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Beta Function

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The beta function B(p,q) is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first kind). It is defined by

B(p,q)=(Gamma(p)Gamma(q))/(Gamma(p+q))=((p-1)!(q-1)!)/((p+q-1)!).
(1)

The beta function B(a,b) is implemented in the Wolfram Language as Beta[a, b].

To derive the integral representation of the beta function, write the product of two factorials as

m!n!=int_0^inftye^(-u)u^mduint_0^inftye^(-v)v^ndv.
(2)

Now, let u=x^2, v=y^2, so

m!n!=4int_0^inftye^(-x^2)x^(2m+1)dxint_0^inftye^(-y^2)y^(2n+1)dy
(3)
=int_(-infty)^inftyint_(-infty)^inftye^(-(x^2+y^2))|x|^(2m+1)|y|^(2n+1)dxdy.
(4)

Transforming to polar coordinates with x=rcostheta, y=rsintheta

m!n!=int_0^(2pi)int_0^inftye^(-r^2)|rcostheta|^(2m+1)|rsintheta|^(2n+1)rdrdtheta
(5)
=int_0^inftye^(-r^2)r^(2m+2n+3)drint_0^(2pi)|cos^(2m+1)thetasin^(2n+1)theta|dtheta
(6)
=4int_0^inftye^(-r^2)r^(2m+2n+3)drint_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta
(7)
=2(m+n+1)!int_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta.
(8)

The beta function is then defined by

B(m+1,n+1)=2int_0^(pi/2)cos^(2m+1)thetasin^(2n+1)thetadtheta
(9)
=(m!n!)/((m+n+1)!).
(10)

Rewriting the arguments then gives the usual form for the beta function,

B(p,q)=(Gamma(p)Gamma(q))/(Gamma(p+q))
(11)
=((p-1)!(q-1)!)/((p+q-1)!).
(12)

By symmetry,

B(p,q)=B(q,p).
(13)

The general trigonometric form is

int_0^(pi/2)sin^nxcos^mxdx=1/2B(1/2(n+1),1/2(m+1)).
(14)

Equation (14) can be transformed to an integral over polynomials by letting u=cos^2theta,

B(m+1,n+1)=(m!n!)/((m+n+1)!)
(15)
=int_0^1u^m(1-u)^ndu
(16)
B(m,n)=(Gamma(m)Gamma(n))/(Gamma(m+n))
(17)
=int_0^1u^(m-1)(1-u)^(n-1)du.
(18)

For any z_1,z_2 with R[z_1],R[z_2]>0,

B(z_1,z_2)=B(z_2,z_1)
(19)

(Krantz 1999, p. 158).

To put it in a form which can be used to derive the Legendre duplication formula, let x=sqrt(u), so u=x^2 and du=2xdx, and

B(m,n)=int_0^1x^(2(m-1))(1-x^2)^(n-1)(2xdx)
(20)
=2int_0^1x^(2m-1)(1-x^2)^(n-1)dx.
(21)

To put it in a form which can be used to develop integral representations of the Bessel functions and hypergeometric function, let u=x^2/(1-x^2), so

B(m+1,n+1)=int_0^infty(u^mdu)/((1+u)^(m+n+2)).
(22)

Derivatives of the beta function are given by

d/(da)B(a,b)=B(a,b)[psi_0(a)-psi_0(a+b)]
(23)
d/(db)B(a,b)=B(a,b)[psi_0(b)-psi_0(a+b)]
(24)
(d^2)/(db^2)B(a,b)=B(a,b){[psi_0(b)-psi_0(a+b)]^2+psi_1(b)-psi_1(a+b)},
(25)
(d^2)/(dadb)B(a,b)=B(a,b){[psi_0(a)-psi_0(a+b)]×[psi_0(b)-psi_0(a+b)]-psi_1(a+b)},
(26)

where psi_n(x) is the polygamma function.

Various identities can be derived using the Gauss multiplication formula

B(np,nq)=(Gamma(np)Gamma(nq))/(Gamma(n(p+q)))
(27)
=n^(-nq)(B(p,q)B(p+1/n,q)...B(p+(n-1)/n,q))/(B(q,q)B(2q,q)...B((n-1)q,q)).
(28)

Additional identities include

B(p,q+1)=(Gamma(p)Gamma(q+1))/(Gamma(p+q+1))
(29)
=q/p(Gamma(p+1)Gamma(q))/(Gamma((p+1)+q))
(30)
=q/pB(p+1,q)
(31)
B(p,q)=B(p+1,q)+B(p,q+1)
(32)
B(p,q+1)=q/(p+q)B(p,q).
(33)

If n is a positive integer, then

B(p,n+1)=(1·2...n)/(p(p+1)...(p+n))
(34)
B(p,p)B(p+1/2,p+1/2)=pi/(2^(4p-1)p)
(35)
B(p+q)B(p+q,r)=B(q,r)B(q+r,p).
(36)

The beta function is also given by the product

B(x,y)=(x+y)/(xy)product_(k=1)^infty(1+(x+y)/k)/((1+x/k)(1+y/k))
(37)

(Andrews et al. 1999, p. 8).

Gosper gave the general formulas

product_(i=0)^(2n)B(i/(2n+1)+a,i/(2n+1)+b) =((2n+1)^((2n+1)/2)pi^nB(n,1/2[(b+a)(2n+1)+1])B(a(2n+1),b(2n+1)))/((n-1)!)
(38)

for odd n, and

product_(i=0)^(2n-1)B(i/(2n)+a,i/(2n)+b) =(n^npi^nB(n,2(a+b)n)B(2an,2bn))/(2^(2(a+b)n-n-1)(n-1)!B((a+b)n,(a+b+1)n)),
(39)

which are an immediate consequence of the analogous identities for gamma functions. Plugging n=1 and n=2 into the above give the special cases

B(a,b)B(a+1/3,b+1/3)B(a+2/3,b+2/3)=(6pisqrt(3)B(3a,3b))/(1+3(a+b))
(40)
B(a,b)B(a+1/4,b+1/4)B(a+1/2,b+1/2)B(a+3/4,b+3/4) =(2^(3-4(a+b))pi^2B(4a,4b))/((a+b)[1+4(a+b)]B(2(a+b),2(a+b+1))).
(41)

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