Strehl Identities

Author

Eric W. Weisstein
December 30, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/StrehlIdentities.nb.

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

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

Sums

(∑_(k=0)^n)(((({{n}, {k}})))^2) (((({{k+n}, {k}})))^2)

Table[Sum[Binomial[n,k]^2Binomial[n+k,k]^2,{k,0,n}],{n,10}]

Table[Sum[Binomial[n,k]Binomial[n+k,k]Binomial[k,j]^3,{j,0,n},{k,0,n}],{n,10}]

FullSimplify[
Sum[Binomial[n,k]Binomial[n+k,k]Binomial[k,j]^3,{j,0,n}],(k|n)∈Integers&&k>0&&n>0]

Sum[(((k+n)! HypergeometricPFQ[{-k,-k,-k},{1,1},-1])/((((k!))^2) Gamma[1-k+n])),{k,0,n}]

Sum[Binomial[n,k]Binomial[n+k,k]Binomial[k,j]^3,{k,0,n}]

FullSimplify[
Sum[Binomial[n,k]Binomial[n+k,k]Binomial[k,j]^3,{k,0,n}],(j|n)∈Integers&&j>0&&n>0]

V7

Sum[Binomial[n,k]^2Binomial[n+k,k]^2,{k,0,n}]

(∑_(k=0)^n)(((({{n}, {k}})))^3) (((({{k+n}, {k}})))^3)

Table[Sum[Binomial[n,k]^3Binomial[n+k,k]^3,{k,0,n}],{n,4}]

V7

Sum[Binomial[n,k]^3Binomial[n+k,k]^3,{k,0,n}]

FullSimplify[%]

(∑_(j=0)^n)((Binomial[n, j])^3)==(∑_(k=0)^n)((Binomial[n, k])^2) (Binomial[2 k, n])==(∑_(k=0)^n)((Binomial[n, k])^2) (Binomial[2 (n-k), n])

FullSimplify[{
Sum[Binomial[n,j]^3,{j,0,n}],Sum[Binomial[n,k]^2Binomial[2k,n],{k,0,n}],
Sum[Binomial[n,k]^2Binomial[2(n-k),n],{k,0,n}]
},n∈Integers]//ColumnForm

Table[Sum[Binomial[n,j]^3,{j,0,n}],{n,0,10}]

Table[(1/(Gamma[1+n]))((((-4))^n) (Sqrt[π]) HypergeometricPFQRegularized[{(1/2)-(n/2),-n,-(n/2)},{1,(1/2)-n},1]),{n,0,10}]

V7

Sum[Binomial[n,j]^3,{j,0,n}]

Sum[Binomial[n,k]^2Binomial[2k,n],{k,0,n}]

Sum[Binomial[n,k]^2Binomial[2(n-k),n],{k,0,n}]

FindSequenceFunction[Table[HypergeometricPFQ[{-n,-n,-n},{1,1},-1],{n,30}]]//FullSimplify