DLMF:18.12.E3 (Q5645): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11581 / rank
 
Normal rank
Property / Symbols used: Q11581 / qualifier
 
test:

F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}

\genhyperF{2}{1}@{\NVar{a},\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: Q11581 / qualifier
 
xml-id: C16.S2.m5adec

Revision as of 14:30, 2 January 2020

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DLMF:18.12.E3
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    Statements

    test
    ( 1 + z ) - α - β - 1 ⁢ F 1 2 ⁡ ( 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) β + 1 ; 2 ⁢ ( x + 1 ) ⁢ z ( 1 + z ) 2 ) = ∑ n = 0 ∞ ( α + β + 1 ) n ( β + 1 ) n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ z n , superscript 1 𝑧 𝛼 𝛽 1 Gauss-hypergeometric-F-as-2F1 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝛽 1 2 𝑥 1 𝑧 superscript 1 𝑧 2 superscript subscript 𝑛 0 Pochhammer 𝛼 𝛽 1 𝑛 Pochhammer 𝛽 1 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({% \tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2% (x+1)z}{(1+z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right% )_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},}}
    0 references
    DLMF id
    DLMF:18.12.E3
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    Symbols used
    Q11581
    test
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    0 references
     
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