DLMF:5.14.E4 (Q2166): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
Property / Symbols used
 
Property / Symbols used: product (locally) / rank
 
Normal rank
Property / Symbols used: product (locally) / qualifier
 
Defining formula:

Δ Δ {\displaystyle{\displaystyle\Delta}}

\Delta
Property / Symbols used: product (locally) / qualifier
 
xml-id: C5.S14.XMD2.m1adec

Latest revision as of 01:06, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:5.14.E4
No description defined

    Statements

    Defining formula
    ∫ [ 0 , 1 ] n t 1 ⁢ t 2 ⁢ ⋯ ⁢ t m ⁢ | Δ ⁢ ( t 1 , … , t n ) | 2 ⁢ c ⁢ ∏ k = 1 n t k a - 1 ⁢ ( 1 - t k ) b - 1 ⁢ d t k = 1 ( Γ ⁡ ( 1 + c ) ) n ⁢ ∏ k = 1 m a + ( n - k ) ⁢ c a + b + ( 2 ⁢ n - k - 1 ) ⁢ c ⁢ ∏ k = 1 n Γ ⁡ ( a + ( n - k ) ⁢ c ) ⁢ Γ ⁡ ( b + ( n - k ) ⁢ c ) ⁢ Γ ⁡ ( 1 + k ⁢ c ) Γ ⁡ ( a + b + ( 2 ⁢ n - k - 1 ) ⁢ c ) , subscript superscript 0 1 𝑛 subscript 𝑡 1 subscript 𝑡 2 ⋯ subscript 𝑡 𝑚 superscript Δ subscript 𝑡 1 … subscript 𝑡 𝑛 2 𝑐 superscript subscript product 𝑘 1 𝑛 superscript subscript 𝑡 𝑘 𝑎 1 superscript 1 subscript 𝑡 𝑘 𝑏 1 subscript 𝑡 𝑘 1 superscript Euler-Gamma 1 𝑐 𝑛 superscript subscript product 𝑘 1 𝑚 𝑎 𝑛 𝑘 𝑐 𝑎 𝑏 2 𝑛 𝑘 1 𝑐 superscript subscript product 𝑘 1 𝑛 Euler-Gamma 𝑎 𝑛 𝑘 𝑐 Euler-Gamma 𝑏 𝑛 𝑘 𝑐 Euler-Gamma 1 𝑘 𝑐 Euler-Gamma 𝑎 𝑏 2 𝑛 𝑘 1 𝑐 {\displaystyle{\displaystyle\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1% },\dots,t_{n})|^{2c}\prod_{k=1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\mathrm{d}t_{k}=% \frac{1}{(\Gamma\left(1+c\right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k% -1)c}\*\prod_{k=1}^{n}\frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c% \right)\Gamma\left(1+kc\right)}{\Gamma\left(a+b+(2n-k-1)c\right)},}}
    0 references
    DLMF id
    DLMF:5.14.E4
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q10815
    Defining formula
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q11276
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C5.S1.XMD2.m1dec
    0 references
    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1cdec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1bdec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1dec
    0 references
    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1dec
    0 references
    product (locally)
    Defining formula
    Δ Δ {\displaystyle{\displaystyle\Delta}}
    xml-id
    C5.S14.XMD2.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2166&oldid=29343"