The Orthogonal Polynomials of Power Series Probability Distributions and Their Uses

Biometrika ◽  
1966 ◽  
Vol 53 (1/2) ◽  
pp. 121
Author(s):  
D. F. I van Heerden ◽  
H. T. Gonin
Keyword(s):  
Power Series ◽  
Orthogonal Polynomials ◽  
Probability Distributions

Padé approximation and orthogonal polynomials

1974 ◽  
Vol 10 (2) ◽  
pp. 263-270 ◽  
Author(s):  
G.D. Allen ◽  
C.K. Chui ◽  
W.R. Madych ◽  
F.J. Narcowich ◽  
P.W. Smith
Keyword(s):  
Power Series ◽  
Variational Method ◽  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
The Past ◽  
Gaussian Quadrature Formulas ◽  
Poles And Zeros ◽  
Formal Power ◽  
Determinant Theory

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

scholarly journals On an Inequality for Legendre Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2044
Author(s):  
Florin Sofonea ◽  
Ioan Ţincu
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Legendre Polynomials ◽  
Probability Distributions ◽  
Upper And Lower Bounds ◽  
Discrete Probability ◽  
Discrete Probability Distributions

This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.

Burmann-series distributions and approximation operators

2005 ◽  
Vol 42 (4) ◽  
pp. 355-369
Author(s):  
J. P. King
Keyword(s):  
Power Series ◽  
Probability Distributions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Operators ◽  
Real Functions

Burmann series are used to give probability distributions which generalize the known class of distributions given by power series. Positive linear operators associated with Burmann-series distribution are described. Convergence of these operators to continuous real functions is studied. Examples are discussed.

scholarly journals From Invariance Under Binomial Thinning to Unification of the Cauchy and the Gołąb–Schinzel-Type Equations

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Karol Baron ◽  
Jacek Wesolowski
Keyword(s):  
Power Series ◽  
Functional Equations ◽  
Probability Distributions ◽  
Binomial Thinning ◽  
Additive Form

AbstractWe point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Gołąb–Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.

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