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=abf(x)¯g(x)dx=f,g:RRabf(x)g(x)dx

Dirac bracket[48.5]

x2|x=x2,x:RRabx2xdx=abx3dx=[x44]ab0

x0⊥̸x1,x1⊥̸x2,

xm|xn=abxmxndx=δmn

1|xn=abx0xndx=δ0nx0=δ(x)=δ(x0)

xm|xn=abxmxndx=δmnxm=(1)mm!δ(m)(x)

|f=1|f=(i|f^if^i|)|f=i|f^if^i|f

|f=1|f=(i|f^if^i|)|f=i|f^if^i|f=1|f=(n|xnxn|)|f=n|xnxn|f=nxn|f|xnxn||f=xn|f=abxnf(x)dx=ab(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

(nk)=Ckn=n!(nk)!k!=n(n1)(nk+1)k!,{nNk({0}N)(rk)={r(r1)(rk+1)k!k0,kZ0k<0,kZ

k=0n(rk)()

k=n(rk)()=(0+0+)+k=0n(rk)()

k=(rk)()


n!=Γ(n+1)=0x(n+1)1exdx

Γ(z)=0xz1exdx

Γ(z+1)=zΓ(z)

Γ(z)Γ(1z)=πsin(πz)

Γ(z)Γ(1z)=πsin(πz)[Γ(z)Γ(1z)]z=n=[πsin(πz)]z=n,nNΓ(n)n!=Γ(n+1)=Γ(n)Γ(1(n))=πsin(π(n))=πsin(nπ)Γ(n)=πn!sin(nπ)=πn!0,nN

(nk)=Ckn=n!(nk)!k!=Γ(n+1)Γ(nk+1)Γ(k+1)

(nk)=Γ(n+1)Γ(nk+1)Γ(k+1)

(nk)=Γ(n+1)Γ(nk+1)Γ(k+1)=k0{Γ(n+1)Γ(n+1)Γ(1)=Γ(n+1)Γ(n+1)1=1k=0Γ(n+1)Γ(nk+1)()=0k1,kZ

beta function = β function

Definition 46.1 beta function = β function

B(p,q)=01xp1(1x)q1dx=Γ(p)Γ(q)Γ(p+q)

(nk)=Γ(n+1)Γ(nk+1)Γ(k+1)[(nk)]{n=a+bk=a=[Γ(n+1)Γ(nk+1)Γ(k+1)]{n=a+bk=a(a+ba)=Γ(a+b+1)Γ(a+ba+1)Γ(a+1)=Γ(a+b+1)Γ(b+1)Γ(a+1)

B(p,q)=Γ(p)Γ(q)Γ(p+q)[B(p,q)]{p=a+1q=b+1=[Γ(p)Γ(q)Γ(p+q)]{p=a+1q=b+1B(a+1,b+1)=Γ(a+1)Γ(b+1)Γ(a+1+b+1)=Γ(a+1)Γ(b+1)Γ([a+b+1]+1)=Γ(a+1)Γ(b+1)[a+b+1]Γ(a+b+1)

(a+ba)=Γ(a+b+1)Γ(b+1)Γ(a+1)=1Γ(b+1)Γ(a+1)Γ(a+b+1)=1[a+b+1]Γ(b+1)Γ(a+1)[a+b+1]Γ(a+b+1)=1[a+b+1]B(a+1,b+1)


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