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Язык оригинала | английский |
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Страницы (с-по) | 233-240 |

Журнал | Vestnik St. Petersburg University: Mathematics |

Номер выпуска | 4 |

DOI | |

Состояние | Опубликовано - 2015 |

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**Asymptotics behavior of integrals of the legendre polynomials and their sums.** / Kholshevnikov, K.V.; Shaidulin, V.S.

Результат исследований: Научные публикации в периодических изданиях › статья

TY - JOUR

T1 - Asymptotics behavior of integrals of the legendre polynomials and their sums

AU - Kholshevnikov, K.V.

AU - Shaidulin, V.S.

PY - 2015

Y1 - 2015

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.

U2 - 10.3103/S1063454115040068

DO - 10.3103/S1063454115040068

M3 - Article

SP - 233

EP - 240

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -