DLMF:18.5.E12 (Q5520): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
Defining formula:

( 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.m1afdec

Revision as of 15:17, 2 January 2020

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DLMF:18.5.E12
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    Statements

    Defining formula
    L n ( α ) ⁡ ( x ) = ∑ ℓ = 0 n ( α + ℓ + 1 ) n - ℓ ( n - ℓ ) ! ⁢ ℓ ! ⁢ ( - x ) ℓ = ( α + 1 ) n n ! ⁢ F 1 1 ⁡ ( - n α + 1 ; x ) . Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript subscript ℓ 0 𝑛 Pochhammer 𝛼 ℓ 1 𝑛 ℓ 𝑛 ℓ ℓ superscript 𝑥 ℓ Pochhammer 𝛼 1 𝑛 𝑛 Kummer-confluent-hypergeometric-M-as-1F1 𝑛 𝛼 1 𝑥 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}% \frac{{\left(\alpha+\ell+1\right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=% \frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x% \right).}}
    0 references
    DLMF id
    DLMF:18.5.E12
    0 references
    Symbols used
    Q11384
    Defining formula
    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.m7adec
    0 references
    Q11333
    Defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1afdec
    0 references
     
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