The Legendre Functions P v (x) and Q v (x)

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

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ROLES OF LOG-CONCAVITY, LOG-CONVEXITY, AND GROWTH ORDER IN WHITE NOISE ANALYSIS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025701000498 ◽

2001 ◽

Vol 04 (01) ◽

pp. 59-84 ◽

Cited By ~ 6

Author(s):

NOBUHIRO ASAI ◽

IZUMI KUBO ◽

HUI-HSIUNG KUO

Keyword(s):

White Noise ◽

Noise Analysis ◽

Holomorphic Functions ◽

Legendre Transform ◽

Legendre Functions ◽

Growth Order ◽

White Noise Analysis ◽

Systematic Method ◽

Growth Functions ◽

Natural Way

In this paper we will develop a systematic method to answer the questions (Q1) (Q2) (Q3) (Q4) (stated in Sec. 1) with complete generality. As a result, we can solve the difficulties (D1) (D2) (discussed in Sec. 1) without uncertainty. For these purposes we will introduce certain classes of growth functions u and apply the Legendre transform to obtain a sequence which leads to the weight sequence {α(n)} first studied by Cochran et al.6 The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on ℰc will also be discussed.

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Legendre Functions of Large Degree: Real Arguments

Asymptotics and Special Functions ◽

10.1201/9781439864548-154 ◽

1997 ◽

pp. 481-487

Keyword(s):

Large Degree ◽

Legendre Functions

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Spatially Restricted Integrals in Gradiometric Boundary Value Problems

Artificial Satellites ◽

10.2478/v10018-009-0025-4 ◽

2009 ◽

Vol 44 (4) ◽

pp. 131-148 ◽

Cited By ~ 9

Author(s):

M. Eshagh

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Legendre Functions ◽

Gravity Gradiometry ◽

Satellite Gravity Gradiometry ◽

Satellite Gravity ◽

Earth’S Gravity Field ◽

Associated Legendre Functions ◽

Tensor Spherical Harmonics ◽

Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

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