Results 231 to 240 of about 80,027 (267)

The Classical Orthogonal Polynomials

1988
In §2 we introduced the polynomials y n (z) of hypergeometric type, which are solutions of $$\sigma \left( z \right)y'' + \tau \left( z \right)y' + \lambda y = 0$$ (1) with \(\lambda = {\lambda _n} = - n\tau ' - \frac{1}{2}n\left( {n - 1} \right)\sigma ''\)
Arnold F. Nikiforov, Vasilii B. Uvarov
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On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials

Ukrainian Mathematical Journal, 2021
If \(\sum_{k=0}^{\infty} c_{k} g_{k}(x)\) is a formal series of orthonormal polynomials \(g_{k}(x)\) on the real line with positive coefficients \(c_{k}\), then its partial sums \(u_{n}(x)\) are associated with Jacobi-type pencils. Therefore, they possess a recurrence relation and special orthonormality conditions.
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Classical Orthogonal Polynomials

1991
Classical orthogonal Polynomials — the Jacobi, Laguerre and Hermite polynomials — form the simplest class of special functions. At the same time, the theory of these polynomials admits wide generalizations. By using the Rodrigues formula for the Jacobi, Laguerre and Hermite polynomials we can come to integral representations for other special functions
Arnold F. Nikiforov, Vasilii B. Uvarov, Sergei K. Suslov   +2 more
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Resonant Equations with Classical Orthogonal Polynomials. II

Ukrainian Mathematical Journal, 2019
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Gavrilyuk, I., Makarov, V.
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Discrete orthogonal polynomials – polynomial modification of a classical functional

Journal of Difference Equations and Applications, 2001
Polynomial modifications of a classical discrete linear functional are examined in detail, in particular when the new linear functional remains classical. New addition formulas are deduced for Charlier, Meixner and Hahn polynomials from the Christoffei representation and results are also given for a particular generalized Meixner family.
Ronveaux, André, Salto, L.
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Classical Orthogonal Polynomials

1999
Example 7.2.5 discussed only one of the many types of the so-called classical orthogonal polynomials. Historically, these polynomials were discovered as solutions to differential equations arising in various physical problems. Such polynomials can be produced by starting with 1,x,x 2,… and employing the Gram-Schmidt process.
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Positive Sums of the Classical Orthogonal Polynomials

SIAM Journal on Mathematical Analysis, 1977
An expansion as a sum of squares of Jacobi polynomials \(P_n^{(\alpha , \beta )}(x)\) is used to prove that if \(0 \leq \lambda \leq \alpha + \beta\) and \(\beta \geq -1/2\), then \[ \sum_{k=1}^{n} \frac{(\lambda +1)_{n-k}}{(n-k)!} \frac{(\lambda +1)_k}{k!} \frac{P_k^{(\alpha ,\beta )}(x)}{P_k^{(\alpha ,\beta )}} = 0,\quad -1\leq x \leq \infty, \tag{\(*
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On the “linearization” of the products of classical orthogonal polynomials

USSR Computational Mathematics and Mathematical Physics, 1979
Abstract IN VARIATIONAL methods of solving quantum-mechanical problems, when calculating the energy matrix, the problem arises of the expansion of the products of classical orthogonal polynomials in polynomials of the same type. Convenient recurrence methods of calculating the coefficients of these expansions are given.
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Discrete classical orthogonal polynomials

Journal of Difference Equations and Applications, 1998
We find necessary and sufficient conditions for the difference equation of hypergeometric type to have polynomial solutions , which are orthogonal, that is Traditionallydμ(x) is a positive measure but here we allow it to be a signed measure. We then show that the usual restrictions on parameters in discrete classical orthogonal polynomials can be ...
Kwon, KH Kwon, Kil Hyun, Lee, DW, Park, SB, Yoo, BH   +3 more
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Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials

SIAM Journal on Algebraic Discrete Methods, 1984
The author considers the problem of writing the product of two polynomials as the sum of other polynomials. The polynomials are given as the sum of orthogonal polynomials, and he uses the comrade matrix to encode some of the calculations. The calculations are done directly using the three term recurrence relation rather than using powers of x as an ...
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