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 end up options of distinction equations of hypergeometric kind on lattices. The authors current a concise creation to this idea, providing whilst equipment of fixing a wide category of distinction equations. They observe the idea to numerous difficulties in clinical computing, chance, queuing concept, coding and knowledge compression. The ebook is an multiplied and revised model of the 1st version, released in Russian (Nauka 1985). scholars and scientists will discover a worthy textbook in numerical research.
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Additional info for Classical Orthogonal Polynomials of a Discrete Variable
4) where Pn(s) are the Legendre polynomials. 5) where N = N + 1(a + ,8) (N - t 00). 5) takes the form ~ntn [~(1+S)-~] =Pn(s)+O(~2) . 6) may be derived in the following way. 1), this relation may be rewritten in the form an a+,8+2N ( x - a-,8+2N-2») 4 hn(x) = Thn+l(X) + 4 ,8n hn(x) + ~n (a +,8 + N + n)(N - n)hn_1(x) , where an, ,8n and In are the coefficients of the recursion relation for the Jacobi polynomials. 6). 8 + 1)/N, 1 - (a + 1)/N]. 8 + n + I)v(s) = 0 , where h = 2/ N. 8+n+ I)u = 0 to the second order of accuracy on the lattice with the step h = 2/ N.
2) and the positivity of the weight function e(Xi) by taking a = 0, b = +00, °< p, < 1, It is convenient to take from probability theory , > °. e to be 1/ r(,). 10) With Bn = p,-n the corresponding polynomials are the Meixner polynomials m~'Y,/L)(x), introduced in [MS]. b) Arguing similarly in the second case, we take a=O, p, = !!. q b=N+l, ,==N, (p > 0, q > 0, p + q = 1), e = qN N! The numbers e(Xi) become the familiar binomial distribution from probability theory, i N! 'I(N _ z')1. 11) With Bn == (_I)n qn/n!
E. 4) CI = C2 = c. 4) enables us to study the behavior of y(x, A) for x -+ a (x -+ b) if ct 10 (c210). 6) we choose the point Xo < b so that it lies to the right of all the zeros of Yn(x). e. C2 = O. Similarly it can be shown that ct = O. 16 Hence if Af. An (n lb = 0, y(x, A)Yn(x)e(x)dx 1, ... 3) for = 0, Xl - a, X2 - b we obtain n = 0, 1, .... By virtue of closure of the systems of classical orthogonal polynomials this equation is possible only when y(x, A) =0 for X E (a, b). e. the solutions Yn(x) and y(x, A) are linearly dependent, which contradicts the hypothesis.
Classical Orthogonal Polynomials of a Discrete Variable by Professor Dr. Arnold F. Nikiforov, Professor Dr. Vasilii B. Uvarov, Sergei K. Suslov (auth.)