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1: 19.2 Definitions
Here a , b , p are real parameters, and k c and x are real or complex variables, with p 0 , k c 0 . … If 1 < k 1 / sin ϕ , then k c is pure imaginary. …
§19.2(iv) A Related Function: R C ( x , y )
When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
2: 26.5 Lattice Paths: Catalan Numbers
C ( n ) is the Catalan number. …(Sixty-six equivalent definitions of C ( n ) are given in Stanley (1999, pp. 219–229).) …
26.5.3 C ( n + 1 ) = k = 0 n C ( k ) C ( n k ) ,
26.5.4 C ( n + 1 ) = 2 ( 2 n + 1 ) n + 2 C ( n ) ,
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
3: 15.8 Transformations of Variable
In (15.8.8) when c a k m is a nonpositive integer ψ ( c a k m ) / Γ ( c a k m ) is interpreted as ( 1 ) m + k + a c + 1 ( m + k + a c ) ! . … Alternatively, if b a is a negative integer, then we interchange a and b in 𝐅 ( a , b ; c ; z ) . In a similar way, when c a b is an integer limits are taken in (15.8.4) and (15.8.5) as follows. If c a b is a nonnegative integer, then … Lastly, if c a b is a negative integer, then we first apply the transformation …
4: 17.13 Integrals
17.13.1 c d ( q x / c ; q ) ( q x / d ; q ) ( a x / c ; q ) ( b x / d ; q ) d q x = ( 1 q ) ( q ; q ) ( a b ; q ) c d ( c / d ; q ) ( d / c ; q ) ( a ; q ) ( b ; q ) ( c + d ) ( b c / d ; q ) ( a d / c ; q ) ,
17.13.2 c d ( q x / c ; q ) ( q x / d ; q ) ( x q α / c ; q ) ( x q β / d ; q ) d q x = Γ q ( α ) Γ q ( β ) Γ q ( α + β ) c d c + d ( c / d ; q ) ( d / c ; q ) ( q β c / d ; q ) ( q α d / c ; q ) .
17.13.4 0 t α 1 ( c t q α + β ; q ) ( c t ; q ) d q t = Γ q ( α ) Γ q ( β ) ( c q α ; q ) ( q 1 α / c ; q ) Γ q ( α + β ) ( c ; q ) ( q / c ; q ) .
5: 15.4 Special Cases
If ( c a b ) > 0 , then … If c = a + b , then … If ( c a b ) = 0 and c a + b , then … If ( c a b ) < 0 , then … If a , b are not integers and ( c + d a b ) > 1 , then …
6: 4.43 Cubic Equations
4.43.2 z 3 + p z + q = 0
  • (a)

    A sin a , A sin ( a + 2 3 π ) , and A sin ( a + 4 3 π ) , with sin ( 3 a ) = 4 q / A 3 , when 4 p 3 + 27 q 2 0 .

  • (b)

    A cosh a , A cosh ( a + 2 3 π i ) , and A cosh ( a + 4 3 π i ) , with cosh ( 3 a ) = 4 q / A 3 , when p < 0 , q < 0 , and 4 p 3 + 27 q 2 > 0 .

  • (c)

    B sinh a , B sinh ( a + 2 3 π i ) , and B sinh ( a + 4 3 π i ) , with sinh ( 3 a ) = 4 q / B 3 , when p > 0 .

  • Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. …
    7: 15.5 Derivatives and Contiguous Functions
    The six functions F ( a ± 1 , b ; c ; z ) , F ( a , b ± 1 ; c ; z ) , F ( a , b ; c ± 1 ; z ) are said to be contiguous to F ( a , b ; c ; z ) . …
    15.5.14 c ( a + ( b c ) z ) F ( a , b ; c ; z ) a c ( 1 z ) F ( a + 1 , b ; c ; z ) + ( c a ) ( c b ) z F ( a , b ; c + 1 ; z ) = 0 ,
    15.5.18 c ( c 1 ) ( z 1 ) F ( a , b ; c 1 ; z ) + c ( c 1 ( 2 c a b 1 ) z ) F ( a , b ; c ; z ) + ( c a ) ( c b ) z F ( a , b ; c + 1 ; z ) = 0 .
    By repeated applications of (15.5.11)–(15.5.18) any function F ( a + k , b + ; c + m ; z ) , in which k , , m are integers, can be expressed as a linear combination of F ( a , b ; c ; z ) and any one of its contiguous functions, with coefficients that are rational functions of a , b , c , and z . …
    15.5.20 z ( 1 z ) ( d F ( a , b ; c ; z ) / d z ) = ( c a ) F ( a 1 , b ; c ; z ) + ( a c + b z ) F ( a , b ; c ; z ) = ( c b ) F ( a , b 1 ; c ; z ) + ( b c + a z ) F ( a , b ; c ; z ) ,
    8: 17.6 ϕ 1 2 Function
    17.6.13 ϕ 1 2 ( a , b ; c ; q , q ) + ( q / c , a , b ; q ) ( c / q , a q / c , b q / c ; q ) ϕ 1 2 ( a q / c , b q / c ; q 2 / c ; q , q ) = ( q / c , a b q / c ; q ) ( a q / c , b q / c ; q ) ,
    17.6.23 q ( 1 a c ) ϕ 1 2 ( a / q , b c ; q , z ) + ( 1 a ) ( 1 a b z c ) ϕ 1 2 ( a q , b c ; q , z ) = ( 1 + q a a q c + a 2 z c a b z c ) ϕ 1 2 ( a , b c ; q , z ) ,
    17.6.24 ( 1 c ) ( q c ) ( a b z c ) ϕ 1 2 ( a , b c / q ; q , z ) + z ( c a ) ( c b ) ϕ 1 2 ( a , b c q ; q , z ) = ( c 1 ) ( c ( q c ) + z ( c a + c b a b a b q ) ) ϕ 1 2 ( a , b c ; q , z ) .
    (17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions a q a , b q b , c q c , followed by lim q 1 . … where | z | < 1 , | ph ( z ) | < π , and the contour of integration separates the poles of ( q 1 + ζ , c q ζ ; q ) / sin ( π ζ ) from those of 1 / ( a q ζ , b q ζ ; q ) , and the infimum of the distances of the poles from the contour is positive. …
    9: 10 Bessel Functions
    10: 4.42 Solution of Triangles
    4.42.5 c 2 = a 2 + b 2 2 a b cos C ,
    4.42.6 a = b cos C + c cos B
    4.42.7 area = 1 2 b c sin A = ( s ( s a ) ( s b ) ( s c ) ) 1 / 2 ,
    where s = 1 2 ( a + b + c ) (the semiperimeter). …
    4.42.8 cos a = cos b cos c + sin b sin c cos A ,