DLMF:18.14.E3 (Q5666)

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DLMF:18.14.E3
No description defined

    Statements

    Defining formula
    ( 1 2 ⁢ ( 1 - x ) ) 1 2 ⁢ α + 1 4 ⁢ ( 1 2 ⁢ ( 1 + x ) ) 1 2 ⁢ β + 1 4 ⁢ | P n ( α , β ) ⁡ ( x ) | ≤ Γ ⁡ ( max ⁡ ( α , β ) + n + 1 ) π 1 2 ⁢ n ! ⁢ ( n + 1 2 ⁢ ( α + β + 1 ) ) max ⁡ ( α , β ) + 1 2 , superscript 1 2 1 𝑥 1 2 𝛼 1 4 superscript 1 2 1 𝑥 1 2 𝛽 1 4 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Euler-Gamma 𝛼 𝛽 𝑛 1 superscript 𝜋 1 2 𝑛 superscript 𝑛 1 2 𝛼 𝛽 1 𝛼 𝛽 1 2 {\displaystyle{\displaystyle\left(\tfrac{1}{2}(1-x)\right)^{\frac{1}{2}\alpha+% \frac{1}{4}}\left(\tfrac{1}{2}(1+x)\right)^{\frac{1}{2}\beta+\frac{1}{4}}|P^{(% \alpha,\beta)}_{n}\left(x\right)|\leq\frac{\Gamma\left(\max(\alpha,\beta)+n+1% \right)}{\pi^{\frac{1}{2}}n!\left(n+\tfrac{1}{2}(\alpha+\beta+1)\right)^{\max(% \alpha,\beta)+\frac{1}{2}}},}}
    0 references
    DLMF id
    DLMF:18.14.E3
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    constraint
    - 1 2 ≤ β ≤ 1 2 1 2 𝛽 1 2 {\displaystyle{\displaystyle-\tfrac{1}{2}\leq\beta\leq\tfrac{1}{2}}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Q11660
    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.m2abdec
    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
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1bdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1bdec
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