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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 22. Orthogonal Polynomials Mathematical Properties 22.1. Definition of Orthogonal Polynomials A system of polynomialsf,(x), degree [f,(x)]=n, is called orthogonal on the interval a < x l b , with respect to the weight function w(x), if 22.1.1 1 w(x>jn (x>jm(x)dx=o (n#m;n, m=O, 1,2,. . .) The weight function w(x)[w(x) 201 determines the systemf,(x) up to a constant factor in each polynomial. The specification of these factors is referred to as standardization. For suitably standardized orthogonal polynomials we set 22.1.2 1 w(z)j~(x)dx=hn,f,(s)=k,s~+k~x"-'+ . . . (n=O, 1,2, . . .) These polynomials satisfy a number of relation- ships of the same general form. The most important ones are: Differential Equation gz ( ~ > f t + g l (x>jL +a j n = o 22.1.3 where gz(x), g l ( x ) are independent of n and an a constant depending only on n. Recurrence Relation 22.1.4 j n + l = (an + ~ b n > j n - c n f * - I where 22.1.5 Rodrigues' Formula where g(x) is a polynomial in x independent of n. The system { g } consists again of orthogonal polynomials. 3 2 -I FIGURE 22.1. Jucobi Polynomials P\$ fl)(x), a=1.5, 8=-.5, n=1(1)5. 773 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.