DLMF:18.11.E1 (Q5635): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
test:

𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
test:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1adec
Property / Symbols used
 
Property / Symbols used: Q11475 / rank
 
Normal rank
Property / Symbols used: Q11475 / qualifier
 
test:

C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}

\ultrasphpoly{\NVar{\lambda}}{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11475 / qualifier
 
xml-id: C18.S3.T1.t1.r3.m2adec
Property / Symbols used
 
Property / Symbols used: Q11730 / rank
 
Normal rank
Property / Symbols used: Q11730 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11730 / qualifier
 
xml-id: C18.S1.XMD5.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.m1dec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

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

Latest revision as of 14:29, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.11.E1
No description defined

    Statements

    test
    𝖯 n m ⁡ ( x ) = ( 1 2 ) m ⁢ ( - 2 ) m ⁢ ( 1 - x 2 ) 1 2 ⁢ m ⁢ C n - m ( m + 1 2 ) ⁡ ( x ) = ( n + 1 ) m ⁢ ( - 2 ) - m ⁢ ( 1 - x 2 ) 1 2 ⁢ m ⁢ P n - m ( m , m ) ⁡ ( x ) , Ferrers-Legendre-P-first-kind 𝑚 𝑛 𝑥 Pochhammer 1 2 𝑚 superscript 2 𝑚 superscript 1 superscript 𝑥 2 1 2 𝑚 ultraspherical-Gegenbauer-polynomial 𝑚 1 2 𝑛 𝑚 𝑥 Pochhammer 𝑛 1 𝑚 superscript 2 𝑚 superscript 1 superscript 𝑥 2 1 2 𝑚 Jacobi-polynomial-P 𝑚 𝑚 𝑛 𝑚 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{% 2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x% \right)={\left(n+1\right)_{m}}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}% \left(x\right),}}
    0 references
    DLMF id
    DLMF:18.11.E1
    0 references
    constraint
    0 ≤ m ≤ n 0 𝑚 𝑛 {\displaystyle{\displaystyle 0\leq m\leq n}}
    0 references
    Symbols used
    Ferrers function of the first kind
    test
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2adec
    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.m2adec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q11475
    test
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2adec
    0 references
    Q11730
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1dec
    0 references
    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1dec
    0 references
    Q11727
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1dec
    0 references
     
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