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DLMF:18.26.E1 - MaRDI portal

DLMF:18.26.E1 (Q6627)

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DLMF:18.26.E1
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    Statements

    DLMF defining formula
    W n ⁡ ( y 2 ; a , b , c , d ) = ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n ⁢ F 3 4 ⁡ ( - n , n + a + b + c + d - 1 , a + i ⁢ y , a - i ⁢ y a + b , a + c , a + d ; 1 ) . Wilson-polynomial-W 𝑛 superscript 𝑦 2 𝑎 𝑏 𝑐 𝑑 Pochhammer 𝑎 𝑏 𝑛 Pochhammer 𝑎 𝑐 𝑛 Pochhammer 𝑎 𝑑 𝑛 Gauss-hypergeometric-pFq 4 3 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑎 𝑖 𝑦 𝑎 𝑖 𝑦 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 1 {\displaystyle{\displaystyle W_{n}\left(y^{2};a,b,c,d\right)={\left(a+b\right)% _{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}\*{{}_{4}F_{3}}\left({-n,n+a+% b+c+d-1,a+iy,a-iy\atop a+b,a+c,a+d};1\right).}}
    0 references
    Symbols used
    alternatively F q p ( a ; b ; z ) or F q p ( a b ; z ) generalized hypergeometric function
    DLMF defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1adec
    0 references
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Wilson polynomial
    DLMF defining formula
    W n ⁡ ( x ; a , b , c , d ) Wilson-polynomial-W 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c},\NVar{d}\right)}}
    xml-id
    C18.S25.T1.t1.r2.m2adec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1dec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1dec
    0 references

    Identifiers

     
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