Polynomial Expansions of Analytic Functions

Boas, R. P.; Buck, R. C.

doi:10.1007/978-3-642-87887-9

Cited by 115 publications

(72 citation statements)

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“…Due to the fact that the defining property (ii) of Appell sets implies automatically a direct link to a corresponding exponential function ( [11]) and combining this fact with a result of [9] about the relationship to Bessel functions we have the following result.…”

Section: Binomial Sums and An Appell Set Of Polynomials In R N+1mentioning

confidence: 95%

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Clifford Analysis between Continuous and Discrete

Malonek¹,

Falcão²

2008

AIP Conference Proceedings

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Some decades ago D. Knuth et al. have coined concrete mathematics as the blending of CONtinuous and disCRETE math, taking into account that problems of standard discrete mathematics can often be solved by methods based on continuous mathematics together with a controlled manipulation of mathematical formulas. Of course, it was not a new idea, but due to the ongoing emergence of computer aided algebraic manipulation tools of that time it emphasized their use for elegant solutions of old problems or even the detection of new important relationships. Our aim is to show that the same philosophy can be successfully applied to Clifford Analysis by taking advantages of its inherent non-commutative algebra to obtain results or develop methods that are different from other ones. In particular, we determine new binomial sums by using a hypercomplex generating function for a special type of monogenic polynomials and develop an algorithm for the determination of their scalar and vector part which illustrates well the differences to the corresponding complex case.

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Section: Binomial Sums and An Appell Set Of Polynomials In R N+1mentioning

confidence: 95%

“…In particular, they behave like power-law functions under the differentiation operation (properties (i) and (ii) of the list, which characterize a set of Appell polynomials ; see e.g. [10,11]). In detail, it holds for arbitrary dimension n ≥ 1…”

Section: Binomial Sums and An Appell Set Of Polynomials In R N+1mentioning

confidence: 99%

Clifford Analysis between Continuous and Discrete

Malonek¹,

Falcão²

2008

AIP Conference Proceedings

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“…This commutation relation shows, in order to find a discrete operator of "x" we have to consider operation of multiplication of analytical function f (x) on function exp(bx) and write this operation in the space of coefficients of the Taylor expansion [5]. The product of two functions f (x) and g(x) is defined as follows…”

Section: )] Under Action Of the Pascal Matrix According To Formulamentioning

confidence: 99%

Pascal Matrix Representation of Evolution of Polynomials

Yamaleev

2015

Int. J. Appl. Comput. Math

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Pascal matrix is an adjoint operator of the differential operator of translation. This feature of the Pascal matrix is used in order to construct evolution equations for coefficients of polynomials induced by shifts of the roots. Under certain initial data solutions of the evolution equations are given by sequences of the Appell polynomials. The Pascal matrix is applied to calculate coefficients of the invariant polynomial and to construct a new algorithm of deflation.

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“…The use of generating functions to characterize sequences is a common resource for transforming problems about sequences into problems involving functions formal power series. Following [6], Appell polynomials are characterized by a generating function of the form…”

Section: Preliminariesmentioning

confidence: 99%

Matrix Approach to Frobenius-Euler Polynomials

Tomaz

Malonek

2014

Computational Science and Its Applications – ICCSA 2014

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Abstract. In the last two years Frobenius-Euler polynomials have gained renewed interest and were studied by several authors. This paper presents a novel approach to these polynomials by treating them as Appell polynomials. This allows to apply an elementary matrix representation based on a nilpotent creation matrix for proving some of the main properties of Frobenius-Euler polynomials in a straightforward way.

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Polynomial Expansions of Analytic Functions

Cited by 115 publications

References 0 publications

Clifford Analysis between Continuous and Discrete

Clifford Analysis between Continuous and Discrete

Pascal Matrix Representation of Evolution of Polynomials

Matrix Approach to Frobenius-Euler Polynomials

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