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1: 18.3 Definitions
§18.3 Definitions
This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. …
Chebyshev
In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. … Formula (18.3.1) can be understood as a Gauss-Chebyshev quadrature, see (3.5.22), (3.5.23). …
2: 15.5 Derivatives and Contiguous Functions
§15.5(i) Differentiation Formulas
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 ) . …
15.5.12 ( b a ) F ( a , b ; c ; z ) + a F ( a + 1 , b ; c ; z ) b F ( a , b + 1 ; c ; 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: 16.12 Products
The following formula is often referred to as Clausen’s formula
16.12.3 ( F 1 2 ( a , b c ; z ) ) 2 = k = 0 ( 2 a ) k ( 2 b ) k ( c 1 2 ) k ( c ) k ( 2 c 1 ) k k ! F 3 4 ( 1 2 k , 1 2 ( 1 k ) , a + b c + 1 2 , 1 2 a + 1 2 , b + 1 2 , 3 2 k c ; 1 ) z k , | z | < 1 .
4: 15.4 Special Cases
F ( a , b ; a ; z ) = ( 1 z ) b ,
F ( a , b ; b ; z ) = ( 1 z ) a ,
5: 18.5 Explicit Representations
Chebyshev
For the definitions of F 1 2 , F 1 1 , and F 0 2 see §16.2. …
Chebyshev
For corresponding formulas for Chebyshev, Legendre, and the Hermite 𝐻𝑒 n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
Chebyshev
6: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
The hypergeometric function F ( a , b ; c ; z ) is defined by the Gauss seriesOn the circle of convergence, | z | = 1 , the Gauss series: … 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 , . … Formula (15.4.6) reads F ( a , b ; a ; z ) = ( 1 z ) b . …
7: 16.3 Derivatives and Contiguous Functions
§16.3(i) Differentiation Formulas
Two generalized hypergeometric functions F q p ( 𝐚 ; 𝐛 ; z ) are (generalized) contiguous if they have the same pair of values of p and q , and corresponding parameters differ by integers. …
16.3.6 z F 1 0 ( ; b + 1 ; z ) + b ( b 1 ) F 1 0 ( ; b ; z ) b ( b 1 ) F 1 0 ( ; b 1 ; z ) = 0 ,
8: 3.5 Quadrature
§3.5(v) Gauss Quadrature
Gauss–Legendre Formula
GaussChebyshev Formula
Gauss–Jacobi Formula
Gauss–Laguerre Formula
9: 15.8 Transformations of Variable
The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). …
15.8.13 F ( a , b 2 b ; z ) = ( 1 1 2 z ) a F ( 1 2 a , 1 2 a + 1 2 b + 1 2 ; ( z 2 z ) 2 ) , | ph ( 1 z ) | < π ,
15.8.14 F ( a , b 2 b ; z ) = ( 1 z ) a / 2 F ( 1 2 a , b 1 2 a b + 1 2 ; z 2 4 z 4 ) , | ph ( 1 z ) | < π .
15.8.15 F ( a , b a b + 1 ; z ) = ( 1 + z ) a F ( 1 2 a , 1 2 a + 1 2 a b + 1 ; 4 z ( 1 + z ) 2 ) , | z | < 1 ,
15.8.16 F ( a , b a b + 1 ; z ) = ( 1 z ) a F ( 1 2 a , 1 2 a b + 1 2 a b + 1 ; 4 z ( 1 z ) 2 ) , | z | < 1 .
10: 16.4 Argument Unity
The function F q q + 1 ( 𝐚 ; 𝐛 ; z ) is well-poised if … The function F q q + 1 with argument unity and general values of the parameters is discussed in Bühring (1992). … For generalizations involving F r + 2 r + 3 functions see Kim et al. (2013). … Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. … Transformations for both balanced F 3 4 ( 1 ) and very well-poised F 6 7 ( 1 ) are included in Bailey (1964, pp. 56–63). …