Article

# Quantum LC - circuits with diffusive modification of the continuity equation

03/2007;

Source: arXiv

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**ABSTRACT:**Special functions of mathematical physics usually arise as solutions of certain differential equations of the second order. These solutions have been well studied, for many of themdetailed tables have been Constructed, as well as effective algorithms for calculating them on computers. Hence up to now the necessity of approximating specific values of the special functions by replacing the differential equations by difference equations has not arisen. However, recently it was discovered [i] that one can find the solution of such equations explicitly with the help of a generalization of the method used in [2] for the original differential equations. Some of these solutions, as it happened, have independent value and were already used in quantum mechanics, the theory of group representations, and computational mathematics for a long time. It suffices to mention the Clebsch--Gordon coefficients and the Racah coefficients, widely used in atomic and nuclear spectroscopy. In this paper we consider the simplest and most distinguished generalizations of the special functions of mathematical physics, the classical orthogonal polynomials of a discrete variable, which are the difference analogs of the Jacobi, Laguerre, and Hermite polynomials to uniform and nonuniform nets. Although the study of the classical orthogonal polynomials of a discrete variable had already begun in the last century [3], up to very recently there was no systematic account of the theory in the literature. It was not even clear which of the orthogonal polynomials introduced by different authors with the help of different arguments belong to a named class of special functions. The difference equation given in this paper has a simple property, analogous to a property of the differential equations for the Jacobi, Laguerre, and Hermite polynomials: as the result of difference differentiation of the original equation we get an equation of the same type. This permits us, with the help of the simplest mathematical means, to get the basic properties of the classical orthogonal polynomials of a discrete variable -- the analog of Rodrigues formula, the orthogonality property, the formula for difference differentiation, and the asymptotic representation, We give a classification of the polynomials which arise. Special or limit cases of them turn out to be the polynomials considered in [1-17].Functional Analysis and Its Applications 01/1985; 19(3):182-193. · 0.53 Impact Factor -
##### Book: Noncommutative geometry.

01/1994; Academic Press., ISBN: 978-0-12-185860-5 - 01/1991;

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