About the Project
NIST

polynomials orthogonal on the unit circle

AdvancedHelp

(0.003 seconds)

1—10 of 19 matching pages

1: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
§18.33(i) Definition
§18.33(iii) Connection with OP’s on the Line
§18.33(v) Biorthogonal Polynomials on the Unit Circle
See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.
2: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
3: Bibliography S
  • B. Simon (2005a) Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
  • B. Simon (2005b) Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
  • 4: 18.34 Bessel Polynomials
    §18.34 Bessel Polynomials
    §18.34(ii) Orthogonality
    Because the coefficients C n in (18.34.4) are not all positive, the polynomials y n ( x ; a ) cannot be orthogonal on the line with respect to a positive weight function. There is orthogonality on the unit circle, however: …
    5: 18.35 Pollaczek Polynomials
    §18.35 Pollaczek Polynomials
    §18.35(i) Definition and Hypergeometric Representation
    §18.35(ii) Orthogonality
    §18.35(iii) Other Properties
    For the polynomials C n ( λ ) ( x ) and P n ( λ ) ( x ; ϕ ) see §§18.3 and 18.19, respectively. …
    6: Bibliography
  • W. A. Al-Salam and L. Carlitz (1965) Some orthogonal q -polynomials. Math. Nachr. 30, pp. 47–61.
  • W. A. Al-Salam and M. E. H. Ismail (1994) A q -beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
  • R. Askey and M. E. H. Ismail (1984) Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
  • R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
  • R. Askey (1975b) Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • 7: 18.10 Integral Representations
    §18.10 Integral Representations
    Ultraspherical
    Legendre
    Jacobi
    See also §18.17.
    8: 18.19 Hahn Class: Definitions
    §18.19 Hahn Class: Definitions
    Hahn, Krawtchouk, Meixner, and Charlier
    Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials Q n ( x ; α , β , N ) , Krawtchouk polynomials K n ( x ; p , N ) , Meixner polynomials M n ( x ; β , c ) , and Charlier polynomials C n ( x ; a ) . … These polynomials are orthogonal on ( - , ) , and with a > 0 , b > 0 are defined as follows. … These polynomials are orthogonal on ( - , ) , and are defined as follows. …
    9: 3.5 Quadrature
    For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). … For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … Below we give for the classical orthogonal polynomials the recurrence coefficients α n and β n in (3.5.30). … Complex orthogonal polynomials p n ( 1 / ζ ) of degree n = 0 , 1 , 2 , , in 1 / ζ that satisfy the orthogonality condition … …
    10: 18.21 Hahn Class: Interrelations
    §18.21 Hahn Class: Interrelations
    §18.21(i) Dualities
    §18.21(ii) Limit Relations and Special Cases
    Hahn Jacobi
    Meixner Laguerre