Chapter 46: hypergeometric function
46.1 linear space of function
https://www.bilibili.com/video/BV1PX4y167RS
Taylor vs. Fourier[@ref(taylor-vs.-fourier)]
f(x)=a0x0+a1x1+a2x2+⋯=∞∑k=0akxk
f(x)=x0x0+x1x1+x2x2+⋯=∞∑k=0xkxk
Def: 48.3
⟨f|g⟩=∫ba¯f(x)g(x)dxf,g:R→R=∫baf(x)g(x)dx
⟨x2|x⟩x2,x:R→R=∫bax2xdx=∫bax3dx=[x44]ba≢0
x0⊥̸x1,x1⊥̸x2,⋯
⟨xm|xn⟩=∫baxmxndx=δmn
⟨1|xn⟩=∫bax0xndx=δ0n⇒x0=δ(x)=δ(x−0)
⟨xm|xn⟩=∫baxmxndx=δmn⇒xm=(−1)mm!δ(m)(x)
|f⟩=1|f⟩=(∑i|ˆfi⟩⟨ˆfi|)|f⟩=∑i|ˆfi⟩⟨ˆfi|f⟩
|f⟩=1|f⟩=(∑i|ˆfi⟩⟨ˆfi|)|f⟩=∑i|ˆfi⟩⟨ˆfi|f⟩=1|f⟩=(∑n|xn⟩⟨xn|)|f⟩=∑n|xn⟩⟨xn|f⟩=∑n⟨xn|f⟩|xn⟩⟨xn||f⟩=⟨xn|f⟩=∫baxnf(x)dx=∫ba(−1)nn!δ(n)(x)f(x)dx=f(n)(0)n!|f⟩=∑n⟨xn|f⟩|xn⟩=∑nf(n)(0)n!|xn⟩|f⟩=∑nf(n)(0)n!|xn⟩⇓f(x)=∑nf(n)(0)n!xn
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
46.4 recursion
https://www.bilibili.com/video/BV1FV4y1Z7jm
https://www.bilibili.com/video/BV1Sg4y1L7DF
46.5 mean and variance of discrete probability distributions
https://www.bilibili.com/video/BV1Tk4y1n7NX
46.6 CORDIC = coordinate rotation digital computer
https://www.bilibili.com/video/BV1ge411e7K7
https://space.bilibili.com/11008987/channel/collectiondetail?sid=2053177