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1: 18.37 Classical OP’s in Two or More Variables
18.37.1 R m , n ( α ) ( r e i θ ) = e i ( m - n ) θ r | m - n | P min ( m , n ) ( α , | m - n | ) ( 2 r 2 - 1 ) P min ( m , n ) ( α , | m - n | ) ( 1 ) , r 0 , θ , α > - 1 .
18.37.3 R m , n ( α ) ( z ) = j = 0 min ( m , n ) c j z m - j z ¯ n - j ,
18.37.6 R m , n ( α ) ( z ) = j = 0 min ( m , n ) ( - 1 ) j ( α + 1 ) m + n - j ( - m ) j ( - n ) j ( α + 1 ) m ( α + 1 ) n j ! z m - j z ¯ n - j .
2: 3.5 Quadrature
Complex orthogonal polynomials p n ( 1 / ζ ) of degree n = 0 , 1 , 2 , , in 1 / ζ that satisfy the orthogonality condition … …
3: Software Index
4: 18.33 Polynomials Orthogonal on the Unit Circle
18.33.8 p n ( 1 2 ( z + z - 1 ) ) = (const.) × ( z - n ϕ 2 n ( z ) + z n ϕ 2 n ( z - 1 ) ) = (const.) × ( z - n + 1 ϕ 2 n - 1 ( z ) + z n - 1 ϕ 2 n - 1 ( z - 1 ) ) ,
18.33.9 q n ( 1 2 ( z + z - 1 ) ) = (const.) × z - n - 1 ϕ 2 n + 2 ( z ) - z n + 1 ϕ 2 n + 2 ( z - 1 ) z - z - 1 = (const.) × z - n ϕ 2 n + 1 ( z ) - z n ϕ 2 n + 1 ( z - 1 ) z - z - 1 .
18.33.10 z - n ϕ 2 n ( z ) = A n p n ( 1 2 ( z + z - 1 ) ) + B n ( z - z - 1 ) q n - 1 ( 1 2 ( z + z - 1 ) ) ,
18.33.11 z - n + 1 ϕ 2 n - 1 ( z ) = C n p n ( 1 2 ( z + z - 1 ) ) + D n ( z - z - 1 ) q n - 1 ( 1 2 ( z + z - 1 ) ) ,
5: 18.38 Mathematical Applications
was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …
6: Bibliography K
  • W. Koepf (1999) Orthogonal polynomials and computer algebra. In Recent developments in complex analysis and computer algebra (Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.), Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.
  • 7: 18.2 General Orthogonal Polynomials
    It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) …
    8: 18.19 Hahn Class: Definitions
    These polynomials are orthogonal on ( - , ) , and with a > 0 , b > 0 are defined as follows. …
    9: 31.15 Stieltjes Polynomials
    §31.15 Stieltjes Polynomials
    §31.15(i) Definitions
    Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). … … are mutually orthogonal over the set Q : …
    10: 2.8 Differential Equations with a Parameter
    in which u is a real or complex parameter, and asymptotic solutions are needed for large | u | that are uniform with respect to z in a point set D in or . For example, u can be the order of a Bessel function or degree of an orthogonal polynomial. … For error bounds, extensions to pure imaginary or complex u , an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … For error bounds, more delicate error estimates, extensions to complex ξ and u , zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … For error bounds, more delicate error estimates, extensions to complex ξ , ν , and u , zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). …