Chapter 46: hypergeometric function

46.1 linear space of function

https://www.bilibili.com/video/BV1PX4y167RS

Taylor vs. Fourier^{[@ref(taylor-vs.-fourier)]}

\[ f\left(x\right)=a_{{\scriptscriptstyle 0}}x^{0}+a_{{\scriptscriptstyle 1}}x^{1}+a_{{\scriptscriptstyle 2}}x^{2}+\cdots=\sum\limits _{k=0}^{\infty}a_{{\scriptscriptstyle k}}x^{k} \]

\[ f\left(x\right)=x_{{\scriptscriptstyle 0}}x^{0}+x_{{\scriptscriptstyle 1}}x^{1}+x_{{\scriptscriptstyle 2}}x^{2}+\cdots=\sum\limits _{k=0}^{\infty}x_{{\scriptscriptstyle k}}x^{k} \]

Def: 48.3

\[ \left\langle f\middle|g\right\rangle =\int_{a}^{b}\overline{f\left(x\right)}g\left(x\right)\mathrm{d}x\overset{f,g:\mathbb{R}\rightarrow\mathbb{R}}{=}\int_{a}^{b}f\left(x\right)g\left(x\right)\mathrm{d}x \]

Dirac bracket^[48.5]

\[ \left\langle x^{2}\middle|x\right\rangle \overset{x^{2},x:\mathbb{R}\rightarrow\mathbb{R}}{=}\int_{a}^{b}x^{2}x\mathrm{d}x=\int_{a}^{b}x^{3}\mathrm{d}x=\left[\dfrac{x^{4}}{4}\right]_{a}^{b}\not\equiv0 \]

\[ x^{0}\not\perp x^{1},x^{1}\not\perp x^{2},\cdots \]

\[ \left\langle x_{{\scriptscriptstyle m}}\middle|x^{n}\right\rangle =\int_{a}^{b}x_{{\scriptscriptstyle m}}x^{n}\mathrm{d}x=\delta_{{\scriptscriptstyle mn}} \]

\[ \left\langle 1\middle|x^{n}\right\rangle =\int_{a}^{b}x_{{\scriptscriptstyle 0}}x^{n}\mathrm{d}x=\delta_{{\scriptscriptstyle 0n}}\Rightarrow x_{{\scriptscriptstyle 0}}=\delta\left(x\right)=\delta\left(x-0\right) \]

\[ \left\langle x_{{\scriptscriptstyle m}}\middle|x^{n}\right\rangle =\int_{a}^{b}x_{{\scriptscriptstyle m}}x^{n}\mathrm{d}x=\delta_{{\scriptscriptstyle mn}}\Rightarrow x_{{\scriptscriptstyle m}}=\dfrac{\left(-1\right)^{m}}{m!}\delta^{\left(m\right)}\left(x\right) \]

\[ \left|f\right\rangle =1\left|f\right\rangle =\left(\sum\limits _{i}\left|\hat{f}_{{\scriptscriptstyle i}}\right\rangle \left\langle \hat{f}_{{\scriptscriptstyle i}}\right|\right)\left|f\right\rangle =\sum\limits _{i}\left|\hat{f}_{{\scriptscriptstyle i}}\right\rangle \left\langle \hat{f}_{{\scriptscriptstyle i}}\middle|f\right\rangle \]

