About the Project
NIST

Hahn polynomials

AdvancedHelp

(0.002 seconds)

1—10 of 20 matching pages

1: 18.21 Hahn Class: Interrelations
18.21.1 Q n ( x ; α , β , N ) = R x ( n ( n + α + β + 1 ) ; α , β , N ) , n , x = 0 , 1 , , N .
For the dual Hahn polynomial R n ( x ; γ , δ , N ) see §18.25. …
18.21.3 lim t Q n ( x ; p t , ( 1 - p ) t , N ) = K n ( x ; p , N ) .
18.21.4 lim N Q n ( x ; b - 1 , N ( c - 1 - 1 ) , N ) = M n ( x ; b , c ) .
18.21.5 lim N Q n ( N x ; α , β , N ) = P n ( α , β ) ( 1 - 2 x ) P n ( α , β ) ( 1 ) .
2: 18.19 Hahn Class: Definitions
§18.19 Hahn Class: Definitions
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 ) . …
Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.
p n ( x ) k n
Q n ( x ; α , β , N ) ( n + α + β + 1 ) n ( α + 1 ) n ( - N ) n
18.19.1 p n ( x ) = p n ( x ; a , b , a ¯ , b ¯ ) ,
3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.1 p n ( x ) = Q n ( x ; α , β , N ) ,
18.22.4 q n ( x ) = p n ( x ; a , b , a ¯ , b ¯ ) / p n ( i a ; a , b , a ¯ , b ¯ ) ,
18.22.19 Δ x Q n ( x ; α , β , N ) = - n ( n + α + β + 1 ) ( α + 1 ) N Q n - 1 ( x ; α + 1 , β + 1 , N - 1 ) ,
4: 18.23 Hahn Class: Generating Functions
§18.23 Hahn Class: Generating Functions
Hahn
18.23.1 F 1 1 ( - x α + 1 ; - z ) F 1 1 ( x - N β + 1 ; z ) = n = 0 N ( - N ) n ( β + 1 ) n n ! Q n ( x ; α , β , N ) z n , x = 0 , 1 , , N .
18.23.6 F 1 1 ( a + i x 2 a ; - i z ) F 1 1 ( b ¯ - i x 2 b ; i z ) = n = 0 p n ( x ; a , b , a ¯ , b ¯ ) ( 2 a ) n ( 2 b ) n z n .
5: 18.20 Hahn Class: Explicit Representations
For the Hahn polynomials p n ( x ) = Q n ( x ; α , β , N ) and …
18.20.3 w ( x ; a , b , a ¯ , b ¯ ) p n ( x ; a , b , a ¯ , b ¯ ) = 1 n ! δ x n ( w ( x ; a + 1 2 n , b + 1 2 n , a ¯ + 1 2 n , b ¯ + 1 2 n ) ) .
§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions
(For symmetry properties of p n ( x ; a , b , a ¯ , b ¯ ) with respect to a , b , a ¯ , b ¯ see Andrews et al. (1999, Corollary 3.3.4).) …
6: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
Hahn Class OP’s
  • Hahn: Q n ( x ; α , β , N ) .

  • Continuous Hahn: p n ( x ; a , b , a ¯ , b ¯ ) .

  • q -Hahn: Q n ( x ; α , β , N ; q ) .

  • 7: 18.26 Wilson Class: Continued
    18.26.5 lim d W n ( x ; a , b , c , d ) ( a + d ) n = S n ( x ; a , b , c ) .
    18.26.6 lim t W n ( ( x + t ) 2 ; a - i t , b - i t , a ¯ + i t , b ¯ + i t ) ( - 2 t ) n n ! = p n ( x ; a , b , a ¯ , b ¯ ) .
    18.26.10 lim δ R n ( x ( x + γ + δ + 1 ) ; α , β , - N - 1 , δ ) = Q n ( x ; α , β , N ) .
    18.26.11 lim t R n ( x ( x + t + 1 ) ; p t , ( 1 - p ) t , N ) = K n ( x ; p , N ) .
    18.26.13 lim N R n ( r ( x ; β , c , N ) ; β - 1 , c - 1 ( 1 - c ) N , N ) = M n ( x ; β , c ) .
    8: 18.24 Hahn Class: Asymptotic Approximations
    §18.24 Hahn Class: Asymptotic Approximations
    When the parameters α and β are fixed and the ratio n / N = c is a constant in the interval(0,1), uniform asymptotic formulas (as n ) of the Hahn polynomials Q n ( z ; α , β , N ) can be found in Lin and Wong (2013) for z in three overlapping regions, which together cover the entire complex plane. … Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
    9: 18.25 Wilson Class: Definitions
    Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) .
    Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
    p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
    S n ( x ; a , b , c ) y 2 ( 0 , ) ( a , b , c ) > 0 ; nonreal parameters in conjugate pairs
    18.25.6 p n ( x ) = S n ( x ; a 1 , a 2 , a 3 ) ,
    Table 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
    10: 18.27 q -Hahn Class
    The generic (top level) cases are the q -Hahn polynomials and the big q -Jacobi polynomials, each of which depends on three further parameters. …
    §18.27(ii) q -Hahn Polynomials
    18.27.3 Q n ( x ) = Q n ( x ; α , β , N ; q ) = ϕ 2 3 ( q - n , α β q n + 1 , x α q , q - N ; q , q ) , n = 0 , 1 , , N .
    §18.27(iii) Big q -Jacobi Polynomials
    §18.27(iv) Little q -Jacobi Polynomials