DLMF:18.34.E6 (Q6036): Difference between revisions

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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.m1edec

Latest revision as of 13:29, 2 January 2020

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DLMF:18.34.E6
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ | z | = 1 z a - 2 ⁢ y n ⁡ ( z ; a ) ⁢ y m ⁡ ( z ; a ) ⁢ e - 2 / z ⁢ d z = ( - 1 ) n + a - 1 ⁢ n ! ⁢  2 a - 1 ( n + a - 2 ) ! ⁢ ( 2 ⁢ n + a - 1 ) ⁢ δ n , m , 1 2 𝜋 imaginary-unit subscript 𝑧 1 superscript 𝑧 𝑎 2 Bessel-polynomial-y 𝑛 𝑧 𝑎 Bessel-polynomial-y 𝑚 𝑧 𝑎 superscript 𝑒 2 𝑧 𝑧 superscript 1 𝑛 𝑎 1 𝑛 superscript  2 𝑎 1 𝑛 𝑎 2 2 𝑛 𝑎 1 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}z^{a-2}y_{n}% \left(z;a\right)y_{m}\left(z;a\right)e^{-2/z}\mathrm{d}z=\frac{(-1)^{n+a-1}n!% \,2^{a-1}}{(n+a-2)!(2n+a-1)}\delta_{n,m},}}
    0 references
    DLMF id
    DLMF:18.34.E6
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    constraint
    a = 1 , 2 , … 𝑎 1 2 … {\displaystyle{\displaystyle a=1,2,\dots}}
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    Symbols used
    Bessel polynomial
    Defining formula
    y n ⁡ ( x ; a ) Bessel-polynomial-y 𝑛 𝑥 𝑎 {\displaystyle{\displaystyle y_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)}}
    xml-id
    C18.S34.E1.m2addec
    0 references
    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1dec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1dec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1edec
    0 references
     
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