DLMF:18.17.E45 (Q5786)

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DLMF:18.17.E45
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    Statements

    Defining formula
    ( n + 1 2 ) ⁢ ( 1 + x ) 1 2 ⁢ ∫ - 1 x ( x - t ) - 1 2 ⁢ P n ⁡ ( t ) ⁢ d t = T n ⁡ ( x ) + T n + 1 ⁡ ( x ) , 𝑛 1 2 superscript 1 𝑥 1 2 superscript subscript 1 𝑥 superscript 𝑥 𝑡 1 2 Legendre-spherical-polynomial 𝑛 𝑡 𝑡 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle(n+\tfrac{1}{2})(1+x)^{\frac{1}{2}}\int_{-1}^{x}(x% -t)^{-\frac{1}{2}}P_{n}\left(t\right)\mathrm{d}t=T_{n}\left(x\right)+T_{n+1}% \left(x\right),}}
    0 references
    DLMF id
    DLMF:18.17.E45
    0 references
    Symbols used
    Q11207
    Defining formula
    T n ⁡ ( x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle T_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r4.m2abdec
    0 references
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aardec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aapdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1ardec
    0 references
     
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