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11: 18.28 Askey–Wilson Class
β–ΊThe Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of q -Racah polynomials, and cases of these families obtained by specialization of parameters. … β–ΊThe polynomials p n ⁑ ( x ; a , b , c , d | q ) are symmetric in the parameters a , b , c , d . … β–Ί
18.28.2 1 1 p n ⁑ ( x ) ⁒ p m ⁑ ( x ) ⁒ w ⁑ ( x ) ⁒ d x = h n ⁒ δ n , m , | a | , | b | , | c | , | d | 1 , a ⁒ b , a ⁒ c , a ⁒ d , b ⁒ c , b ⁒ d , c ⁒ d 1 ,
β–Ί
18.28.6 1 1 p n ⁑ ( x ) ⁒ p m ⁑ ( x ) ⁒ w ⁑ ( x ) ⁒ d x + β„“ p n ⁑ ( x β„“ ) ⁒ p m ⁑ ( x β„“ ) ⁒ Ο‰ β„“ = h n ⁒ Ξ΄ n , m , a ⁒ b , a ⁒ c , a ⁒ d , b ⁒ c , b ⁒ d , c ⁒ d { z β„‚ ∣ | z | 1 , z 1 } ,
β–ΊDefine dual parameters a ~ , b ~ , c ~ , d ~ in terms of a , b , c , d by …
12: 19.16 Definitions
β–ΊThus R a ⁑ ( 𝐛 ; 𝐳 ) is symmetric in the variables z j and z β„“ if the parameters b j and b β„“ are equal. … β–Ί
19.16.9 R a ⁑ ( 𝐛 ; 𝐳 ) = 1 B ⁑ ( a , a ) ⁒ 0 t a 1 ⁒ j = 1 n ( t + z j ) b j ⁒ d t = 1 B ⁑ ( a , a ) ⁒ 0 t a 1 ⁒ j = 1 n ( 1 + t ⁒ z j ) b j ⁒ d t , b 1 + β‹― + b n > a > 0 , b j ℝ , z j β„‚ βˆ– ( , 0 ] ,
β–Ί
19.16.12 R a ⁑ ( b 1 , , b 4 ; c 1 , c k 2 , c , c Ξ± 2 ) = 2 ⁒ ( sin 2 ⁑ Ο• ) 1 a B ⁑ ( a , a ) ⁒ 0 Ο• ( sin ⁑ ΞΈ ) 2 ⁒ a 1 ⁒ ( sin 2 ⁑ Ο• sin 2 ⁑ ΞΈ ) a 1 ⁒ ( cos ⁑ ΞΈ ) 1 2 ⁒ b 1 ⁒ ( 1 k 2 ⁒ sin 2 ⁑ ΞΈ ) b 2 ⁒ ( 1 Ξ± 2 ⁒ sin 2 ⁑ ΞΈ ) b 4 ⁒ d ΞΈ ,
β–Ί R a ⁑ ( 𝐛 ; 𝐳 ) is an elliptic integral iff the z ’s are distinct and exactly four of the parameters a , a , b 1 , , b n are half-odd-integers, the rest are integers, and none of a , a , a + a is zero or a negative integer. …
13: 10.32 Integral Representations
β–Ί
10.32.13 K Ξ½ ⁑ ( z ) = ( 1 2 ⁒ z ) Ξ½ 4 ⁒ Ο€ ⁒ i ⁒ c i ⁒ c + i ⁒ Ξ“ ⁑ ( t ) ⁒ Ξ“ ⁑ ( t Ξ½ ) ⁒ ( 1 2 ⁒ z ) 2 ⁒ t ⁒ d t , c > max ⁑ ( ⁑ Ξ½ , 0 ) , | ph ⁑ z | < 1 2 ⁒ Ο€ .
14: 25.11 Hurwitz Zeta Function
β–Ί
25.11.9 ΞΆ ⁑ ( 1 s , a ) = 2 ⁒ Ξ“ ⁑ ( s ) ( 2 ⁒ Ο€ ) s ⁒ n = 1 1 n s ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ s 2 ⁒ n ⁒ Ο€ ⁒ a ) , ⁑ s > 0 if 0 < a < 1 ; ⁑ s > 1 if a = 1 .
15: 10.40 Asymptotic Expansions for Large Argument
β–Ί
10.40.3 I Ξ½ ⁑ ( z ) e z ( 2 ⁒ Ο€ ⁒ z ) 1 2 ⁒ k = 0 ( 1 ) k ⁒ b k ⁑ ( Ξ½ ) z k , | ph ⁑ z | 1 2 ⁒ Ο€ Ξ΄ ,
β–Ί
10.40.4 K Ξ½ ⁑ ( z ) ( Ο€ 2 ⁒ z ) 1 2 ⁒ e z ⁒ k = 0 b k ⁑ ( Ξ½ ) z k , | ph ⁑ z | 3 2 ⁒ Ο€ Ξ΄ .
β–Ί
10.40.6 I Ξ½ ⁑ ( z ) ⁒ K Ξ½ ⁑ ( z ) 1 2 ⁒ z ⁒ ( 1 1 2 ⁒ ΞΌ 1 ( 2 ⁒ z ) 2 + 1 3 2 4 ⁒ ( ΞΌ 1 ) ⁒ ( ΞΌ 9 ) ( 2 ⁒ z ) 4 β‹― ) ,
β–Ί
10.40.7 I Ξ½ ⁑ ( z ) ⁒ K Ξ½ ⁑ ( z ) 1 2 ⁒ z ⁒ ( 1 + 1 2 ⁒ ΞΌ 3 ( 2 ⁒ z ) 2 1 2 4 ⁒ ( ΞΌ 1 ) ⁒ ( ΞΌ 45 ) ( 2 ⁒ z ) 4 + β‹― ) ,
β–Ί
10.40.12 𝒱 z , ⁑ ( t β„“ ) { | z | β„“ , | ph ⁑ z | 1 2 ⁒ Ο€ , Ο‡ ⁑ ( β„“ ) ⁒ | z | β„“ , 1 2 ⁒ Ο€ | ph ⁑ z | Ο€ , 2 ⁒ Ο‡ ⁑ ( β„“ ) ⁒ | ⁑ z | β„“ , Ο€ | ph ⁑ z | < 3 2 ⁒ Ο€ ,
16: 10.22 Integrals
β–Ί
10.22.8 0 x J ν ⁑ ( t ) ⁒ d t = 2 ⁒ k = 0 J ν + 2 ⁒ k + 1 ⁑ ( x ) , ⁑ ν > 1 .
β–Ί
10.22.41 0 J ν ⁑ ( t ) ⁒ d t = 1 , ⁑ ν > 1 ,
β–ΊEquation (10.22.70) also remains valid if the order Ξ½ + 1 of the J functions on both sides is replaced by Ξ½ + 2 ⁒ n 3 , n = 1 , 2 , , and the constraint ⁑ Ξ½ > 3 2 is replaced by ⁑ Ξ½ > n + 1 2 . … β–Ί
10.22.76 g ⁑ ( y ) = 0 f ⁑ ( x ) ⁒ J ν ⁑ ( x ⁒ y ) ⁒ ( x ⁒ y ) 1 2 ⁒ d x .
β–Ί
10.22.77 f ⁑ ( y ) = 0 g ⁑ ( x ) ⁒ J ν ⁑ ( x ⁒ y ) ⁒ ( x ⁒ y ) 1 2 ⁒ d x .
17: 19.25 Relations to Other Functions
β–Ί β–Ί β–Ί
19.25.4 Π ⁑ ( α 2 , k ) = 1 3 ⁒ ( k 2 / α 2 ) ⁒ R J ⁑ ( 0 , 1 k 2 , 1 , 1 ( k 2 / α 2 ) ) , < k 2 < 1 < α 2 .
β–Ί β–ΊThe three changes of parameter of Ξ  ⁑ ( Ο• , Ξ± 2 , k ) in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). …
18: Errata
β–Ί
  • §19.25(i)

