About the Project
NIST

q-zAl-Salam--Chihara polynomials

AdvancedHelp

Did you mean q-zill-Salam--Chihara polynomials ?

(0.002 seconds)

1—10 of 245 matching pages

1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.2 Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) = H ( a , q n , m ; - n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
3: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
4: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
5: 18 Orthogonal Polynomials
Chapter 18 Orthogonal Polynomials
6: 18.30 Associated OP’s
§18.30 Associated OP’s
Associated Jacobi Polynomials
Associated Legendre Polynomials
For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). …
7: 18.28 Askey–Wilson Class
§18.28(iii) Al-SalamChihara Polynomials
§18.28(iv) q - 1 -Al-SalamChihara Polynomials
For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006).
§18.28(v) Continuous q -Ultraspherical Polynomials
These polynomials are also called Rogers polynomials. …
8: 18.1 Notation
Classical OP’s
Hahn Class OP’s
Wilson Class OP’s
  • Al-SalamChihara: Q n ( x ; a , b | q ) .

  • Nor do we consider the shifted Jacobi polynomials: …
    9: 18.38 Mathematical Applications
    Approximation Theory
    Integrable Systems
    Ultraspherical polynomials are zonal spherical harmonics. …
    Group Representations
    For applications of Krawtchouk polynomials K n ( x ; p , N ) and q -Racah polynomials R n ( x ; α , β , γ , δ | q ) to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).
    10: 18.2 General Orthogonal Polynomials
    §18.2 General Orthogonal Polynomials
    §18.2(ii) x -Difference Operators
    This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials18.20(i)). …
    §18.2(vi) Zeros