Hilbert Matrix

Author

Eric W. Weisstein
September 14, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/HilbertMatrix.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HilbertMatrix.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Hilbert Matrix

MatrixForm/@Table[HilbertMatrix[n],{n,5}]

Plot

ListDensityPlot[Reverse[HilbertMatrix[256]],Mesh->False,ColorFunction->Hue]//Timing

Inverse

MatrixForm/@Table[Inverse[HilbertMatrix[n]],{n,5}]

Inverse

h[n_,{i_,j_}]:=(-1)^(i+j)(i+j-1)Binomial[n+i-1,n-j]Binomial[n+j-1,n-i]Binomial[i+j-2,i-1]^2

HInv[n_]:=Table[h[n,{i,j}],{i,n},{j,n}]

MatrixForm/@Table[HInv[n],{n,5}]

MatrixForm/@Table[HInv[n]-Inverse[HilbertMatrix[n]],{n,5}]

Determinant

l=Table[1/Det[HilbertMatrix[n]],{n,10}]

l//N

FindSequenceFunction[l][n]

FullSimplify[%]

Products

(∏_k^(n-1))(2 k+1) ((Binomial[2 k, k])^2)==(((A^3) (2^(n (2 n-1)-(1/12))) (π^(-n)) G(n+(1/2)) G(n+(3/2)))/((e^(1/4)) ((G(n+1))^2)))

Product[(2k+1)((Binomial[2k,k])^2),{k,n-1}]//FullSimplify

Table[(((2^(-(1/12)+n (-1+2 n))) (Glaisher^3) (π^(-n)) BarnesG[(1/2)+n] BarnesG[(3/2)+n])/((E^(1/4)) ((BarnesG[1+n])^2))),{n,10}]

(((n^n) (∏_k^(n-1))((((n^2)-(k^2)))^(n-k)))/((∏_(k=0)^(n-1))(((k!))^2)))==(((n^n) (∏_k^(n-1))((((n^2)-(k^2)))^(n-k)))/((G(n+1))^2))

n^n ((Product[((((n^2)-(k^2)))^(n-k)),{k,n-1}])/(Product[(((k!))^2),{k,0,n-1}]))//InputForm

Table[(((n^n) (∏_k^(-1+n))(((-(k^2)+(n^2)))^(-k+n)))/((BarnesG[1+n])^2)),{n,10}]

((A (2^(n (2 n-1)-(1/12))) (π^(-n)) ((Γ(n+(1/2)))^(2 n)) (e^(-2 Derivative[1,0][ζ](-1,n+(1/2))-(1/12))))/((G(n+1))^2))

FunctionExpand[(((2^(-(1/12)+n (-1+2 n))) (E^(1/4)) (E^(-(1/3)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) Glaisher (π^(-n)) ((Gamma[(1/2)+n])^(2 n)))/((BarnesG[1+n])^2)),n∈Integers&&n>0]//FullSimplify

Table[(((2^(-(1/12)+n (-1+2 n))) (E^(1/4)) (E^(-(1/3)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) Glaisher (π^(-n)) ((Gamma[(1/2)+n])^(2 n)))/((BarnesG[1+n])^2)),{n,10}]//FunctionExpand//FullSimplify

Simpification

l=Table[BarnesG[n+3/2]BarnesG[n+1/2],{n,10}]

l=Table[((BarnesG[n+3/2]BarnesG[n+1/2])/((E^(1/4))/(Glaisher^3)/(2^(11/12))(π^n))),{n,10}]//Simplify

Through[{Numerator,Denominator}[%]]

Product[(2*k-1)^(2*n-2*k+1),{k,1,n}]

Table[(e^(1/4))/(Glaisher^3)/(2^(11/12))(π^n)(2^(-(1/12)+(n^2))) (E^(-(1/12)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) Glaisher (π^(-n)) ((Gamma[(1/2)+n])^(2 n))(2^(-((n^2)-1))),{n,5}]//FullSimplify

FullSimplify[(2^(-(1/12)+n (-1+2 n))) (Glaisher^3) (π^(-n))(e^(1/4))/(Glaisher^3)/(2^(11/12))(π^n)(2^(-(1/12)+(n^2))) (E^(-(1/12)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) Glaisher (π^(-n)) ((Gamma[(1/2)+n])^(2 n))(2^(-((n^2)-1)))(1/((E^(1/4)) ((BarnesG[1+n])^2)))]

Table[(((2^(-(1/12)+n (-1+2 n))) (E^(1/4)) (E^(-(1/3)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) Glaisher (π^(-n)) ((Gamma[(1/2)+n])^(2 n)))/((BarnesG[1+n])^2)),{n,10}]//FunctionExpand//FullSimplify

FunctionExpand[(((e^(1/4)) (E^(-(1/12)-2 Derivative[1,0][Zeta][-1,(1/2)+n])) ((Gamma[(1/2)+n])^(2 n)))/(Glaisher^2))]

Table[ (E^(-(1/12)-2 Derivative[1,0][Zeta][-1,(1/2)+n])),{n,5}]//FunctionExpand//FullSimplify

Table[((2^(n^2))/(3^(n^2))),{n,5}]

FunctionExpand[BarnesG[n+3/2]/BarnesG[n+1/2],n∈Integers&&n>0]

Table[%,{n,10}]

Table[(((2n-1)!!(Sqrt[π]))/(2^n)),{n,10}]