\HyperpFq r s @ @ a 1 , … , a r b 1 , … , b s z := ∑ k = 0 ∞ ( a 1 , … , a r ) k ( b 1 , … , b s ) k z k k ! assign \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑧 superscript subscript 𝑘 0 Pochhammer-symbol subscript 𝑎 1 … subscript 𝑎 𝑟 𝑘 Pochhammer-symbol subscript 𝑏 1 … subscript 𝑏 𝑠 𝑘 superscript 𝑧 𝑘 𝑘 {\displaystyle{\displaystyle{\displaystyle{}\HyperpFq{r}{s}@@{a_{1},\ldots,a_{% r}}{b_{1},\ldots,b_{s}}{z}:=\sum\limits_{k=0}^{\infty}\frac{{\left(a_{1},% \ldots,a_{r}\right)_{k}}}{{\left(b_{1},\ldots,b_{s}\right)_{k}}}\frac{z^{k}}{k% !}}}} {\displaystyle \index{Hypergeometric function} \HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}:= \sum\limits_{k=0}^{\infty}\frac{\pochhammer{a_1,\ldots,a_r}{k}}{\pochhammer{b_1,\ldots,b_s}{k}} \frac{z^k}{k!} } ( a 1 , … , a r ) k := ( a 1 ) k ⋯ ( a r ) k assign Pochhammer-symbol subscript 𝑎 1 … subscript 𝑎 𝑟 𝑘 Pochhammer-symbol subscript 𝑎 1 𝑘 ⋯ Pochhammer-symbol subscript 𝑎 𝑟 𝑘 {\displaystyle{\displaystyle{\displaystyle{\left(a_{1},\ldots,a_{r}\right)_{k}% }:={\left(a_{1}\right)_{k}}\cdots{\left(a_{r}\right)_{k}}}}} {\displaystyle \pochhammer{a_1,\ldots,a_r}{k}:=\pochhammer{a_1}{k}\cdots\pochhammer{a_r}{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. } \HyperpFq r s @ @ a 1 , … , a r - 1 , μ b 1 , … , b s - 1 , μ z = \HyperpFq r - 1 s - 1 @ @ a 1 , … , a r - 1 b 1 , … , b s - 1 z formulae-sequence \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜇 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜇 𝑧 \HyperpFq 𝑟 1 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝑧 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{r}{s}@@{a_{1},\ldots,a_{r-% 1},\mu}{b_{1},\ldots,b_{s-1},\mu}{z}=\HyperpFq{r-1}{s-1}@@{a_{1},\ldots,a_{r-1% }}{b_{1},\ldots,b_{s-1}}{z}}}} {\displaystyle \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z} =\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z} } lim λ → ∞ \HyperpFq r s @ @ a 1 , … , a r - 1 , λ a r b 1 , … , b s z λ = \HyperpFq r - 1 s @ @ a 1 , … , a r - 1 b 1 , … , b s a r z formulae-sequence subscript → 𝜆 \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜆 subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑧 𝜆 \HyperpFq 𝑟 1 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 subscript 𝑎 𝑟 𝑧 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\HyperpFq{r}{s}@@{a_{1},\ldots,a_{r-1},\lambda a_{r}}{b_{1},\ldots,b_{s% }}{\frac{z}{\lambda}}=\HyperpFq{r-1}{s}@@{a_{1},\ldots,a_{r-1}}{b_{1},\ldots,b% _{s}}{a_{r}z}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} =\HyperpFq{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz} } lim λ → ∞ \HyperpFq r s @ @ a 1 , … , a r b 1 , … , b s - 1 , λ b s λ z = \HyperpFq r s - 1 @ @ a 1 , … , a r b 1 , … , b s - 1 z b s formulae-sequence subscript → 𝜆 \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜆 subscript 𝑏 𝑠 𝜆 𝑧 \HyperpFq 𝑟 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝑧 subscript 𝑏 𝑠 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\HyperpFq{r}{s}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s-1},\lambda b_{s% }}{\lambda z}=\HyperpFq{r}{s-1}@@{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{s-1}}{% \frac{z}{b_{s}}}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= \HyperpFq{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}} } lim λ → ∞ \HyperpFq r s @ @ a 1 , … , a r - 1 , λ a r b 1 , … , b s - 1 , λ b s z = \HyperpFq r - 1 s - 1 @ @ a 1 , … , a r - 1 b 1 , … , b s - 1 a r z b s formulae-sequence subscript → 𝜆 \HyperpFq 𝑟 𝑠 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 𝜆 subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 