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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: 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 ) ,
3: 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 ,
4: 15.13 Zeros
Let N ( a , b , c ) denote the number of zeros of F ( a , b ; c ; z ) in the sector | ph ( 1 z ) | < π . If a , b , c are real, a , b , c , c a , c b 0 , 1 , 2 , , and, without loss of generality, b a , c a + b (compare (15.8.1)), then
15.13.1 N ( a , b , c ) = { 0 , a > 0 , a + 1 2 ( 1 + S ) , a < 0 , c a > 0 , a + 1 2 ( 1 + S ) + a c + 1 S , a < 0 , c a < 0 ,
where S = sign ( Γ ( a ) Γ ( b ) Γ ( c a ) Γ ( c b ) ) . … If a , b , c , c a , or c b { 0 , 1 , 2 , } , then F ( a , b ; c ; z ) is not defined, or reduces to a polynomial, or reduces to ( 1 z ) c a b times a polynomial. …
5: 18.30 Associated OP’s
In the recurrence relation (18.2.8) assume that the coefficients A n , B n , and C n + 1 are defined when n is a continuous nonnegative real variable, and let c be an arbitrary positive constant. …Then the associated orthogonal polynomials p n ( x ; c ) are defined by …  (18.30.3) continues to hold, except that when n = 0 , B c is replaced by an arbitrary real constant. Then the polynomials p n ( x , c ) generated in this manner are called corecursive associated OP’s. … where p n ( x ; c ) is given by (18.30.2) and (18.30.3), with A n , B n , and C n as in (18.9.2). …
6: 15.1 Special Notation
7: 15.7 Continued Fractions
t n = c + n ,
u 2 n + 1 = ( a + n ) ( c b + n ) ,
u 2 n = ( b + n ) ( c a + n ) .
v n = c + n + ( b a + n + 1 ) z ,
w n = ( b + n ) ( c a + n ) z .
8: 28.14 Fourier Series
The coefficients satisfy
28.14.4 q c 2 m + 2 ( a ( ν + 2 m ) 2 ) c 2 m + q c 2 m 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
28.14.7 c 2 m ν ( q ) = c 2 m ν ( q ) ,
28.14.8 c 2 m ν ( q ) = ( 1 ) m c 2 m ν ( q ) .
c 0 ν ( 0 ) = 1 ,
9: 15.2 Definitions and Analytical Properties
In general, F ( a , b ; c ; z ) does not exist when c = 0 , 1 , 2 , . … For all values of c
  • (c)

    Diverges when ( c a b ) 1 .

  • The principal branch of 𝐅 ( a , b ; c ; z ) is an entire function of a , b , and c . …The same properties hold for F ( a , b ; c ; z ) , except that as a function of c , F ( a , b ; c ; z ) in general has poles at c = 0 , 1 , 2 , . …
    10: 26.11 Integer Partitions: Compositions
    c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . …
    26.11.1 c ( 0 ) = c ( T , 0 ) = 1 .
    26.11.2 c m ( 0 ) = δ 0 , m ,