DLMF:18.28.E9 (Q5984)

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DLMF:18.28.E9
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    Statements

    Defining formula
    Q n ⁡ ( 1 2 ⁢ ( a ⁢ q - y + a - 1 ⁢ q y ) ; a , b | q - 1 ) = ( - 1 ) n ⁢ b n ⁢ q - 1 2 ⁢ n ⁢ ( n - 1 ) ⁢ ( ( a ⁢ b ) - 1 ; q ) n ⁢ ϕ 1 3 ⁡ ( q - n , q - y , a - 2 ⁢ q y ( a ⁢ b ) - 1 ; q , q n ⁢ a ⁢ b - 1 ) . q-inverse-AlSalam-Chihara-polynomial-Q 𝑛 1 2 𝑎 superscript 𝑞 𝑦 superscript 𝑎 1 superscript 𝑞 𝑦 𝑎 𝑏 superscript 𝑞 1 superscript 1 𝑛 superscript 𝑏 𝑛 superscript 𝑞 1 2 𝑛 𝑛 1 q-Pochhammer-symbol superscript 𝑎 𝑏 1 𝑞 𝑛 q-hypergeometric-rphis 3 1 superscript 𝑞 𝑛 superscript 𝑞 𝑦 superscript 𝑎 2 superscript 𝑞 𝑦 superscript 𝑎 𝑏 1 𝑞 superscript 𝑞 𝑛 𝑎 superscript 𝑏 1 {\displaystyle{\displaystyle Q_{n}\left(\tfrac{1}{2}(aq^{-y}+a^{-1}q^{y});a,b% \,|\,q^{-1}\right)=(-1)^{n}b^{n}q^{-\frac{1}{2}n(n-1)}\*\left((ab)^{-1};q% \right)_{n}{{}_{3}\phi_{1}}\left({q^{-n},q^{-y},a^{-2}q^{y}\atop(ab)^{-1}};q,q% ^{n}ab^{-1}\right).}}
    0 references
    DLMF id
    DLMF:18.28.E9
    0 references
    Symbols used
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2afdec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2abdec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1dec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1fdec
    0 references
     
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