1 𝜆 subscript 𝑏 𝑠 𝑧 \HyperpFq 𝑟 1 𝑠 1 @ @ subscript 𝑎 1 … subscript 𝑎 𝑟 1 subscript 𝑏 1 … subscript 𝑏 𝑠 1 subscript 𝑎 𝑟 𝑧 subscript 𝑏 𝑠 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}\HyperpFq{r}{s}@@{a_{1},\ldots,a_{r-1},\lambda a_{r}}{b_{1},\ldots,b_{s% -1},\lambda b_{s}}{z}=\HyperpFq{r-1}{s-1}@@{a_{1},\ldots,a_{r-1}}{b_{1},\ldots% ,b_{s-1}}{\frac{a_{r}z}{b_{s}}}}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} =\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}} } Q N ( x ; α , β , N ) = ∑ k = 0 N ( N + α + β + 1 ) k ( - x ) k ( α + 1 ) k k ! Hahn-polynomial-Q 𝑁 𝑥 𝛼 𝛽 𝑁 superscript subscript 𝑘 0 𝑁 Pochhammer-symbol 𝑁 𝛼 𝛽 1 𝑘 Pochhammer-symbol 𝑥 𝑘 Pochhammer-symbol 𝛼 1 𝑘 𝑘 {\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(x;\alpha,\beta,N\right% )=\sum_{k=0}^{N}\frac{{\left(N+\alpha+\beta+1\right)_{k}}{\left(-x\right)_{k}}% }{{\left(\alpha+1\right)_{k}}k!}}}} {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N}=\sum_{k=0}^N\frac{\pochhammer{N+\alpha+\beta+1}{k}\pochhammer{-x}{k}}{\pochhammer{\alpha+1}{k}k!} } [ f ( t ) ] N := ∑ k = 0 N f ( k ) ( 0 ) k ! t k assign subscript delimited-[] 𝑓 𝑡 𝑁 superscript subscript 𝑘 0 𝑁 superscript 𝑓 𝑘 0 𝑘 superscript 𝑡 𝑘 {\displaystyle{\displaystyle{\displaystyle\left[f(t)\right]_{N}:=\sum_{k=0}^{N% }\frac{f^{(k)}(0)}{k!}t^{k}}}} {\displaystyle \left[f(t)\right]_N:=\sum_{k=0}^N\frac{f^{(k)}(0)}{k!}t^k } e t \HyperpFq 11 @ @ - x - N - t p = ∑ k = 0 ∞ t k k ! ∑ m = 0 x ( - x ) m ( - N ) m m ! ( - t p ) m 𝑡 \HyperpFq 11 @ @ 𝑥 𝑁 𝑡 𝑝 superscript subscript 𝑘 0 superscript 𝑡 𝑘 𝑘 superscript subscript 𝑚 0 𝑥 Pochhammer-symbol 𝑥 𝑚 Pochhammer-symbol 𝑁 𝑚 𝑚 superscript 𝑡 𝑝 𝑚 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{1}{1}@@{% -x}{-N}{-\frac{t}{p}}=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\sum_{m=0}^{x}\frac{{% \left(-x\right)_{m}}}{{\left(-N\right)_{m}}m!}\left(-\frac{t}{p}\right)^{m}}}} {\displaystyle \expe^t\,\HyperpFq{1}{1}@@{-x}{-N}{-\frac{t}{p}}= \sum_{k=0}^{\infty}\frac{t^k}{k!}\sum_{m=0}^x\frac{\pochhammer{-x}{m}}{\pochhammer{-N}{m}m!} \left(-\frac{t}{p}\right)^m } ∑ n = 0 N K n ( x ; p , N ) n ! t n superscript subscript 𝑛 0 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\frac{K_{n}\!\left(x;p% ,N\right)}{n!}t^{n}}}} {\displaystyle \sum_{n=0}^N\frac{\Krawtchouk{n}@{x}{p}{N}}{n!}t^n } e z = \HyperpFq 00 @ @ - - z fragments 𝑧 \HyperpFq 00 @ @ z {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{z}}=\HyperpFq{0}{0}@@{-% }{-}{z}}}} {\displaystyle \expe^z=\HyperpFq{0}{0}@@{-}{-}{z} } cos z = \HyperpFq 01 @ @ - 1 2 - z 2 4 𝑧 \HyperpFq 01 @ @ 1 2 superscript 𝑧 2 4 {\displaystyle{\displaystyle{\displaystyle\cos z=\HyperpFq{0}{1}@@{-}{\frac{1}% {2}}{-\frac{z^{2}}{4}}}}} {\displaystyle \cos@@{z}=\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{-\frac{z^2}{4}} } sin z = z \HyperpFq 01 @ @ - 3 2 - z 2 4 𝑧 𝑧 \HyperpFq 01 @ @ 3 2 superscript 𝑧 2 4 {\displaystyle{\displaystyle{\displaystyle\sin z=z\,\HyperpFq{0}{1}@@{-}{\frac% {3}{2}}{-\frac{z^{2}}{4}}}}} {\displaystyle \sin@@{z}=z\,\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{-\frac{z^2}{4}} } J ν ( z ) := ( 1 2 z ) ν Γ ( ν + 1 ) \HyperpFq 01 @ @ - ν + 1 - z 2 4 assign Bessel-J 𝜈 𝑧 superscript 1 2 𝑧 𝜈 Euler-Gamma 𝜈 1 \HyperpFq 01 @ @ 𝜈 1 superscript 𝑧 2 4 {\displaystyle{\displaystyle{\displaystyle{}J_{\nu}\left(z\right):=\frac{\left% (\frac{1}{2}z\right)^{\nu}}{\Gamma\left(\nu+1\right)}\ \HyperpFq{0}{1}@@{-}{% \nu+1}{-\frac{z^{2}}{4}}}}} {\displaystyle \index{Bessel function} \BesselJ{\nu}@{z}:=\frac{\left(\frac{1}{2}z\right)^{\nu}}{\EulerGamma@{\nu+1}}\ \HyperpFq{0}{1}@@{-}{\nu+1}{-\frac{z^2}{4}} }