DLMF:18.17.E1 (Q5742)

From DRMF
Revision as of 15:44, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:18.17.E1
No description defined

    Statements

    Defining formula
    2 ⁢ n ⁢ ∫ 0 x ( 1 - y ) α ⁢ ( 1 + y ) β ⁢ P n ( α , β ) ⁡ ( y ) ⁢ d y = P n - 1 ( α + 1 , β + 1 ) ⁡ ( 0 ) - ( 1 - x ) α + 1 ⁢ ( 1 + x ) β + 1 ⁢ P n - 1 ( α + 1 , β + 1 ) ⁡ ( x ) . 2 𝑛 superscript subscript 0 𝑥 superscript 1 𝑦 𝛼 superscript 1 𝑦 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑦 𝑦 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 0 superscript 1 𝑥 𝛼 1 superscript 1 𝑥 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle 2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(% \alpha,\beta)}_{n}\left(y\right)\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(% 0\right)-(1-x)^{\alpha+1}(1+x)^{\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x% \right).}}
    0 references
    DLMF id
    DLMF:18.17.E1
    0 references
    Symbols used
    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.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1dec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5742&oldid=60387"