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1: 18.19 Hahn Class: Definitions
18.19.1 p n ( x ) = p n ( x ; a , b , a ¯ , b ¯ ) ,
18.19.2 w ( z ; a , b , a ¯ , b ¯ ) = Γ ( a + i z ) Γ ( b + i z ) Γ ( a ¯ - i z ) Γ ( b ¯ - i z ) ,
18.19.3 w ( x ) = w ( x ; a , b , a ¯ , b ¯ ) = | Γ ( a + i x ) Γ ( b + i x ) | 2 ,
18.19.4 h n = 2 π Γ ( n + a + a ¯ ) Γ ( n + b + b ¯ ) | Γ ( n + a + b ¯ ) | 2 ( 2 n + 2 ( a + b ) - 1 ) Γ ( n + 2 ( a + b ) - 1 ) n ! ,
2: 18.20 Hahn Class: Explicit Representations
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 ) ) .
(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).) …
3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.4 q n ( x ) = p n ( x ; a , b , a ¯ , b ¯ ) / p n ( i a ; a , b , a ¯ , b ¯ ) ,
A ( x ) = ( x + i a ¯ ) ( x + i b ¯ ) ,
18.22.27 δ x ( p n ( x ; a , b , a ¯ , b ¯ ) ) = ( n + 2 ( a + b ) - 1 ) p n - 1 ( x ; a + 1 2 , b + 1 2 , a ¯ + 1 2 , b ¯ + 1 2 ) ,
18.22.28 δ x ( w ( x ; a + 1 2 , b + 1 2 , a ¯ + 1 2 , b ¯ + 1 2 ) p n ( x ; a + 1 2 , b + 1 2 , a ¯ + 1 2 , b ¯ + 1 2 ) ) = - ( n + 1 ) w ( x ; a , b , a ¯ , b ¯ ) p n + 1 ( x ; a , b , a ¯ , b ¯ ) .
4: 4.3 Graphics
Corresponding points share the same letters, with bars signifying complex conjugates. …
See accompanying text
A B C C ¯ D D ¯ E E ¯ F
Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify
5: 12.21 Software
6: 36.8 Convergent Series Expansions
36.8.5 f n ( ζ , ζ ¯ ) = c n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + c n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) c n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) + d n ( ζ ) d n ( ζ ¯ ) Ai ( ζ ) Bi ( ζ ¯ ) ,
7: 10.34 Analytic Continuation
I ν ( z ¯ ) = I ν ( z ) ¯ ,
K ν ( z ¯ ) = K ν ( z ) ¯ .
For complex ν replace ν by ν ¯ on the right-hand sides.
8: 12.3 Graphics
See accompanying text
Figure 12.3.5: U ( - 8 , x ) , U ¯ ( - 8 , x ) , F ( - 8 , x ) , - 4 2 x 4 2 . Magnify
See accompanying text
Figure 12.3.6: U ( - 8 , x ) , U ¯ ( - 8 , x ) , G ( - 8 , x ) , - 4 2 x 4 2 . Magnify
9: 18.33 Polynomials Orthogonal on the Unit Circle
where the bar signifies complex conjugate. …
18.33.3 ϕ n * ( z ) = z n ϕ n ( z ¯ - 1 ) ¯ = κ n + = 1 n κ ¯ n , n - z ,
where the bar again signifies compex conjugate. …
10: 12.2 Differential Equations
Standard solutions are U ( a , ± z ) , V ( a , ± z ) , U ¯ ( a , ± x ) (not complex conjugate), U ( - a , ± i z ) for (12.2.2); W ( a , ± x ) for (12.2.3); D ν ( ± z ) for (12.2.4), where …
§12.2(vi) Solution U ¯ ( a , x ) ; Modulus and Phase Functions
When z ( = x ) is real the solution U ¯ ( a , x ) is defined by …unless a = 1 2 , 3 2 , , in which case U ¯ ( a , x ) is undefined. …Properties of U ¯ ( a , x ) follow immediately from those of V ( a , x ) via (12.2.21). …