Chu-Vandermonde Identity

Author

Eric W. Weisstein
January 15, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Chu-VandermondeIdentity.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identity

FullSimplify[Hypergeometric2F1[-n,b,c,1]-Pochhammer[c-b,n]/Pochhammer[c,n],n∈Integers&&n>0]

Numeric Check

Hypergeometric2F1[-n,b,c,1]-Pochhammer[c-b,n]/Pochhammer[c,n]/.Thread[{n,b,c}->#]&/@Table[{Random[Integer,{1,10}],Random[Complex],Random[Complex]},{200}]//Chop

Pochhammer Binomial Sum

Sum[Binomial[n,k]Pochhammer[x,k]Pochhammer[a,n-k],{k,0,∞}]

FullSimplify[%,n∈Integers]

Subtract@@@Table[{(((((-1))^n) Gamma[1-a-x])/(Gamma[1-a-n-x])),Pochhammer[x+a,n]}/.{x->Random[Complex,5(I+1){-1,1},30],n->Random[Integer,{-10,10}],a->Random[Complex,5(I+1){-1,1},30]},{10}]//Chop//Union

{(((((-1))^n) Gamma[1-a-x])/(Gamma[1-a-n-x])),Pochhammer[x+a,n]}/.n->0

FunctionExpand[Pochhammer[x+a,n]]

FullSimplify[(((((-1))^n) Gamma[1-a-x])/(Gamma[1-a-n-x]))-((Gamma[a+n+x])/(Gamma[a+x])),n∈Integers]

Alternate Form

(({{a+x}, {n}}))=(∑_(k=0)^n)(({{x}, {k}})) (({{a}, {n-k}}))

FullSimplify[Sum[Binomial[x,k]Binomial[a,n-k],{k,0,n}],n∈Integers]

FunctionExpand[Binomial[x+a,n]]

Special Case

(∑_(l=0)^k)(({{m}, {k-l}})) (({{n}, {l}}))=(({{m+n}, {k}}))

Sum[Binomial[m,k-l]Binomial[n,l],{l,0,k}]

Sum[Binomial[m,k-l]Binomial[n,l],{l,0,n}]

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

FunctionExpand[Sum[Binomial[n,l]^2,{l,0,n}]]

FullSimplify[FunctionExpand[Binomial[2n,n]]]