Discrete classical orthogonal polynomials

Kwon, Kil Hyun; Dw, Lee; Sb, Park; Yoo, Byueng–Su

doi:10.1080/10236199808808134

Cited by 15 publications

(11 citation statements)

References 11 publications

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“…(1.3)] Hahn introduced the difference operator D q,ω , that extends both Jackson's q-difference operator, D q , and the difference operator Δ ω ; see definitions below. Hahn's operator has been considered from computational points of view, in particular in the determination of new families of orthogonal polynomials; see, e.g., [2][3][4][5]. Calculi, difference equations and special functions based on the q-difference operator, as well as the difference operator Δ ω , have been considered extensively; see, e.g., the monographs [6][7][8][9][10][11] and the papers [1,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Annaby

Hamza

Aldwoah

2012

J Optim Theory Appl

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This paper is devoted for a rigorous investigation of Hahn's difference operator and the associated calculus. Hahn's difference operator generalizes both the difference operator and Jackson's q-difference operator. Unlike these two operators, the calculus associated with Hahn's difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn's difference operator. We construct a right inverse of Hahn's operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.Keywords Hahn's operator · Difference equations · q-Difference equations · Nörlund sums · Jackson q-integral M.H. Annaby is on leave from:

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Section: Introductionmentioning

confidence: 99%

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Annaby

Hamza

Aldwoah

2012

J Optim Theory Appl

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“…(1.2) Orthogonal polynomials satisfying (1.1) are known as discrete classical orthogonal polynomials and they are well studied [6,13,15,16,19,23]. Like classical orthogonal polynomials satisfying second-order differential equations of hypergeometric type, discrete classical orthogonal polynomials can be characterized in many different ways (see [1 5, 7, 8, 10, 14, 18]).…”

Section: Introductionmentioning

confidence: 99%

New Characterizations of Discrete Classical Orthogonal Polynomials

Kwon

Lee

Park

1997

Journal of Approximation Theory

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We prove that if both [P n (x)] n=0 and [{ r P n (x)] n=r are orthogonal polynomials for any fixed integer r 1, then [P n (x)] n=0 must be discrete classical orthogonal polynomials. This result is a discrete version of the classical Hahn's theorem stating that if both [P n (x)] n=0 and [(dÂdx) r P n (x)] n=r are orthogonal polynomials, then [P n (x)] n=0 are classical orthogonal polynomials. We also obtain several other characterizations of discrete classical orthogonal polynomials.1997 Academic Press

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“…Lately we encounter the growth of interest to the study of orthogonal polynomials relative an arbitrary (but still symmetric and nondegenerate) form, cf. [4,7,8] and references therein. In these approaches, however, the bilinear forms are introduced "by hands" and the differential or difference equations the orthogonal polynomials satisfy are of high degree.…”

Section: Introductionmentioning

confidence: 99%

Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate

Сергеев¹

2001

JNMP

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Let A be an associative simple (central) superalgebra over C and L an invariant linear functional on it (trace). Let a → a t be an antiautomorphism of A such that (2))/m, where m is any maximal ideal of U (sl(2)), Leites and I have constructed orthogonal basis in A whose elements turned out to be, essentially, Chebyshev (Hahn) polynomials in one discrete variable. Here I take A = U (osp(1|2))/m for any maximal ideal m and apply a similar procedure. As a result we obtain either Hahn polynomials over C[τ ], where τ 2 ∈ C, or a particular case of Meixner polynomials, or -when A = Mat(n + 1|n) -dual Hahn polynomials of even degree, or their (hopefully, new) analogs of odd degree. Observe that the nondegenerate bilinear forms we consider for orthogonality are, as a rule, not sign definite.

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Discrete classical orthogonal polynomials

Cited by 15 publications

References 11 publications

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

New Characterizations of Discrete Classical Orthogonal Polynomials

Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate

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