Formula:KLS:01.13:20

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\qHyperrphis 21 @ @ q - n , b c q z = ( b - 1 c ; q ) n ( c ; q ) n \qHyperrphis 32 @ @ q - n , b , b c - 1 q - n z b c - 1 q 1 - n , 0 q q formulae-sequence \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑏 𝑐 𝑞 𝑧 q-Pochhammer-symbol superscript 𝑏 1 𝑐 𝑞 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑏 𝑏 superscript 𝑐 1 superscript 𝑞 𝑛 𝑧 𝑏 superscript 𝑐 1 superscript 𝑞 1 𝑛 0 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}% {z}{}=\frac{\left(b^{-1}c;q\right)_{n}}{\left(c;q\right)_{n}}\,\qHyperrphis{3}% {2}@@{q^{-n},b,bc^{-1}q^{-n}z}{bc^{-1}q^{1-n},0}{q}{q}}}}

Proof

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Symbols List

ϕ s r subscript subscript italic-ϕ 𝑠 𝑟 {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}  : basic hypergeometric (or q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 1.13 of KLS.

URL links

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