Some Interpolators Properties of Laguerre Polynomials*

1967 ◽  
Vol 10 (4) ◽  
pp. 559-571 ◽  
Author(s):  
J. Prasad ◽  
R. B. Saxena
Keyword(s):  
Legendre Polynomial ◽  
Laguerre Polynomials ◽  
Explicit Representation ◽  
Second Derivatives

In 1955, J. Suranyi and P. Turán [8] introduced the nomenclature (0, 2)-interpolation for the problem of finding polynomials of degree ≤ 2n-1 whose values and second derivatives are prescribed in certain given nodes. In a series of papers ([2], [3], [8]) Professor Turán and his associates discussed the problems of existence, uniqueness, explicit representation and convergence of such interpolatory 2 polynomials when the nodes are the zeros of (1-x2) P′n- 1(x), Pn- 1(x) being the Legendre polynomial of degree n - 1.

An Analogue of a Problem of J. Balázs and P. Turán

1969 ◽  
Vol 21 ◽  
pp. 54-63 ◽  
Author(s):  
J. Prasad ◽  
A. K. Varma
Keyword(s):  
Differential Equation ◽  
Modulus Of Continuity ◽  
Legendre Polynomial ◽  
Existence And Uniqueness ◽  
Explicit Representation ◽  
Lacunary Interpolation ◽  
First Derivative ◽  
Second Derivatives ◽  
Significant Application

In 1955, J. Surányi and P. Turán (8) initiated the problem of existence and uniqueness of interpolatory polynomials of degrees less than or equal to 2n — 1 when their values and second derivatives are prescribed on n given nodes. This kind of interpolation was termed (0, 2)-interpolation. Later, Balázs and Turán (1) gave the explicit representation of the interpolatory polynomials for the case when the n given nodes (n even) are taken to be the zeros of πn(x) = (1 — x2)Pn′(x), where Pn–i(x) is the Legendre polynomial of degree n — 1. In this case the explicit representation of interpolatory polynomials turns out to be simple and elegant.Balázs and Turán (2) proved the convergence of these polynomials when f(x) has a continuous first derivative satisfying certain conditions of modulus of continuity. They noted (1) that a significant application of lacunary interpolation could possibly be given in the theory of a differential equation of the form y′ + A(x)y= 0.

scholarly journals Two variables generalized Laguerre polynomials

2021 ◽  
Vol 13 (1) ◽  
pp. 134-141
Author(s):  
A. Asad
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Laguerre Polynomial ◽  
Laguerre Polynomials ◽  
Explicit Representation ◽  
Generalized Hypergeometric Function ◽  
Generalized Laguerre Polynomials

The objective of this paper is to introduce and study the generalized Laguerre polynomial for two variables. We prove that these polynomials are characterized by the generalized hypergeometric function. An explicit representation, generating functions and some recurrence relations are shown. Moreover, these polynomials appear as solutions of some differential equations.

scholarly journals On Weighted Pál Type (0,2)-interpolation on the Unit Circle

2021 ◽  
Author(s):  
Swarnima Bahadur ◽  
Sariya Bano
Keyword(s):  
Unit Circle ◽  
Legendre Polynomial ◽  
Pairwise Disjoint ◽  
Explicit Representation ◽  
Disjoint Sets ◽  
Ams Classification

Abstract In this paper, we study the explicit representation of weighted Pál-type (0,2)-interpolation on two pairwise disjoint sets of nodes on the unit circle, which are obtained by projecting vertically the zeros of (1−x2)Pn(x) and Pn′′(x) respectively, where Pn(x) stands for nth Legendre polynomial.AMS Classification (2000): 41A05, 30E10.

scholarly journals Polynomials of the Laguerre type

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 261-267
Author(s):  
Gospava Djordjevic
Keyword(s):  
Real Number ◽  
Hypergeometric Function ◽  
Generating Function ◽  
Laguerre Polynomials ◽  
Explicit Representation ◽  
Convolution Type ◽  
Generalized Laguerre Polynomials ◽  
Special Cases

In this noteweshall study a class of polynomials {f c,r n,m(x)}, where c is some real number, r ? N?{0}, m ? N. These polynomials are defined by the generating function. Also, for these polynomials we find an explicit representation in the form of the hypergeometric function; some identities of the convolution type are presented; some special cases are shown. The special cases of these polynomials are: Panda?s polynomials [2], [4]; the generalized Laguerre polynomials [1], [6]; the Celine Fasenmyer polynomials [3].

Extended State Space of the Rational sl(2) Gaudin Model in Terms of Laguerre Polynomials

2013 ◽  
Vol 58 (11) ◽  
pp. 1084-1091
Author(s):  
Yu.V. Bezvershenko ◽  
◽  
P.I. Holod ◽  
Keyword(s):  
State Space ◽  
Laguerre Polynomials ◽  
Gaudin Model ◽  
Extended State

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

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