\[ \begin{aligned} \left|f\right\rangle = & 1\left|f\right\rangle =\left(\sum\limits _{i}\left|\hat{f}_{{\scriptscriptstyle i}}\right\rangle \left\langle \hat{f}_{{\scriptscriptstyle i}}\right|\right)\left|f\right\rangle =\sum\limits _{i}\left|\hat{f}_{{\scriptscriptstyle i}}\right\rangle \left\langle \hat{f}_{{\scriptscriptstyle i}}\middle|f\right\rangle \\ = & 1\left|f\right\rangle =\left(\sum\limits _{n}\left|x^{n}\right\rangle \left\langle x^{n}\right|\right)\left|f\right\rangle =\sum\limits _{n}\left|x^{n}\right\rangle \left\langle x^{n}\middle|f\right\rangle =\sum\limits _{n}\left\langle x^{n}\middle|f\right\rangle \left|x^{n}\right\rangle \\ \left\langle x^{n}\right|\left|f\right\rangle = & \left\langle x^{n}\middle|f\right\rangle =\int_{a}^{b}x_{{\scriptscriptstyle n}}f\left(x\right)\mathrm{d}x=\int_{a}^{b}\dfrac{\left(-1\right)^{n}}{n!}\delta^{\left(n\right)}\left(x\right)f\left(x\right)\mathrm{d}x=\dfrac{f^{\left(n\right)}\left(0\right)}{n!}\\ \left|f\right\rangle = & \sum\limits _{n}\left\langle x^{n}\middle|f\right\rangle \left|x^{n}\right\rangle =\sum\limits _{n}\dfrac{f^{\left(n\right)}\left(0\right)}{n!}\left|x^{n}\right\rangle \\ \left|f\right\rangle = & \sum\limits _{n}\dfrac{f^{\left(n\right)}\left(0\right)}{n!}\left|x^{n}\right\rangle \\ \Downarrow\\ f\left(x\right)= & \sum\limits _{n}\dfrac{f^{\left(n\right)}\left(0\right)}{n!}x^{n} \end{aligned} \]

46.2 beta function

https://www.bilibili.com/video/BV1pa4y1P7Da

\[ \begin{aligned} \dbinom{n}{k}=\mathrm{C}_{k}^{n}= & \dfrac{n!}{\left(n-k\right)!k!}\\ = & \dfrac{n\left(n-1\right)\cdots\left(n-k+1\right)}{k!},\begin{cases} n\in\mathbb{N}\\ k\in\left(\left\{ 0\right\} \cup\mathbb{N}\right) \end{cases}\\ \dbinom{r}{k}= & \begin{cases} \dfrac{r\left(r-1\right)\cdots\left(r-k+1\right)}{k!} & k\ge0,k\in\mathbb{Z}\\ 0 & k<0,k\in\mathbb{Z} \end{cases} \end{aligned} \]

\[ \sum\limits _{k=0}^{n}\dbinom{r}{k}\left(\cdot\right) \]

\[ \sum\limits _{k=-\infty}^{n}\dbinom{r}{k}\left(\cdot\right)=\left(0+0+\cdots\right)+\sum\limits _{k=0}^{n}\dbinom{r}{k}\left(\cdot\right) \]

\[ \sum\limits _{k=-\infty}^{\infty}\dbinom{r}{k}\left(\cdot\right) \]

\[ n!=\Gamma\left(n+1\right)=\int_{0}^{\infty}x^{\left(n+1\right)-1}\mathrm{e}^{-x}\mathrm{d}x \]

\[ \Gamma\left(z\right)=\int_{0}^{\infty}x^{z-1}\mathrm{e}^{-x}\mathrm{d}x \]

\[ \Gamma\left(z+1\right)=z\Gamma\left(z\right) \]

\[ \Gamma\left(z\right)\Gamma\left(1-z\right)=\dfrac{\pi}{\sin\left(\pi z\right)} \]

\[ \begin{aligned} \Gamma\left(z\right)\Gamma\left(1-z\right)= & \dfrac{\pi}{\sin\left(\pi z\right)}\\ \left[\Gamma\left(z\right)\Gamma\left(1-z\right)\right]_{z=-n}= & \left[\dfrac{\pi}{\sin\left(\pi z\right)}\right]_{z=-n},n\in\mathbb{N}\\ \Gamma\left(-n\right)n!=\Gamma\left(n+1\right)=\Gamma\left(-n\right)\Gamma\left(1-\left(-n\right)\right)= & \dfrac{\pi}{\sin\left(\pi\left(-n\right)\right)}=\dfrac{\pi}{-\sin\left(n\pi\right)}\\ \Gamma\left(-n\right)= & \dfrac{-\pi}{n!\sin\left(n\pi\right)}=\dfrac{-\pi}{n!0}\rightarrow-\infty,n\in\mathbb{N} \end{aligned} \]

