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11: 5.21 Methods of Computation
For the computation of the q -gamma and q -beta functions see Gabutti and Allasia (2008).
12: 5.1 Special Notation
The main functions treated in this chapter are the gamma function Γ ( z ) , the psi function (or digamma function) ψ ( z ) , the beta function B ( a , b ) , and the q -gamma function Γ q ( z ) . …
13: 5.16 Sums
For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
14: 35.1 Special Notation
a , b complex variables.
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( 𝐓 ) and (of the second kind) B ν ( 𝐓 ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; 𝐓 ) or F 1 1 ( a b ; 𝐓 ) and (of the second kind) Ψ ( a ; b ; 𝐓 ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; 𝐓 ) or F 1 2 ( a 1 , a 2 b ; 𝐓 ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) or F q p ( a 1 , , a p b 1 , , b q ; 𝐓 ) . … Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( 𝐓 ) = A ν ( 𝐓 ) / A ν ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | 𝐒 , 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
15: 8.28 Software
§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters
§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters
16: 19.16 Definitions
19.16.9 R a ( 𝐛 ; 𝐳 ) = 1 B ( a , a ) 0 t a 1 j = 1 n ( t + z j ) b j d t = 1 B ( a , a ) 0 t a 1 j = 1 n ( 1 + t z j ) b j d t , b 1 + + b n > a > 0 , b j , z j ( , 0 ] ,
where B ( x , y ) is the beta function5.12) and …
19.16.12 R a ( b 1 , , b 4 ; c 1 , c k 2 , c , c α 2 ) = 2 ( sin 2 ϕ ) 1 a B ( a , a ) 0 ϕ ( sin θ ) 2 a 1 ( sin 2 ϕ sin 2 θ ) a 1 ( cos θ ) 1 2 b 1 ( 1 k 2 sin 2 θ ) b 2 ( 1 α 2 sin 2 θ ) b 4 d θ ,
19.16.19 R a ( b 1 , , b n ; 0 , z 2 , , z n ) = B ( a , a b 1 ) B ( a , a ) R a ( b 2 , , b n ; z 2 , , z n ) , a + a > 0 , a > b 1 .
19.16.24 R a ( 𝐛 ; 𝐳 ) = z 1 a b 1 B ( b 1 , a b 1 ) 0 t b 1 1 ( t + z 1 ) a R a ( 𝐛 ; 0 , t + z 2 , , t + z n ) d t , a > b 1 , a + a > b 1 > 0 .
17: 19.23 Integral Representations
Also, in (19.23.8) and (19.23.10) B denotes the beta function5.12). …
19.23.8 R a ( 𝐛 ; 𝐳 ) = 2 B ( b 1 , b 2 ) 0 π / 2 ( z 1 cos 2 θ + z 2 sin 2 θ ) a ( cos θ ) 2 b 1 1 ( sin θ ) 2 b 2 1 d θ , b 1 , b 2 > 0 ; z 1 , z 2 > 0 .
19.23.10 R a ( 𝐛 ; 𝐳 ) = 1 B ( a , a ) 0 1 u a 1 ( 1 u ) a 1 j = 1 n ( 1 u + u z j ) b j d u , a , a > 0 ; a + a = j = 1 n b j ; z j ( , 0 ] .
18: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
The function ϕ ( ρ , β ; z ) is defined by
10.46.1 ϕ ( ρ , β ; z ) = k = 0 z k k ! Γ ( ρ k + β ) , ρ > 1 .
10.46.2 I ν ( z ) = ( 1 2 z ) ν ϕ ( 1 , ν + 1 ; 1 4 z 2 ) .
For asymptotic expansions of ϕ ( ρ , β ; z ) as z in various sectors of the complex z -plane for fixed real values of ρ and fixed real or complex values of β , see Wright (1935) when ρ > 0 , and Wright (1940b) when 1 < ρ < 0 . … The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …
19: Guide to Searching the DLMF
Table 1: Query Examples
Query Matching records contain
Euler the word ”Euler” or any of the various Euler terms such as Euler Gamma function Γ , Euler Beta function B , etc.
20: 5.13 Integrals
§5.13 Integrals