Jorge Sánchez-Ruiz 2003 J. Phys. A: Math. Gen. 36 4857 doi:10.1088/0305-4470/36/17/312
Information entropy of Gegenbauer polynomials and Gaussian quadrature
Jorge Sánchez-Ruiz
Show affiliationsIn a recent paper (Buyarov V S, López-Artés P, Martínez-Finkelshtein A and Van Assche W 2000 J. Phys. A: Math. Gen. 33 6549–60), an efficient method was provided for evaluating in closed form the information entropy of the Gegenbauer polynomials C(λ)n(x) in the case when λ = l 
. For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, P(x) and H(x), of degrees 2l − 2 and 2l − 4, respectively. Here it is shown that P(x) is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights wl(x) = (1 − x2)l−1/2, and this fact is used to obtain the explicit expression of P(x). From this result, an explicit formula is also given for the polynomial S(x) = limn→∞ P(1 − x/(2n2)), which is relevant to the study of the asymptotic (n → ∞ with l fixed) behaviour of the entropy.
- PACS
- MSC
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94A17 Measures of information, entropy
- Subjects
- Dates
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Issue 17 (2 May 2003)
Received 28 October 2002, in final form 4 March 2003
Published 16 April 2003
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Information entropy of Gegenbauer polynomials and Gaussian quadrature
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