About the Project
NIST

hypergeometric functions

AdvancedHelp

(0.012 seconds)

1—10 of 219 matching pages

1: 19.15 Advantages of Symmetry
Elliptic integrals are special cases of a particular multivariate hypergeometric function called Lauricella’s F D (Carlson (1961b)). …
2: 17.1 Special Notation
§17.1 Special Notation
k , j , m , n , r , s nonnegative integers.
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . Another function notation used is the “idem” function: …
3: 13.2 Definitions and Basic Properties
13.2.2 M ( a , b , z ) = s = 0 ( a ) s ( b ) s s ! z s = 1 + a b z + a ( a + 1 ) b ( b + 1 ) 2 ! z 2 + ,
13.2.3 M ( a , b , z ) = s = 0 ( a ) s Γ ( b + s ) s ! z s ,
13.2.6 U ( a , b , z ) z - a , z , | ph z | 3 2 π - δ ,
Kummer’s Transformations
4: 13.14 Definitions and Basic Properties
13.14.2 M κ , μ ( z ) = e - 1 2 z z 1 2 + μ M ( 1 2 + μ - κ , 1 + 2 μ , z ) ,
13.14.3 W κ , μ ( z ) = e - 1 2 z z 1 2 + μ U ( 1 2 + μ - κ , 1 + 2 μ , z ) ,
Except when z = 0 , each branch of the functions M κ , μ ( z ) / Γ ( 2 μ + 1 ) and W κ , μ ( z ) is entire in κ and μ . …
13.14.26 𝒲 { M κ , μ ( z ) , W κ , μ ( z ) } = - Γ ( 1 + 2 μ ) Γ ( 1 2 + μ - κ ) ,
13.14.28 𝒲 { M κ , - μ ( z ) , W κ , μ ( z ) } = - Γ ( 1 - 2 μ ) Γ ( 1 2 - μ - κ ) ,
5: 16.4 Argument Unity
Denote, formally, the bilateral hypergeometric function
16.4.16 H q p ( a 1 , , a p b 1 , , b q ; z ) = k = - ( a 1 ) k ( a p ) k ( b 1 ) k ( b q ) k z k .
6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; T ) = 1 Γ m ( a ) Ω etr ( - T X ) | X | a - 1 2 ( m + 1 ) | I + X | b - a - 1 2 ( m + 1 ) d X , ( a ) > 1 2 ( m - 1 ) , T Ω .
Laguerre Form
§35.6(ii) Properties
§35.6(iv) Asymptotic Approximations
7: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(i) Definition
Jacobi Form
Confluent Form
Integral Representation
8: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
15.2.1 F ( a , b ; c ; z ) = s = 0 ( a ) s ( b ) s ( c ) s s ! z s = 1 + a b c z + a ( a + 1 ) b ( b + 1 ) c ( c + 1 ) 2 ! z 2 + = Γ ( c ) Γ ( a ) Γ ( b ) s = 0 Γ ( a + s ) Γ ( b + s ) Γ ( c + s ) s ! z s ,
15.2.2 F ( a , b ; c ; z ) = s = 0 ( a ) s ( b ) s Γ ( c + s ) s ! z s , | z | < 1 ,
§15.2(ii) Analytic Properties
9: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
35.8.1 F q p ( a 1 , , a p b 1 , , b q ; T ) = k = 0 1 k ! | κ | = k [ a 1 ] κ [ a p ] κ [ b 1 ] κ [ b q ] κ Z κ ( T ) .
Convergence Properties
Confluence
10: 16.2 Definition and Analytic Properties
§16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
16.2.5 F q p ( a ; b ; z ) = F q p ( a 1 , , a p b 1 , , b q ; z ) / ( Γ ( b 1 ) Γ ( b q ) ) = k = 0 ( a 1 ) k ( a p ) k Γ ( b 1 + k ) Γ ( b q + k ) z k k ! ;