DLMF:17.6.E23 (Q5390)

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DLMF:17.6.E23
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    Statements

    Defining formula
    q ⁢ ( 1 - a c ) ⁢ ϕ 1 2 ⁡ ( a / q , b c ; q , z ) + ( 1 - a ) ⁢ ( 1 - a ⁢ b ⁢ z c ) ⁢ ϕ 1 2 ⁡ ( a ⁢ q , b c ; q , z ) = ( 1 + q - a - a ⁢ q c + a 2 ⁢ z c - a ⁢ b ⁢ z c ) ⁢ ϕ 1 2 ⁡ ( a , b c ; q , z ) , 𝑞 1 𝑎 𝑐 q-hypergeometric-rphis 2 1 𝑎 𝑞 𝑏 𝑐 𝑞 𝑧 1 𝑎 1 𝑎 𝑏 𝑧 𝑐 q-hypergeometric-rphis 2 1 𝑎 𝑞 𝑏 𝑐 𝑞 𝑧 1 𝑞 𝑎 𝑎 𝑞 𝑐 superscript 𝑎 2 𝑧 𝑐 𝑎 𝑏 𝑧 𝑐 q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 {\displaystyle{\displaystyle q\left(1-\frac{a}{c}\right){{}_{2}\phi_{1}}\left(% {a/q,b\atop c};q,z\right)+(1-a)\left(1-\frac{abz}{c}\right){{}_{2}\phi_{1}}% \left({aq,b\atop c};q,z\right)=\left(1+q-a-\frac{aq}{c}+\frac{a^{2}z}{c}-\frac% {abz}{c}\right){{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right),}}
    0 references
    DLMF id
    DLMF:17.6.E23
    0 references
    Symbols used
    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.m2avdec
    0 references
     
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