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Spaces of functions satisfying simple differential equations. KonradZuseZentrum Berlin (ZIB
, 1994
"... In [6]–[9] the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series ∞∑ akzk at the origin and vice versa, implementations of which exist in Mathematica [19], (s. [9]), Maple [12] (s. [4]) and Reduce [5] ..."
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In [6]–[9] the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series ∞∑ akzk at the origin and vice versa, implementations of which exist in Mathematica [19], (s. [9]), Maple [12] (s. [4]) and Reduce [5] (s. [13]). One main part of this procedure is an algorithm to derive a homogeneous linear differential equation with polynomial coefficients for the given function. We call this type of ordinary differential equations simple. Whereas the opposite question to find functions satisfying given differential equations is studied in great detail, our question to find differential equations that are satisfied by given functions seems to be rarely posed. In this paper we consider the family F of functions satisfying a simple differential equation generated by the rational, the algebraic, and certain transcendental functions. It turns out that F forms a linear space of transcendental functions. Further F is closed under multiplication and under the composition with rational functions and rational powers. These results had been published by Stanley who had proved them by theoretical algebraic considerations. In contrast our treatment is purely algorithmically oriented. We present algorithms that generate simple differential equation for f +g, f ·g, f ◦r (r rational), and f ◦x p/q (p, q ∈ IN0), given simple differential equations for f, and g, and give a priori estimates for the order of the resulting differential equations. We show that all order estimates are sharp. After finishing this article we realized that in independent work Salvy and Zimmermann published similar algorithms. Our treatment gives a detailed description of those algorithms and their validity. k=0 1 1 Simple
On the De Branges Theorem
 COMPLEX VARIABLES
, 1995
"... Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary p ..."
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Cited by 9 (8 self)
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Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. In this article, we present an elementary proof of this statement based on the defining differential equations system rather than the closed representation of de Branges' function system. Our proof does neither use special functions (like Wilf's) nor the residue theorem (like Todorov's) nor the closed representation (like both), but is purely algebraic. On the other hand, by a similar algebraic treatment, the closed representation of de Branges' function system is derived. Our whole contribution can be looked at as the study of properties of the Koebe function. Therefore, in a very elementary manner it is shown that the known proofs of the Bieberbach and Milin conjectures can be understood as a consequence of the Lowner differential ...
WKB Approximation And KrallType Orthogonal Polynomials
 Acta Applicandae Mathematicae
, 1996
"... We give an unified approach to the Kralltype polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one "perturbed" by the addition of one or more delta Dirac functions. Some examples studied by different authors are considered from an unique point of view. ..."
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We give an unified approach to the Kralltype polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one "perturbed" by the addition of one or more delta Dirac functions. Some examples studied by different authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The threeterm recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order differential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied. 1 Introduction. In this work we present a survey and some new results relative to the Krall type orthogonal polynomials, i.e., polynomials with are orthogonal with respect to an absolutely continuous measu...
The Algebra of Holonomic Equations
, 1997
"... In this article algorithmic methods are presented that have essentially been introduced into computer algebra systems like Maple or Mathematica within the last decade. The main ideas are due to Stanley and Zeilberger. Some of them had already been discovered in the last century by Beke, but because ..."
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In this article algorithmic methods are presented that have essentially been introduced into computer algebra systems like Maple or Mathematica within the last decade. The main ideas are due to Stanley and Zeilberger. Some of them had already been discovered in the last century by Beke, but because of their complexity the underlying algorithms have fallen into oblivion. We give a survey of these techniques, show how they can be used to identify transcendental functions, and present implementations of these algorithms in computer algebra systems.
On Families of Iterated Derivatives
"... : We give an overview of an approach on special functions due to Truesdell, and show how it can be used to develop certain type of identities for special functions. Once obtained, these identities may be verified by an independent algorithmic method for which we give some examples. 1 The Fequation ..."
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: We give an overview of an approach on special functions due to Truesdell, and show how it can be used to develop certain type of identities for special functions. Once obtained, these identities may be verified by an independent algorithmic method for which we give some examples. 1 The Fequation and Ffunctions Truesdell [11] studied solutions of the functional equation @ @z F (z; ff) = F (z; ff + 1) (1) satisfying the initial condition F (z 0 ; ff) = \Phi(ff) (2) where F is a function of the two variables z and ff. Here z is assumed to be a real or complex variable, ff is such that either ff = ff 0 + k (k 2 IN 0 ) ; or ff = ff 0 + k (k 2 ZZ) ; or ff ff 0 (ff 2 IR) ; or Re ff ff 0 (ff 2 C) ; (3) (ff 0 may equal \Gamma1), and \Phi is a given function of ff. Equation (1) is called the F equation, and we call a solution of the Fequation that satisfies the initial condition (2) an F function corresponding to the initial function \Phi. Truesdell showed how a functional equat...
