DLMF:18.18.E1 (Q5791): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1dec
Property / Symbols used
 
Property / Symbols used: Q11762 / rank
 
Normal rank
Property / Symbols used: Q11762 / qualifier
 
Defining formula:

f ⁢ ( z ) 𝑓 𝑧 {\displaystyle{\displaystyle f(z)}}

f(z)
Property / Symbols used: Q11762 / qualifier
 
xml-id: C18.S18.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1dec

Latest revision as of 01:21, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.18.E1
No description defined

    Statements

    Defining formula
    a n = n ! ⁢ ( 2 ⁢ n + α + β + 1 ) ⁢ Γ ⁡ ( n + α + β + 1 ) 2 α + β + 1 ⁢ Γ ⁡ ( n + α + 1 ) ⁢ Γ ⁡ ( n + β + 1 ) ⁢ ∫ - 1 1 f ⁢ ( x ) ⁢ P n ( α , β ) ⁡ ( x ) ⁢ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ d x . subscript 𝑎 𝑛 𝑛 2 𝑛 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 𝛽 1 superscript 2 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 1 Euler-Gamma 𝑛 𝛽 1 superscript subscript 1 1 𝑓 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 𝑥 {\displaystyle{\displaystyle a_{n}=\frac{n!(2n+\alpha+\beta+1)\Gamma\left(n+% \alpha+\beta+1\right)}{2^{\alpha+\beta+1}\Gamma\left(n+\alpha+1\right)\Gamma% \left(n+\beta+1\right)}\*\int_{-1}^{1}f(x)P^{(\alpha,\beta)}_{n}\left(x\right)% (1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:18.18.E1
    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.m2adec
    0 references
    Q11660
    Defining formula
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1dec
    0 references
    Q11762
    Defining formula
    f ⁢ ( z ) 𝑓 𝑧 {\displaystyle{\displaystyle f(z)}}
    xml-id
    C18.S18.XMD1.m1dec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1dec
    0 references
     
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