Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

Godoy, E.; Ronveaux, A.; Zarzo, A.; Area, Iván

doi:10.1016/s0377-0427(97)00137-4

Cited by 56 publications

(42 citation statements)

References 10 publications

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“…In order to derive the explicit formulae for A n,m (s) we use the method known as the Navima algorithm [15,25]. The main feature of this method is to obtain a recurrence relation for the connection coefficients.…”

Section: Theorem 34 the Connection Coefficientsmentioning

confidence: 99%

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

et al. 2004

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Section: Theorem 34 the Connection Coefficientsmentioning

confidence: 99%

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

et al. 2004

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“…As such, the connection problem has been studied extensively in the literature, see for instance [9][10][11][12][13][14][15]. One of the most complete solutions to the connection problem currently is that in [16] (and continued in [17]).…”

Section: Introductionmentioning

confidence: 99%

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

Bella¹,

Reis²

2015

Mathematics

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Abstract:The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change-of-basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change-of-basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix Mathematics 2015, 3 383 are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).

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“…In particular, positivity of the connection coefficients c k = a k [P n ], or the linearization coefficients l k = a k [P m P n ] is of great importance. See [1,2,4,5,9,12,13,[19][20][21][22][23][24][25][26][27][28]33]. Usually, determination of the expansion coefficients a k [f ] requires a deep knowledge of special (hypergeometric) functions.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we give an algorithmic description of the method generalizing ideas of the papers [11]- [13], to construct the recurrence (1.2) provided f is a solution of the differential equation Notice that an alternative approach is proposed in [5,6]; it should be stressed that there exists a class of important problems for which our method gives more refined results, i.e. lower-order recurrences of the type (1.2).…”

Section: Introductionmentioning

confidence: 99%

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Lewanowicz

2002

Appl. Math. (Warsaw)

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Abstract. Let {P k } 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 a k in f = k a k P k . A systematic use of the basic properties (including some nonstandard ones) of the polynomials {P k } results in obtaining a low order of the recurrence.

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Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

Cited by 56 publications

References 10 publications

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

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