DLMF:18.15.E22 (Q5717)

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DLMF:18.15.E22
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    Statements

    Defining formula
    L n ( α ) ⁡ ( ν ⁢ x ) = ( - 1 ) n ⁢ e 1 2 ⁢ ν ⁢ x 2 α - 1 2 ⁢ x 1 2 ⁢ α + 1 4 ⁢ ( ζ x - 1 ) 1 4 ⁢ ( Ai ⁡ ( ν 2 3 ⁢ ζ ) ν 1 3 ⁢ ∑ m = 0 M - 1 E m ⁢ ( ζ ) ν 2 ⁢ m + Ai ′ ⁡ ( ν 2 3 ⁢ ζ ) ν 5 3 ⁢ ∑ m = 0 M - 1 F m ⁢ ( ζ ) ν 2 ⁢ m + envAi ⁡ ( ν 2 3 ⁢ ζ ) ⁢ O ⁡ ( 1 ν 2 ⁢ M - 2 3 ) ) , Laguerre-polynomial-L 𝛼 𝑛 𝜈 𝑥 superscript 1 𝑛 superscript 𝑒 1 2 𝜈 𝑥 superscript 2 𝛼 1 2 superscript 𝑥 1 2 𝛼 1 4 superscript 𝜁 𝑥 1 1 4 Airy-Ai superscript 𝜈 2 3 𝜁 superscript 𝜈 1 3 superscript subscript 𝑚 0 𝑀 1 subscript 𝐸 𝑚 𝜁 superscript 𝜈 2 𝑚 diffop Airy-Ai 1 superscript 𝜈 2 3 𝜁 superscript 𝜈 5 3 superscript subscript 𝑚 0 𝑀 1 subscript 𝐹 𝑚 𝜁 superscript 𝜈 2 𝑚 envelope-Airy-Ai superscript 𝜈 2 3 𝜁 Big-O 1 superscript 𝜈 2 𝑀 2 3 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{% e^{\frac{1}{2}\nu x}}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}% \*\left(\frac{\zeta}{x-1}\right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(\nu% ^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(% \zeta)}{\nu^{2m}}+\frac{\mathrm{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{5}{3}}}\sum_{m=0}^{M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\mathrm{envAi}\left% (\nu^{\frac{2}{3}}\zeta\right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)% \right),}}
    0 references
    DLMF id
    DLMF:18.15.E22
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    Symbols used
    Q10919
    Defining formula
    Ai ⁡ ( z ) Airy-Ai 𝑧 {\displaystyle{\displaystyle\mathrm{Ai}\left(\NVar{z}\right)}}
    xml-id
    C9.S2.SS1.m1adec
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    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2afdec
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    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
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    Q11082
    Defining formula
    envAi ⁡ ( x ) envelope-Airy-Ai 𝑥 {\displaystyle{\displaystyle\mathrm{envAi}\left(\NVar{x}\right)}}
    xml-id
    C2.S8.SS3.p4.m4adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1ndec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1mdec
    0 references
    DLMF:18.15.E17
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C18.S15.XMD14.m1bdec
    0 references
    Q11758
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C18.S15.XMD18.m1adec
    0 references
    Q11759
    Defining formula
    E m subscript 𝐸 𝑚 {\displaystyle{\displaystyle E_{m}}}
    xml-id
    C18.S15.XMD19.m1dec
    0 references
    coefficient (locally)
    Defining formula
    F m subscript 𝐹 𝑚 {\displaystyle{\displaystyle F_{m}}}
    xml-id
    C18.S15.XMD20.m1dec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1gdec
    0 references
     
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