scholarly journals On Legendre numbers

1985 ◽  
Vol 8 (2) ◽  
pp. 407-411 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
General Formula ◽  
Legendre Polynomials ◽  
Rational Numbers ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Infinite Set ◽  
Derivatives Of

The Legendre numbers, an infinite set of rational numbers are defined from the associated Legendre functions and several elementary properties are presented. A general formula for the Legendre numbers is given. Applications include summing certain series of Legendre numbers and evaluating certain integrals. Legendre numbers are used to obtain the derivatives of all orders of the Legendre polynomials atx=1.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

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.

scholarly journals Some applications of Legendre numbers

1988 ◽  
Vol 11 (2) ◽  
pp. 405-412 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
Legendre Polynomials ◽  
Legendre Functions ◽  
Linear Combinations ◽  
Associated Legendre Functions ◽  
In Series ◽  
Derivatives Of

The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials atx=0are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials andxnin series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.

scholarly journals On Legendre numbers of the second kind

1988 ◽  
Vol 11 (4) ◽  
pp. 815-822 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
Explicit Formula ◽  
Rational Numbers ◽  
The Other ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Infinite Set ◽  
Natural Way

The Legendre numbers of the second kind, an infinite set of rational numbers, are defined from the associated Legendre functions. An explicit formula and a partial table for these numbers are given and many elementary properties are presented. A connection is shown between Legendre numbers of the first and second kinds. Extended Legendre numbers of the first and second kind are defined in a natural way and these are expressed in terms of those of the second and first kind, respectively. Two other sets of rational numbers are defined from the associated Legendre functions by taking derivatives and evaluating these atx=0. One of these sets is connected to Legendre numbers of the first find while the other is connected to Legendre numbers of the second kind. Some series are also discussed.

scholarly journals Simplification of Geopotential Perturbing Force Acting on A Satellite

2008 ◽  
Vol 43 (2) ◽  
pp. 45-64 ◽  
Author(s):  
M. Eshagh ◽  
M. Abdollahzadeh ◽  
M. Najafi-Alamdari
Keyword(s):  
Maximum Degree ◽  
Legendre Functions ◽  
Orbital Elements ◽  
Associated Legendre Functions ◽  
Geopotential Models ◽  
Orbit Integration ◽  
Vector Differential Equation ◽  
Derivatives Of ◽  
Geopotential Coefficients ◽  
Short Arc

Simplification of Geopotential Perturbing Force Acting on A SatelliteOne of the aspects of geopotential models is orbit integration of satellites. The geopotential acceleration has the largest influence on a satellite with respect to the other perturbing forces. The equation of motion of satellites is a second-order vector differential equation. These equations are further simplified and developed in this study based on the geopotential force. This new expression is much simpler than the traditional one as it does not derivatives of the associated Legendre functions and the transformations are included in the equations. The maximum degree and order of the geopotential harmonic expansion must be selected prior to the orbit integration purposes. The values of the maximum degree and order of these coefficients depend directly on the satellite's altitude. In this article, behaviour of orbital elements of recent geopotential satellites, such as CHAMP, GRACE and GOCE is considered with respect to the different degree and order of geopotential coefficients. In this case, the maximum degree 116, 109 and 175 were derived for the Earth gravitational field in short arc orbit integration of the CHAMP, GRACE and GOCE, respectively considering millimeter level in perturbations.

V.—The Addition Theorem for the Legendre Functions of the Second Kind

1927 ◽  
Vol 46 ◽  
pp. 30-35
Author(s):  
T. M. MacRobert
Keyword(s):  
Positive Integer ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Associated Legendre Functions

In a previous paper the author has employed certain formulæ of Dr Dougall's connecting the Associated Legendre Functions Pnm, where m is a positive integer and n is not integral, with the polynomials Ppm in which p is a positive integer, to deduce the Addition Theorem for the Legendre Functions of the first kind from the corresponding theorem for the Legendre Polynomials.

Toxicity of Para-Chloro Substituted Benzene Derivatives in the Microtox™ Test

1985 ◽  
Vol 20 (2) ◽  
pp. 36-43 ◽  
Author(s):  
Klaus L.E. Kaiser ◽  
Juan M. Ribo ◽  
Brian M. Zaruk
Keyword(s):  
General Formula ◽  
Functional Group ◽  
Structural Parameters ◽  
Systematic Investigation ◽  
Photobacterium Phosphoreum ◽  
Benzene Derivatives ◽  
Quantitative Structure ◽  
Chlorinated Benzenes ◽  
Chloro Derivatives ◽  
Derivatives Of

Abstract This paper gives the results of part of a systematic investigation into contaminant toxicity to Photobacterium phosphoreum in the Microtox™ test. Reported are the toxicity values for 39 para-chloro substituted benzene derivatives of the general formula l-Cl-C6h4-4-X=CH2CH(NH2)COOH, F, SO2NH2, OCH2COOH, CH2COOH, CONHNH2, NHCOCH3, CONH2, CH=CHCOOH, SeOOH, CH2NH2, CH2CH2NH2, NO2, H, CF3, CHO, CH2OH, OH, CH3, CCl3, COCH3, COOH, NH2, SO2C6H5, Cl, CH2COCH3, COCl, CN, OCH3, NCO, NHCH3, I, COC6H5, CH2Cl, SH, CH2SH, NCS, CH2CN and SO2C6H4Cl. Except for the last compound, whose solubility is below the required concentration, the toxicities increase in the presented order with a total range of more than three orders of magnitude. The data are discussed in terms of quantitative structure-toxicity correlations with compound-specific structural parameters. In combination with a previously developed submodel on chlorinated benzenes, phenols, nitrobenzenes and anilines, the observed relationships allow the prediction of the toxicity of some 780 possible chloro derivatives of the general formula C6H5-nClnX, where n=<5 and X is a functional group as listed above.

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

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