Some hypergeometric polynomials associated with the Lauricella function of several variables. II
L. Carlitz, H. M. Srivastava (1976)
Matematički Vesnik
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L. Carlitz, H. M. Srivastava (1976)
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Joe Callaghan (2007)
Annales Polonici Mathematici
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Let K be any subset of . We define a pluricomplex Green’s function for θ-incomplete polynomials. We establish properties of analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute when K is a compact...
Stanislaw Lewanowicz (2002)
Applicationes Mathematicae
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Let 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 in . A systematic use of the basic properties (including some nonstandard ones) of the polynomials results in obtaining a low order of the recurrence.
Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be the uniform norm in the unit disk. We study the quantities where the infimum is taken over all polynomials of degree with and . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that . We find the exact values of and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
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 group. In order to get the two identities, we first present two known -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two -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...
Thomas Ernst (2007)
Rendiconti del Seminario Matematico della Università di Padova
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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.
Wojciech Banaszczyk, Artur Lipnicki (2015)
Annales Polonici Mathematici
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The paper deals with the approximation by polynomials with integer coefficients in , 1 ≤ p ≤ ∞. Let be the space of polynomials of degree ≤ n which are divisible by the polynomial , r ≥ 0, and let be the set of polynomials with integer coefficients. Let be the maximal distance of elements of from in . We give rather precise quantitative estimates of for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of for p ≠ 2. It follows that as n → ∞. The results...
Peter Borwein, Tamás Erdélyi, Géza Kós (2013)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , 1/paj ∈ ℂsuch that divides P(x). For n ∈ ℕ and L > 0 let be the largest possible value of k for which there is a polynomial P ≠ 0 of the form , , , such that divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...
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 in the special case where is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that is monic and linear.
Tamás Erdélyi (2001)
Colloquium Mathematicae
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Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials of the form , , by , (here 0/0 is interpreted as 1). We define the norms of the truncation operators by , . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...
Iwona Naraniecka, Jan Szynal, Anna Tatarczak (2013)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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The extremal functions realizing the maxima of some functionals (e.g. , and ) within the so-called universal linearly invariant family (in the sense of Pommerenke [10]) have such a form that looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials of a real variable as coefficients of where the parameters , , satisfy the conditions:...
Pradipto Banerjee, Michael Filaseta, Carrie E. Finch, J. Russell Leidy (2013)
Journal de Théorie des Nombres de Bordeaux
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We show that the discriminant of the generalized Laguerre polynomial is a non-zero square for some integer pair , with , if and only if belongs to one of explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of over is the alternating group . For example, we establish that for all but finitely many positive integers , the only for which the Galois group of over is is...
Giacomo Gigante (2010)
Colloquium Mathematicae
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Let be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞ , . Then, for any non-negative integer n, . When , this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of allow one to write all these type of formulas...
Maurizio Monge (2014)
Journal de Théorie des Nombres de Bordeaux
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Let be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree over , and extend it to when the base fields is an unramified extension of . When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We...
M. Đurić (1973)
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M. K. Aouf (1988)
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Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)
Acta Arithmetica
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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in of degree d for which s consecutive coefficients are fixed. Our estimate asserts that , where . We also prove that , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of of degree d with s consecutive coefficients fixed as above. Finally, we show that , where ₂(d,0) denotes the average second moment for...
K. Orlov (1981)
Matematički Vesnik
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(2016)
Acta Arithmetica
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For n ∈ ℕ, L > 0, and p ≥ 1 let be the largest possible value of k for which there is a polynomial P ≢ 0 of the form , , , such that divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that . We find the size of and for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...
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 is a polynomial with coefficients in the field of complex numbers, of positive degree , then has at least one root a with the following property: if , where is the multiplicity of , then (such a root is said to be a "free" root of ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree ) with coefficients in a field of positive characteristic (Sudbery's Conjecture). In this paper it...