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1: 4.35 Identities
§4.35(iii) Multiples of the Argument
2: 4.21 Identities
§4.21(iii) Multiples of the Argument
3: 17.4 Basic Hypergeometric Functions
17.4.6 Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) = m , n 0 ( a ; q ) m + n ( b ; q ) m ( b ; q ) n x m y n ( q , c ; q ) m ( q , c ; q ) n ,
17.4.7 Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) = m , n 0 ( a , b ; q ) m ( a , b ; q ) n x m y n ( q ; q ) m ( q ; q ) n ( c ; q ) m + n ,
17.4.8 Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) = m , n 0 ( a , b ; q ) m + n x m y n ( q , c ; q ) m ( q , c ; q ) n .
4: 35.10 Methods of Computation
§35.10 Methods of Computation
Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on O ( m ) applied to a generalization of the integral (35.5.8). …
5: 17.11 Transformations of q -Appell Functions
17.11.1 Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) = ( a , b x , b y ; q ) ( c , x , y ; q ) ϕ 2 3 ( c / a , x , y b x , b y ; q , a ) ,
17.11.2 Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) = ( b , a x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( c / b , x ; q ) r b r y n ( q , c ; q ) n ( q ; q ) r ( a x ; q ) n + r ,
17.11.3 Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) = ( a , b x ; q ) ( c , x ; q ) n , r 0 ( a , b ; q ) n ( x ; q ) r ( c / a ; q ) n + r a r y n ( q , c / a ; q ) n ( q , b x ; q ) r .
6: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
Products
§10.40(ii) Error Bounds for Real Argument and Order
§10.40(iii) Error Bounds for Complex Argument and Order
7: Errata
  • Equation (18.27.6)

    18.27.6 P n ( α , β ) ( x ; c , d ; q ) = c n q - ( α + 1 ) n ( q α + 1 , - q α + 1 c - 1 d ; q ) n ( q , - q ; q ) n P n ( q α + 1 c - 1 x ; q α , q β , - q α c - 1 d ; q )

    Originally the first argument to the big q -Jacobi polynomial on the right-hand side was written incorrectly as q α + 1 c - 1 d x .

    Reported 2017-09-27 by Tom Koornwinder.

  • 8: 10.18 Modulus and Phase Functions
    §10.18(iii) Asymptotic Expansions for Large Argument
    The remainder after k terms in (10.18.17) does not exceed the ( k + 1 ) th term in absolute value and is of the same sign, provided that k > ν - 1 2 .
    9: 18.27 q -Hahn Class
    18.27.6 P n ( α , β ) ( x ; c , d ; q ) = c n q - ( α + 1 ) n ( q α + 1 , - q α + 1 c - 1 d ; q ) n ( q , - q ; q ) n P n ( q α + 1 c - 1 x ; q α , q β , - q α c - 1 d ; q ) .
    10: 1.10 Functions of a Complex Variable
    An analytic function f ( z ) has a zero of order (or multiplicity) m ( 1 ) at z 0 if the first nonzero coefficient in its Taylor series at z 0 is that of ( z - z 0 ) m . … … If - n is the first negative integer (counting from - ) with a - n 0 , then z 0 is a pole of order (or multiplicity) n . …
    Phase (or Argument) Principle
    each location again being counted with multiplicity equal to that of the corresponding zero or pole. …