Asymptotics behavior of integrals of the legendre polynomials and their sums

Результат исследований: Научные публикации в периодических изданияхстатья

2 Цитирования (Scopus)

Выдержка

© 2015, Allerton Press, Inc.Asymptotic representations of the integrals Pnk(cosθ) of the Legendre polynomials and their sums with respect to the first subscript from n + 1 to infinity are investigated. Here, (Formula Presented) It is shown that the asymptotic behavior of Pnk as n → ∞ is similar to that of Pn. However, the summand of order n–k–m–1/2 can be represented as a linear combination of m, rather than one, cosines of the form (Formula Presented) where s1 and s2 are integers depending only on k and m. For the sum Pnk with respect to the first subscript from 0 to ∞, closed expressions are obtained. The asymptotic behavior of sums from n + 1 to ∞ is described. Its form differs from that of Pnk only by the additional multiplier cotθ/2. As a byproduct, a Mehler–Dirichlet-type integral for Pnk(cosθ) is obtained.
Язык оригиналаанглийский
Страницы (с-по)233-240
ЖурналVestnik St. Petersburg University: Mathematics
Номер выпуска4
DOI
СостояниеОпубликовано - 2015

Отпечаток

Legendre polynomial
Subscript
Asymptotic Behavior
cos integral
Asymptotic Representation
Multiplier
Linear Combination
Infinity
Closed
Integer

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abstract = "{\circledC} 2015, Allerton Press, Inc.Asymptotic representations of the integrals Pnk(cosθ) of the Legendre polynomials and their sums with respect to the first subscript from n + 1 to infinity are investigated. Here, (Formula Presented) It is shown that the asymptotic behavior of Pnk as n → ∞ is similar to that of Pn. However, the summand of order n–k–m–1/2 can be represented as a linear combination of m, rather than one, cosines of the form (Formula Presented) where s1 and s2 are integers depending only on k and m. For the sum Pnk with respect to the first subscript from 0 to ∞, closed expressions are obtained. The asymptotic behavior of sums from n + 1 to ∞ is described. Its form differs from that of Pnk only by the additional multiplier cotθ/2. As a byproduct, a Mehler–Dirichlet-type integral for Pnk(cosθ) is obtained.",
author = "K.V. Kholshevnikov and V.S. Shaidulin",
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Asymptotics behavior of integrals of the legendre polynomials and their sums. / Kholshevnikov, K.V.; Shaidulin, V.S.

В: Vestnik St. Petersburg University: Mathematics, № 4, 2015, стр. 233-240.

Результат исследований: Научные публикации в периодических изданияхстатья

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N2 - © 2015, Allerton Press, Inc.Asymptotic representations of the integrals Pnk(cosθ) of the Legendre polynomials and their sums with respect to the first subscript from n + 1 to infinity are investigated. Here, (Formula Presented) It is shown that the asymptotic behavior of Pnk as n → ∞ is similar to that of Pn. However, the summand of order n–k–m–1/2 can be represented as a linear combination of m, rather than one, cosines of the form (Formula Presented) where s1 and s2 are integers depending only on k and m. For the sum Pnk with respect to the first subscript from 0 to ∞, closed expressions are obtained. The asymptotic behavior of sums from n + 1 to ∞ is described. Its form differs from that of Pnk only by the additional multiplier cotθ/2. As a byproduct, a Mehler–Dirichlet-type integral for Pnk(cosθ) is obtained.

AB - © 2015, Allerton Press, Inc.Asymptotic representations of the integrals Pnk(cosθ) of the Legendre polynomials and their sums with respect to the first subscript from n + 1 to infinity are investigated. Here, (Formula Presented) It is shown that the asymptotic behavior of Pnk as n → ∞ is similar to that of Pn. However, the summand of order n–k–m–1/2 can be represented as a linear combination of m, rather than one, cosines of the form (Formula Presented) where s1 and s2 are integers depending only on k and m. For the sum Pnk with respect to the first subscript from 0 to ∞, closed expressions are obtained. The asymptotic behavior of sums from n + 1 to ∞ is described. Its form differs from that of Pnk only by the additional multiplier cotθ/2. As a byproduct, a Mehler–Dirichlet-type integral for Pnk(cosθ) is obtained.

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