DLMF:18.20.E9 (Q5851)

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DLMF:18.20.E9
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    Statements

    Defining formula
    p n ⁡ ( x ; a , b , a ¯ , b ¯ ) = i n ⁢ ( a + a ¯ ) n ⁢ ( a + b ¯ ) n n ! ⁢ F 2 3 ⁡ ( - n , n + 2 ⁢ ℜ ⁡ ( a + b ) - 1 , a + i ⁢ x a + a ¯ , a + b ¯ ; 1 ) . continuous-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑎 𝑏 imaginary-unit 𝑛 Pochhammer 𝑎 𝑎 𝑛 Pochhammer 𝑎 𝑏 𝑛 𝑛 Gauss-hypergeometric-pFq 3 2 𝑛 𝑛 2 𝑎 𝑏 1 𝑎 imaginary-unit 𝑥 𝑎 𝑎 𝑎 𝑏 1 {\displaystyle{\displaystyle p_{n}\left(x;a,b,\overline{a},\overline{b}\right)% =\frac{{\mathrm{i}^{n}}{\left(a+\overline{a}\right)_{n}}{\left(a+\overline{b}% \right)_{n}}}{n!}\*{{}_{3}F_{2}}\left({-n,n+2\Re(a+b)-1,a+\mathrm{i}x\atop a+% \overline{a},a+\overline{b}};1\right).}}
    0 references
    DLMF id
    DLMF:18.20.E9
    0 references
    Symbols used
    Q11125
    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.m1abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    complex conjugate
    Defining formula
    z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}
    xml-id
    C1.S9.E11.m2aadec
    0 references
    Q11763
    Defining formula
    p n ⁡ ( x ; a , b , a ¯ , missing ) continuous-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑎 missing {\displaystyle{\displaystyle p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{\overline{a}},\NVar{missing}\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C18.S19.Px2.p1.m4aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1idec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1idec
    0 references
     
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