\qHyperrphis r s @ @ a 1 , … , a r b 1 , … , b s q z := ∑ k = 0 ∞ ( a 1 , … , a r ; q ) k ( b 1 , … , b s ; q ) k ( - 1 ) ( 1 + s - r ) k q ( 1 + s - r ) \binomial k 2 z k ( q ; q ) k assign \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 superscript subscript 𝑘 0 q-Pochhammer-symbol subscript 𝑎 1 … subscript 𝑎 𝑟 𝑞 𝑘 q-Pochhammer-symbol subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑘 superscript 1 1 𝑠 𝑟 𝑘 superscript 𝑞 1 𝑠 𝑟 \binomial 𝑘 2 superscript 𝑧 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle{}\qHyperrphis{r}{s}@@{a_{1},\ldots,% a_{r}}{b_{1},\ldots,b_{s}}{q}{z}{}:=\sum\limits_{k=0}^{\infty}\frac{\left(a_{1% },\ldots,a_{r};q\right)_{k}}{\left(b_{1},\ldots,b_{s};q\right)_{k}}(-1)^{(1+s-% r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^{k}}{\left(q;q\right)_{k}}}}} {\displaystyle \index{Basic hypergeometric function} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{z} {}:=\sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_r}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}} (-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^k}{\qPochhammer{q}{q}{k}} }
ρ = { ∞ if r < s + 1 1 if r = s + 1 0 if r > s + 1 . 𝜌 cases if 𝑟 𝑠 1 1 if 𝑟 𝑠 1 0 if 𝑟 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\rho=\left\{\begin{array}[]{ll}% \displaystyle\infty&\quad\textrm{if}\quad r<s+1\\ \displaystyle 1&\quad\textrm{if}\quad r=s+1\\ \displaystyle 0&\quad\textrm{if}\quad r>s+1.\end{array}\right.}}} {\displaystyle \rho=\left\{\begin{array}{ll} \displaystyle \infty & \quad\textrm{if}\quad r < s+1\[5mm] \displaystyle 1 & \quad\textrm{if}\quad r = s+1\[5mm] \displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right. } \qHyperrphis s + 1 s @ @ a 1 , … , a s + 1 b 1 , … , b s q z = ∑ k = 0 ∞ ( a 1 , … , a s + 1 ; q ) k ( b 1 , … , b s ; q ) k z k ( q ; q ) k \qHyperrphis 𝑠 1 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑠 1 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 superscript subscript 𝑘 0 q-Pochhammer-symbol subscript 𝑎 1 … subscript 𝑎 𝑠 1 𝑞 𝑘 q-Pochhammer-symbol subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑘 superscript 𝑧 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{s+1}{s}@@{a_{1},\ldots,% a_{s+1}}{b_{1},\ldots,b_{s}}{q}{z}=\sum\limits_{k=0}^{\infty}\frac{\left(a_{1}% ,\ldots,a_{s+1};q\right)_{k}}{\left(b_{1},\ldots,b_{s};q\right)_{k}}\frac{z^{k% }}{\left(q;q\right)_{k}}}}} {\displaystyle \qHyperrphis{s+1}{s}@@{a_1,\ldots,a_{s+1}}{b_1,\ldots,b_s}{q}{z}= \sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_{s+1}}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}} \frac{z^k}{\qPochhammer{q}{q}{k}} }
lim q → 1 \qHyperrphis r s @ @ q a 1 , … , q a r q b 1 , … , q b s q ( q - 1 ) 1 + s - r z = \HyperpFq r s @ @ a 1 , … , a r b 1 , … , b s z formulae-sequence subscript → 𝑞 1 \qHyperrphis 𝑟 𝑠 @ @ superscript 𝑞 subscript 𝑎 1 … superscript 𝑞 subscript 𝑎 𝑟 superscript 𝑞 subscript 𝑏 1 … superscript 𝑞 subscript 𝑏 𝑠 𝑞 superscript 𝑞 1 1 𝑠 𝑟 𝑧 \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}% \qHyperrphis{r}{s}@@{q^{a_{1}},\ldots,q^{a_{r}}}{q^{b_{1}},\ldots,q^{b_{s}}}{q% }{(q-1)^{1+s-r}z}=\HyperpFq{r}{s}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s}}{z}% }}} {\displaystyle \lim\limits_{q\rightarrow 1} \qHyperrphis{r}{s}@@{q^{a_1},\ldots,q^{a_r}}{q^{b_1},\ldots,q^{b_s}}{q}{(q-1)^{1+s-r}z} =\HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z} } lim a r → ∞ \qHyperrphis r s @ @ a 1 , … , a r b 1 , … , b s q z a r = \qHyperrphis r - 1 s @ @ a 1 , … , a r - 1 b 1 , … , b s q z formulae-sequence subscript → subscript 𝑎 𝑟 \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 subscript 𝑎 𝑟 \qHyperrphis 𝑟 1 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{a_{r}\rightarrow\infty}% \qHyperrphis{r}{s}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s}}{q}{\frac{z}{a_{r}% }}=\qHyperrphis{r-1}{s}@@{a_{1},\ldots,a_{r-1}}{b_{1},\ldots,b_{s}}{q}{z}}}} {\displaystyle \lim\limits_{a_r\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{\frac{z}{a_r}}= \qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{z} } \qHyperrphis r s @ @ a 1 , … , a r - 1 , μ b 1 , … , b s - 1 , μ q z = \qHyperrphis r - 1 s - 1 @ @ a 1 , … , a r - 1 b 1 , … , b s - 1 q z formulae-sequence \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜇 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜇 𝑞 𝑧 \qHyperrphis 𝑟 1 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝑞 𝑧 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{r}{s}@@{a_{1},\ldots,a_% {r-1},\mu}{b_{1},\ldots,b_{s-1},\mu}{q}{z}=\qHyperrphis{r-1}{s-1}@@{a_{1},% \ldots,a_{r-1}}{b_{1},\ldots,b_{s-1}}{q}{z}}}} {\displaystyle \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{q}{z}= \qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{z} } lim λ → ∞ \qHyperrphis r s @ @ a 1 , … , a r - 1 , λ a r b 1 , … , b s q z λ = \qHyperrphis r - 1 s @ @ a 1 , … , a r - 1 b 1 , … , b s q a r z formulae-sequence subscript → 𝜆 \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜆 subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 𝜆 \qHyperrphis 𝑟 1 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 subscript 𝑎 𝑟 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\qHyperrphis{r}{s}@@{a_{1},\ldots,a_{r-1},\lambda a_{r}}{b_{1},\ldots,b% _{s}}{q}{\frac{z}{\lambda}}=\qHyperrphis{r-1}{s}@@{a_{1},\ldots,a_{r-1}}{b_{1}% ,\ldots,b_{s}}{q}{a_{r}z}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{q}{\frac{z}{\lambda}} =\qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{a_rz} } lim λ → ∞ \qHyperrphis r s @ @ a 1 , … , a r b 1 , … , b s - 1 , λ b s q λ z = \qHyperrphis r s - 1 @ @ a 1 , … , a r b 1 , … , b s - 1 q z b s formulae-sequence subscript → 𝜆 \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜆 subscript 𝑏 𝑠 𝑞 𝜆 𝑧 \qHyperrphis 𝑟 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝑞 𝑧 subscript 𝑏 𝑠 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\qHyperrphis{r}{s}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s-1},\lambda b% _{s}}{q}{\lambda z}=\qHyperrphis{r}{s-1}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_% {s-1}}{q}{\frac{z}{b_{s}}}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{\lambda z}= \qHyperrphis{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{q}{\frac{z}{b_s}} } lim λ → ∞ \qHyperrphis r s @ @ a 1 , … , a r - 1 , λ a r b 1 , … , b s - 1 , λ b s q z = \qHyperrphis r - 1 s - 1 @ @ a 1 , … , a r - 1 b 1 , … , b s - 1 q a r z b s formulae-sequence subscript → 𝜆 \qHyperrphis 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜆 subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜆 subscript 𝑏 𝑠 𝑞 𝑧 \qHyperrphis 𝑟 1 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝑞 subscript 𝑎 𝑟 𝑧 subscript 𝑏 𝑠 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\qHyperrphis{r}{s}@@{a_{1},\ldots,a_{r-1},\lambda a_{r}}{b_{1},\ldots,b% _{s-1},\lambda b_{s}}{q}{z}=\qHyperrphis{r-1}{s-1}@@{a_{1},\ldots,a_{r-1}}{b_{% 1},\ldots,b_{s-1}}{q}{\frac{a_{r}z}{b_{s}}}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{z} =\qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{\frac{a_rz}{b_s}} } Q N ( q - x ; α , β , N ; q ) = ∑ k = 0 N ( α β q N + 1 ; q ) k ( q - x ; q ) k ( α q ; q ) k ( q ; q ) k q k q-Hahn-polynomial-Q 𝑁 superscript 𝑞 𝑥 𝛼 𝛽 𝑁 𝑞 superscript subscript 𝑘 0 𝑁 q-Pochhammer-symbol 𝛼 𝛽 superscript 𝑞 𝑁 1 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑞 𝑥 𝑞 𝑘 q-Pochhammer-symbol 𝛼 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(q^{-x};\alpha,\beta,N;% q\right)=\sum_{k=0}^{N}\frac{\left(\alpha\beta q^{N+1};q\right)_{k}\left(q^{-x% };q\right)_{k}}{\left(\alpha q;q\right)_{k}\left(q;q\right)_{k}}q^{k}}}} {\displaystyle \qHahn{N}@{q^{-x}}{\alpha}{\beta}{N}{q}=\sum_{k=0}^N \frac{\qPochhammer{\alpha\beta q^{N+1}}{q}{k}\qPochhammer{q^{-x}}{q}{k}}{\qPochhammer{\alpha q}{q}{k}\qPochhammer{q}{q}{k}}q^k }