By Professor Dr. Arnold F. Nikiforov, Professor Dr. Vasilii B. Uvarov, Sergei K. Suslov (auth.)
Whereas classical orthogonal polynomials seem as recommendations to hypergeometric differential equations, these of a discrete variable come to be ideas of distinction equations of hypergeometric kind on lattices. The authors current a concise advent to this concept, proposing whilst equipment of fixing a wide category of distinction equations. They follow the speculation to numerous difficulties in clinical computing, likelihood, queuing conception, coding and knowledge compression. The ebook is an improved and revised model of the 1st variation, released in Russian (Nauka 1985). scholars and scientists will discover a necessary textbook in numerical research.
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Additional info for Classical Orthogonal Polynomials of a Discrete Variable
Bn . :1 [a(x)gn-dx)] = 7 n-l(X)gn-l(X). 5) we obtain rr B n -l ( , n + k = An-l "nBn, 7 n -l = n 7 + 2 n. 1. The polynomial solutions Yn(x) have the orthogonality property under certain restrictions on coefficients of Eq. 4). To derive this property we use the equations for Yn(x) and Ym(x) in self-adjoint form (by analogy with the 26 Sturm-Liouville problem) Ll[a(x)e(x)VYn(x)] Ll[a(x)e(x)VYm(X)] + Ane(X)Yn(x) = 0, + Ame(X)Ym(x) = o. Multiply the first equation by Ym(x) and the second by Yn(x), and subtract the second from the first.
1. We have considered a general method of studying the classical orthogonal polynomials of a discrete variable as solutions of the difference equation of hypergeometric type on uniform lattices. In particular a representation of these 30 solutions in the form of the Rodrigues formula was obtained and their orthogonality property under certain conditions was proved. 1). 17) in the form e(x + 1) e(x) = a(x) + r(x) a(x + I) . 1) It is easily verified that the solution of the difference equation e(x + 1) e(x) = f( ) x , whose right-hand side can be expressed as a product or quotient of two functions, has the following simple property: if the functions el(X) and e2(X) are solutions of the equations el(x+l) =f( ) 1 X , el ( x ) e2(X + 1) = ez(x) f () 2 X , then the solution of the equation e(x + I) e(x) = f(x) with f(x) = fl(x)h(x) is e(x) e(x) = el(X)/ez(x).
As a result, Eq. 4) can be reduced to the fonn Ll[u(x)g(X)\7y(x)] + ,\g(x)y(x) =0 . 18) Similarly Eq. 19) where the function gm(x) satisfies the equation Ll[u(x)gm(X)] = Tm(X)gm(x) . 20) Like Eq. 1) up to the second order of accuracy with respect to the mesh h, Eq. 1) is reduced to the self-adjoint form [o-(x)e(x)y'(x)]' + >'e(x)y(x) = 0 . 21) up to the second order of accuracy in h has the form 1 h [o-(x + h)e(x + h) - o-(x)e(x)] = H7'(x + h)e(x + h) + 7'(x)e(x)] . After the replacement x --+ 1 hx, h2o-(hx) --+ 1 h7'(hx) o-(x) , --+ 7'(x) , which transforms Eq.