Chapter 46: hypergeometric function

46.1 linear space of function

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

quantum state[48.2.3]

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:RR=baf(x)g(x)dx

Dirac bracket[48.5]

x2|xx2,x:RR=bax2xdx=bax3dx=[x44]ba0

x0⊥̸x1,x1⊥̸x2,

xm|xn=baxmxndx=δmn

1|xn=bax0xndx=δ0nx0=δ(x)=δ(x0)

xm|xn=baxmxndx=δmnxm=(1)mm!δ(m)(x)

|f=1|f=(i|ˆfiˆfi|)|f=i|ˆfiˆfi|f

|f=1|f=(i|ˆfiˆfi|)|f=i|ˆfiˆfi|f=1|f=(n|xnxn|)|f=n|xnxn|f=nxn|f|xnxn||f=xn|f=baxnf(x)dx=ba(1)nn!δ(n)(x)f(x)dx=f(n)(0)n!|f=nxn|f|xn=nf(n)(0)n!|xn|f=nf(n)(0)n!|xnf(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

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

46.5 mean and variance of discrete probability distributions

https://www.bilibili.com/video/BV1Tk4y1n7NX