scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Über Orthogonalpolynome, die q-Differenzengleichungen genügen

01 Jan 1949-Mathematische Nachrichten (John Wiley & Sons, Ltd)-Vol. 2, pp 4-34
About: This article is published in Mathematische Nachrichten.The article was published on 1949-01-01. It has received 402 citations till now.
Citations
More filters
Posted Content
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
Abstract: We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

1,459 citations


Additional excerpts

  • ...[8], [10], [18], [19], [24], [25], [31], [111], [114], [121], [127], [134], [140], [145], [158], [160], [161], [163], [172], [176], [180], [197], [212], [214]....

    [...]

  • ...[10], [25], [43], [45], [88], [111], [114], [121], [142], [145], [158], [160], [180], [197], [215], [217], [218]....

    [...]

  • ...[8], [10], [25], [114], [121], [127], [140], [142], [160], [163], [176], [180], [181], [212]....

    [...]

Book ChapterDOI
01 Jan 1985
TL;DR: The classical orthogonal polynomials have been defined in this paper, and a number of orthogonality relations for some of the classical polynomial classes have been established.
Abstract: There have been a number of definitions of the classical orthogonal polynomials, but each definition has left out some important orthogonal polynomials which have enough nice properties to justify including them in the category of classical orthogonal polynomials. We summarize some of the previous work on classical orthogonal polynomials, state our definition, and give a few new orthogonality relations for some of the classical orthogonal polynomials.

532 citations

01 Jan 2000
TL;DR: In this paper, the authors considered the problem of additive number theory for the Riemann surface and showed that the solution of the problem can be expressed as a polynomial series.
Abstract: Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

245 citations


Cites background or methods from "Über Orthogonalpolynome, die q-Diff..."

  • ...These functions play in the theory of q-difference equations the rôle of the arbitrary constants of the differential equations [399], [166]....

    [...]

  • ...We can say the following about the convergence of the q-series pφr Hahn 1949 [401]: The series of type (p, p−1) converge for |z| < 1 when 0 < |q| < 1; because the coefficient of zn is bounded and tends to 1....

    [...]

Journal ArticleDOI
TL;DR: It is shown how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction and incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals.
Abstract: This paper shows how Hahn moments provide a unified understanding of the recently introduced Chebyshev and Krawtchouk moments. The two latter moments can be obtained as particular cases of Hahn moments with the appropriate parameter settings and this fact implies that Hahn moments encompass all their properties. The aim of this paper is twofold: (1) To show how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction and (2) to show how Hahn moments can be incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals.

192 citations

References
More filters
Book
01 Jul 1965
TL;DR: Koornwinder as discussed by the authors gave identitity (2.5) with N = 0 and formulas (5.3), 5.3, and 5.4) substituted.
Abstract: This also gives in the paper T. H. Koornwinder, Orthogonal polynomials with weight function (1− x)α(1 + x)β + Mδ(x + 1) + Nδ(x− 1), Canad. Math. Bull. 27 (1984), 205–214 the identitity (2.5) with N = 0 and formulas (5.3), (5.4) substituted. p.95, §10.4, formula (7): second line: replace in denominator (v + n− 1)(w + n− 1) by Γ(v + n− 1)Γ(w + n− 1); third line: replace in denominator Γ(v + n− 1) by (v + n− 1); fifth line: replace in denominator Γ(w + n− 1) by (w + n− 1).

1,562 citations

Journal ArticleDOI
TL;DR: In this paper, the authors show how Riemann mit der nahe verwandten Gamsischen Reihe F(a, b, c, x) gethan hat.
Abstract: welche derselbe Bd. 34, pag. 285 dieses Journals ausführlich abgehandelt hat, von einem ähnlichen Gesichtspunkte aus betrachtet werden, wie es Riemann mit der nahe verwandten Gamsischen Reihe F(a, b, c, x) gethan hat. Bei dieser Behandlung ergiebt sich die von Herrn Heine in der angeführten Abhandlung (pag. 325) gefundene Transformation als eine unmittelbare Folge der Definition, und gelingt es (Art. 4), den noch fehlenden Zusammenhang zwischen je drei beliebigen Integralen der Differenzengleichung, welcher die Reihe genügt, aufzufinden, und sowohl hierdurch, als auch durch directe Darstellung (Art, 5, Gl. 22), in jedem Falle die Fortsetzung einer Heineschen Reihe als Function ihres fünften Elementes über ihr Convergenzgebiet hioaus anzugeben, was in speciellen Fällen direct, oder, wenn man die Relationen (Art. 6) zwischen contiguen Funetionen anwendet, auch allgemein durch die Gleichung (78.) der Abhandlung des Herrn Heine geleistet werden kann. Die Formen, unter welchen sich die Heinesche Reihe darstellen lässt, konnten hier vermehrt werden (Art. 5, Gl. 22,23.). Die von Herrn Heine eingeführte Bezeichnung ist hier und da, abgeändert worden, damit die Analogie mit der Cra«mischen Reihe, namentlich mit deren Behandlung durch Riemann (im 7 Bande der Abhandlungen der Königl. Gesellschaft der Wissenschaften zu Göttingen), welche hier stärker hervortreten soll, auch äusserlich werde. Die angefügte Kettenbruchentwicklung stimmt zwar in ihren Resultaten mit den von Herrn Heine erhaltenen *) überein, durfte aber der Vollständigkeit halber nicht fehlen, und ist eine directe Uebertragung der schönen Methode

101 citations