Power Series, Bieberbach Conjecture and the de Branges and Weinstein Functions
, 2003
"... It is wellknown that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. These hypergeometric polynomials had been already studied by Askey and Gaspe ..."
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It is wellknown that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had introduced in his work. These hypergeometric polynomials had been already studied by Askey and Gasper who had realized their positiveness. This fact was the essential tool in de Branges' proof. In this article, we show that many identities, e.g. the representation of their generating function, for these polynomials, which are intimately related to the Koebe function and therefore to univalent functions, can be automatically detected from power series computations by a method developed by the author and accessible in several computer algebra systems. In other words, in this paper a new and interesting application of the FPS algorithm is given. As working engine we use a Maple implementation by Dominik Gruntz and the author. In particular, the hypergeometric representation of the de Branges and Weinstein functions are determined by successive power series computations. In a final section we show how algebraic computation enables the fast verification of the positivity results using Sturm sequences or similar approaches.
Computer Algebra Algorithms for Orthogonal Polynomials and Special Functions
"... ples. Let us remark that as a general reference we use the book [11], the computer algebra system Maple [16], [4] and the Maple packages FPS [9], [7], gfun [19], hsum [11], infhsum [22], hsols [21], qsum [2] and retode [13]. E. Koelink and W. Van Assche (Eds.): LNM 1817, pp. 124, 2003. Springer ..."
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ples. Let us remark that as a general reference we use the book [11], the computer algebra system Maple [16], [4] and the Maple packages FPS [9], [7], gfun [19], hsum [11], infhsum [22], hsols [21], qsum [2] and retode [13]. E. Koelink and W. Van Assche (Eds.): LNM 1817, pp. 124, 2003. SpringerVerlag Berlin Heidelberg 2003 functions .................................................. 2 1.1 Hypergeometricseries........................................ 3 1.2 Holonomicdi#erentialequations............................... 4 1.3 Algebraofholonomicfunctions................................ 4 1.4 Hypergeometricpowerseries.................................. 5 1.5 Identificationofhypergeometricfunctions ...................... 5 2 Summation of hypergeometric series ...................... 6 2.1 Fasenmyer'smethod ......................................... 6 2.2 IndefinitesummationandGosper'salgorithm................... 7 2.3 Zeilberger'salgorithm........................................ 8 2.4 Agene
On Some Polynomial Mappings for Measures. Applications
, 1997
"... Let fPn gn0 be a given system of monic orthogonal polynomials, ß k and ` m two fixed monic polynomials of degrees k and m (0 m k \Gamma 1), respectively, and fQngn0 a simple set of monic polynomials such that Q kn+m (x) = ` m (x)Pn (ß k (x)); for all n = 0; 1; 2; : : :. The cases when T (x) is a ..."
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Let fPn gn0 be a given system of monic orthogonal polynomials, ß k and ` m two fixed monic polynomials of degrees k and m (0 m k \Gamma 1), respectively, and fQngn0 a simple set of monic polynomials such that Q kn+m (x) = ` m (x)Pn (ß k (x)); for all n = 0; 1; 2; : : :. The cases when T (x) is a polynomial of degree 2 or 3 is considered. We find necessary and sufficient conditions in order that fQn gn0 be a system of orthogonal polynomials and give the relation between the moment functionals associated with the orthogonal systems fPn gn0 and fQngn0 . In particular, we characterize the positive definite case. Some applications to the solution of the eigenvalue problem for a tridiagonal 2 or 3Toeplitz matrix, as well as to a quantum physics model are given. Key words and Phrases: Orthogonal Polynomials, Recurrence coefficients, Polynomial mappings, Toeplitz matrices. 1 Introduction and preliminaries Let us propose the general problem: P. Let fP n g n0 be a given system of monic or...
Photocopying permitted by license only On the De Branges Theorem
, 2009
"... Reprints available directly from the publisher ..."
Federal Republic of Germany
, 1994
"... In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities l ..."
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In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i. e. they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the generalized hypergeometric functions, and therefore most of the special functions that can be found in mathematical dictionaries. The types of identities for which we present verification algorithms cover differential equations, power series representations, identities of the Rodrigues type, hypergeometric representations, and algorithms containing symbolic sums. The current implementations of special functions in existing Computer Algebra systems do not meet these high standards as we shall show in examples. They should be modified, and we show results of our implementations. 1