\[ \begin{aligned} \dbinom{n}{k}=\mathrm{C}_{k}^{n}= & \dfrac{n!}{\left(n-k\right)!k!}\\ = & \dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)} \end{aligned} \]

\[ \dbinom{n}{k}=\dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)} \]

\[ \dbinom{n}{k}=\dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)}\overset{k\le0}{=}\begin{cases} \dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n+1\right)\Gamma\left(1\right)}=\dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n+1\right)1}=1 & k=0\\ \dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\left(-\infty\right)}=0 & k\le-1,k\in\mathbb{Z} \end{cases} \]

beta function = \(\beta\) function

Definition 46.1 beta function = \(\beta\) function

\[ B\left(p,q\right)=\int_{0}^{1}x^{p-1}\left(1-x\right)^{q-1}\mathrm{d}x=\dfrac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(p+q\right)} \]

\[ \begin{aligned} \dbinom{n}{k}= & \dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)}\\ \left[\dbinom{n}{k}\right]_{{\scriptscriptstyle \begin{cases} n=a+b\\ k=a \end{cases}}}= & \left[\dfrac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)}\right]_{{\scriptscriptstyle \begin{cases} n=a+b\\ k=a \end{cases}}}\\ \dbinom{a+b}{a}= & \dfrac{\Gamma\left(a+b+1\right)}{\Gamma\left(a+b-a+1\right)\Gamma\left(a+1\right)}\\ = & \dfrac{\Gamma\left(a+b+1\right)}{\Gamma\left(b+1\right)\Gamma\left(a+1\right)} \end{aligned} \]

\[ \begin{aligned} B\left(p,q\right)= & \dfrac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(p+q\right)}\\ \left[B\left(p,q\right)\right]_{{\scriptscriptstyle \begin{cases} p=a+1\\ q=b+1 \end{cases}}}= & \left[\dfrac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(p+q\right)}\right]_{{\scriptscriptstyle \begin{cases} p=a+1\\ q=b+1 \end{cases}}}\\ B\left(a+1,b+1\right)= & \dfrac{\Gamma\left(a+1\right)\Gamma\left(b+1\right)}{\Gamma\left(a+1+b+1\right)}\\ = & \dfrac{\Gamma\left(a+1\right)\Gamma\left(b+1\right)}{\Gamma\left(\left[a+b+1\right]+1\right)}=\dfrac{\Gamma\left(a+1\right)\Gamma\left(b+1\right)}{\left[a+b+1\right]\Gamma\left(a+b+1\right)} \end{aligned} \]

\[ \begin{aligned} \dbinom{a+b}{a}= & \dfrac{\Gamma\left(a+b+1\right)}{\Gamma\left(b+1\right)\Gamma\left(a+1\right)}=\dfrac{1}{\dfrac{\Gamma\left(b+1\right)\Gamma\left(a+1\right)}{\Gamma\left(a+b+1\right)}}\\ = & \dfrac{1}{\left[a+b+1\right]\dfrac{\Gamma\left(b+1\right)\Gamma\left(a+1\right)}{\left[a+b+1\right]\Gamma\left(a+b+1\right)}}\\ = & \dfrac{1}{\left[a+b+1\right]B\left(a+1,b+1\right)} \end{aligned} \]

https://en.wikipedia.org/wiki/Beta_function

https://en.wikipedia.org/wiki/Beta_function#Other_identities_and_formulas

https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function

https://www.bilibili.com/video/BV1pa4y1P7Da/?t=4m