DLMF:18.5.E8 (Q5516): Difference between revisions

From DRMF
Jump to navigation Jump to search
Newer edit →
‎Created a new Item: Wikidata Toolkit example test item creation
(No difference)

Revision as of 16:02, 29 December 2019

No description defined
Language Label Description Also known as
English
DLMF:18.5.E8
No description defined

    Statements

    Defining formula
    P n ( α , β ) ⁡ ( x ) = 2 - n ⁢ ∑ ℓ = 0 n ( n + α ℓ ) ⁢ ( n + β n - ℓ ) ⁢ ( x - 1 ) n - ℓ ⁢ ( x + 1 ) ℓ = ( α + 1 ) n n ! ⁢ ( x + 1 2 ) n ⁢ F 1 2 ⁡ ( - n , - n - β α + 1 ; x - 1 x + 1 ) , Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 2 𝑛 superscript subscript ℓ 0 𝑛 binomial 𝑛 𝛼 ℓ binomial 𝑛 𝛽 𝑛 ℓ superscript 𝑥 1 𝑛 ℓ superscript 𝑥 1 ℓ Pochhammer 𝛼 1 𝑛 𝑛 superscript 𝑥 1 2 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝑛 𝛽 𝛼 1 𝑥 1 𝑥 1 {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{% \ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+% \beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!% }\left(\frac{x+1}{2}\right)^{n}{{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};% \frac{x-1}{x+1}\right),}}
    0 references
    DLMF id
    DLMF:18.5.E8
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5516&oldid=11515"