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Another Characterization of the Classical Orthogonal Polynomials
SIAM Journal on Mathematical Analysis, 1972The classical orthogonal polynomials of Jacobi, Laguerre and Hermite are characterized as the only orthogonal polynomials with a differentiation formula of the form \[ \pi (x)P'_n (x) = \left( {\alpha _n x + \beta _n } \right)P_n (x) + \gamma _n P_{n - 1} (x),\quad n \geqq 1,\] where $\pi (x)$ is a polynomial.
Al-Salam, W. A., Chihara, T. S.
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On Series of Orthogonal Polynomials and Systems of Classical Type Polynomials
Ukrainian Mathematical Journal, 2021If \(\sum_{k=0}^{\infty} c_{k} g_{k}(x)\) 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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The Classical Orthogonal Polynomials
1988In §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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Resonant Equations with Classical Orthogonal Polynomials. II
Ukrainian Mathematical Journal, 2019zbMATH 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, 2001Polynomial 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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Positive Sums of the Classical Orthogonal Polynomials
SIAM Journal on Mathematical Analysis, 1977An 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 bivariate classical orthogonal polynomials
Applied Mathematics and Computation, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Francisco Marcellán +3 more
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On the “linearization” of the products of classical orthogonal polynomials
USSR Computational Mathematics and Mathematical Physics, 1979Abstract 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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Classical Orthogonal Polynomials
1999Example 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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Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials
SIAM Journal on Algebraic Discrete Methods, 1984The 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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