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DLMF:18.26.E7 - MaRDI portal

DLMF:18.26.E7 (Q6633)

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DLMF:18.26.E7
No description defined

    Statements

    DLMF defining formula
    lim t → ∞ ⁡ W n ⁡ ( 1 2 ⁢ ( 1 - x ) ⁢ t 2 ; 1 2 ⁢ α + 1 2 , 1 2 ⁢ α + 1 2 , 1 2 ⁢ β + 1 2 + i ⁢ t , 1 2 ⁢ β + 1 2 - i ⁢ t ) t 2 ⁢ n ⁢ n ! = P n ( α , β ) ⁡ ( x ) . subscript → 𝑡 Wilson-polynomial-W 𝑛 1 2 1 𝑥 superscript 𝑡 2 1 2 𝛼 1 2 1 2 𝛼 1 2 1 2 𝛽 1 2 𝑖 𝑡 1 2 𝛽 1 2 𝑖 𝑡 superscript 𝑡 2 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle\lim_{t\to\infty}\frac{W_{n}\left(\tfrac{1}{2}(1-x% )t^{2};\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{% 1}{2}\beta+\tfrac{1}{2}+it,\tfrac{1}{2}\beta+\tfrac{1}{2}-it\right)}{t^{2n}n!}% =P^{(\alpha,\beta)}_{n}\left(x\right).}}
    0 references
    Symbols used
    Jacobi polynomial
    DLMF defining formula
    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
    Wilson polynomial
    DLMF defining formula
    W n ⁡ ( x ; a , b , c , d ) Wilson-polynomial-W 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c},\NVar{d}\right)}}
    xml-id
    C18.S25.T1.t1.r2.m2acdec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1fdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1bdec
    0 references

    Identifiers

     
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