... , N , are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial 𝐩 by a matrix...

L p inequalities for the growth of polynomials with restricted zeros

Nisar A. Rather, Suhail Gulzar, Aijaz A. Bhat (2022)

Archivum Mathematicum

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Let P ( z ) = ν = 0 n a ν z ν be a polynomial of degree at most n which does not vanish in the disk | z | < 1 , then for 1 p < and R > 1 , Boas and Rahman proved P ( R z ) p ( R n + z p / 1 + z p ) P p . In this paper, we improve the above inequality for 0 p < by involving some of the coefficients of the polynomial P ( z ) . Analogous result for the class of polynomials P ( z ) having no zero in | z | > 1 is also given.

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial p ( z ) = c n z n + j = μ n c n - j z n - j , 1 μ n , having all its zeros on | z | = k , k 1 , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

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).

Linear maps preserving elements annihilated by the polynomial X Y - Y X

Jianlian Cui, Jinchuan Hou (2006)

Studia Mathematica

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Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by T the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies Φ ( A ) Φ ( B ) = Φ ( B ) Φ ( A ) for all A, B ∈ ℬ(H) with A B = B A if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that Φ ( A ) = c U A U for all A ∈ ℬ(H).

The multiplicity of the zero at 1 of polynomials with constrained coefficients

Peter Borwein, Tamás Erdélyi, Géza Kós (2013)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p 1/p , aj ∈ ℂ , such that ( x - 1 ) k divides P(x). For n ∈ ℕ and L > 0 let κ ( n , L ) be the largest possible value of k for which there is a polynomial P ≠ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L m a x 1 j n | a j | , a j , such that ( x - 1 ) k divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that c 1 ( n / L ) - 1 κ ( n , L ) c 2 ( n / L ) for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number q > 1 and a positive integer m , let Y m ( q ) : = i = 0 n ϵ i q i : ϵ i 0 , ± 1 , ... , ± m , n = 0 , 1 , ... . In this paper, we show that Y m ( q ) is dense in if and only if q < m + 1 and q is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

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 ) + + a ( z ) f ' + D ( z ) f = 0 , (1) where D ( z ) = Q ( z ) e P ( z ) + Q ( z ) e P ( z ) + Q ( z ) e P ( 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.

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

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Let P be a polynomial with integral coefficients. Shapiro showed that if the values of P at infinitely many blocks of consecutive integers are of the form Q ( m ) , where Q is a polynomial with integral coefficients, then P ( x ) = Q ( R ( x ) ) for some polynomial R . In this paper, we show that if the values of P at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form m q where q is an integer greater than 1, then P ( x ) = ( R ( x ) ) q for some polynomial R ( x ) .

On nonsingular polynomial maps of ℝ²

Nguyen Van Chau, Carlos Gutierrez (2006)

Annales Polonici Mathematici

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We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity ( J ) : There does not exist a sequence ( p k , q k ) ² of complex singular points of F such that the imaginary parts ( ( p k ) , ( q k ) ) tend to (0,0), the real parts ( ( p k ) , ( q k ) ) tend to ∞ and F ( ( p k ) , ( q k ) ) ) a ² . It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition ( J ) and if, in addition, the restriction of F to every real level set P - 1 ( c ) is proper for values of |c| large enough.

On the distribution of the roots of polynomial z k - z k - 1 - - 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 - - z - 1 for k 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.

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...

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality p ' [ - 1 , 1 ] 1 2 p [ - 1 , 1 ] for a constrained polynomial p of degree at most n , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval ( - 1 , 1 ) and establish a new asymptotically sharp inequality. ...

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

Didier D&amp;#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 | f | 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 .

Maximum number of limit cycles for generalized Liénard polynomial differential systems

Aziza Berbache, Ahmed Bendjeddou, Sabah Benadouane (2021)

Mathematica Bohemica

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We consider limit cycles of a class of polynomial differential systems of the form x ˙ = y , y ˙ = - x - ε ( g 21 ( x ) y 2 α + 1 + f 21 ( x ) y 2 β ) - ε 2 ( g 22 ( x ) y 2 α + 1 + f 22 ( x ) y 2 β ) , where β and α are positive integers, g 2 j and f 2 j have degree m and n , respectively, for each j = 1 , 2 , and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center x ˙ = y , y ˙ = - x using the averaging theory of first and second order.

On a generalization of the Beiter Conjecture

Bartłomiej Bzdęga (2016)

Acta Arithmetica

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We prove that for every ε > 0 and every nonnegative integer w there exist primes p 1 , . . . , p w such that for n = p 1 . . . p w the height of the cyclotomic polynomial Φ n is at least ( 1 - ε ) c w M n , where M n = i = 1 w - 2 p i 2 w - 1 - i - 1 and c w is a constant depending only on w; furthermore l i m w c w 2 - w 0 . 71 . In our construction we can have p i > h ( p 1 . . . p i - 1 ) for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Entire functions of exponential type not vanishing in the half-plane 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 1 , and let Q ( z ) : = z n P ( 1 / 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 ) | 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.

An a b c d theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u , v , in term of the degree of u , v , which is sharp whenever u , v have few distinct zeros and poles compared to their degree. This sharpens the “ a b c d -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic...

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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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 μ k n , where μ is the multiplicity of α , then f ( k ) ( α ) 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...

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Generalizations of Milne’s U ( n + 1 ) q -Chu-Vandermonde summation

Jian-Ping Fang (2016)

Czechoslovak Mathematical Journal

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We derive two identities for multiple basic hyper-geometric series associated with the unitary U ( n + 1 ) group. In order to get the two identities, we first present two known q -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two U ( n + 1 ) q -Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial...