DLMF:18.17.E33 (Q5774): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11660 / rank
 
Normal rank
Property / Symbols used: Q11660 / qualifier
 
test:

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)}}

\JacobipolyP{\NVar{\alpha}}{\NVar{\beta}}{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11660 / qualifier
 
xml-id: C18.S3.T1.t1.r2.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11384 / rank
 
Normal rank
Property / Symbols used: Q11384 / qualifier
 
test:

F 1 1 ⁡ ( a ; b ; z ) Kummer-confluent-hypergeometric-M-as-1F1 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}% \right)}}

\genhyperF{1}{1}@{\NVar{a}}{\NVar{b}}{\NVar{z}}
Property / Symbols used: Q11384 / qualifier
 
xml-id: C16.S2.m7aadec
Property / Symbols used
 
Property / Symbols used: Q10923 / rank
 
Normal rank
Property / Symbols used: Q10923 / qualifier
 
test:

ℂ {\displaystyle{\displaystyle\mathbb{C}}}

\Complexes
Property / Symbols used: Q10923 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aafdec
Property / Symbols used
 
Property / Symbols used: Q10784 / rank
 
Normal rank
Property / Symbols used: Q10784 / qualifier
 
test:

∈ {\displaystyle{\displaystyle\in}}

\in
Property / Symbols used: Q10784 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r10.m2adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2apdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5aidec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aafdec
Property / Symbols used
 
Property / Symbols used: Q11750 / rank
 
Normal rank
Property / Symbols used: Q11750 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11750 / qualifier
 
xml-id: C18.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1afdec

Latest revision as of 00:13, 2 January 2020

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Language Label Description Also known as
English
DLMF:18.17.E33
No description defined

    Statements

    test
    ∫ - 1 1 e - ( x + 1 ) ⁢ z ⁢ P n ( α , β ) ⁡ ( x ) ⁢ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ d x = ( - 1 ) n ⁢ 2 α + β + n + 1 ⁢ Γ ⁡ ( α + n + 1 ) ⁢ Γ ⁡ ( β + n + 1 ) Γ ⁡ ( α + β + 2 ⁢ n + 2 ) ⁢ n ! ⁢ z n ⁢ F 1 1 ⁡ ( β + n + 1 α + β + 2 ⁢ n + 2 ; - 2 ⁢ z ) , superscript subscript 1 1 superscript 𝑒 𝑥 1 𝑧 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 𝑥 superscript 1 𝑛 superscript 2 𝛼 𝛽 𝑛 1 Euler-Gamma 𝛼 𝑛 1 Euler-Gamma 𝛽 𝑛 1 Euler-Gamma 𝛼 𝛽 2 𝑛 2 𝑛 superscript 𝑧 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝛽 𝑛 1 𝛼 𝛽 2 𝑛 2 2 𝑧 {\displaystyle{\displaystyle\int_{-1}^{1}e^{-(x+1)z}P^{(\alpha,\beta)}_{n}% \left(x\right)(1-x)^{\alpha}(1+x)^{\beta}\mathrm{d}x=\frac{(-1)^{n}2^{\alpha+% \beta+n+1}\Gamma\left(\alpha+n+1\right)\Gamma\left(\beta+n+1\right)}{\Gamma% \left(\alpha+\beta+2n+2\right)n!}z^{n}{{}_{1}F_{1}}\left({\beta+n+1\atop\alpha% +\beta+2n+2};-2z\right),}}
    0 references
    DLMF id
    DLMF:18.17.E33
    0 references
    constraint
    z ∈ ℂ 𝑧 {\displaystyle{\displaystyle z\in\mathbb{C}}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aldec
    0 references
    Q11660
    test
    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.m2aedec
    0 references
    Q11384
    test
    F 1 1 ⁡ ( a ; b ; z ) Kummer-confluent-hypergeometric-M-as-1F1 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}% \right)}}
    xml-id
    C16.S2.m7aadec
    0 references
    Q10923
    test
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aafdec
    0 references
    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2apdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aidec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aafdec
    0 references
    Q11750
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1dec
    0 references
    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1afdec
    0 references
     
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