    The constraint Ο€ < ⁑ Ο• Ο€ was added just above (19.25.1).

  • β–Ί
  • §20.10(i)

    The general constraint ⁑ s > 2 has been extended to ⁑ s > 1 for (20.10.1), (20.10.2) and to ⁑ s > 0 for (20.10.3).

  • β–Ί
  • Equation (25.10.3)

    The constraint m = t / ( 2 ⁒ Ο€ ) was added.

    Reported by GergΕ‘ Nemes on 2021-08-23

  • β–Ί
  • Equation (5.11.14)

    The previous constraint ⁑ ( b a ) > 0 was removed, see Fields (1966, (3)).

  • β–Ί
  • Equation (4.8.14)

    The constraint a 0 was added.

  • 19: 25.2 Definition and Expansions
    β–Ί
    25.2.4 ΢ ⁑ ( s ) = 1 s 1 + n = 0 ( 1 ) n n ! ⁒ γ n ⁒ ( s 1 ) n ,
    β–Ί
    25.2.8 ΢ ⁑ ( s ) = k = 1 N 1 k s + N 1 s s 1 s ⁒ N x x x s + 1 ⁒ d x , ⁑ s > 0 , N = 1 , 2 , 3 , .
    20: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    β–Ί
    Transformations of Parameters
    β–Ί
    35.7.8 F 1 2 ⁑ ( a , b c ; 𝐓 ) = Ξ“ m ⁑ ( c ) ⁒ Ξ“ m ⁑ ( c a b ) Ξ“ m ⁑ ( c a ) ⁒ Ξ“ m ⁑ ( c b ) ⁒ F 1 2 ⁑ ( a , b a + b c + 1 2 ⁒ ( m + 1 ) ; 𝐈 𝐓 ) , 𝟎 < 𝐓 < 𝐈 ; 1 2 ⁒ ( j + 1 ) a β„• for some j = 1 , , m ; 1 2 ⁒ ( j + 1 ) c β„• and c a b 1 2 ⁒ ( m j ) β„• for all j = 1 , , m .
    β–Ί β–Ί