Szekély Identity

Author

Eric W. Weisstein
December 31, 2008

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

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

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

Sum

(∑_(k=-∞)^∞)(Binomial[a+d, d+k]) (Binomial[b+c, c+k]) (Binomial[a+b+c+d+e-k, e-k])==(Binomial[a+c+d+e, a+c]) (Binomial[b+c+d+e, c+e])

(sum=Sum[Binomial[a+b+c+d+e-k,e-k]Binomial[a+d,k+d]Binomial[b+c,k+c],{k,-∞,∞}])//Timing

FullSimplify[sum]

FullSimplify[sum-Binomial[a+c+d+e,a+c]Binomial[b+c+d+e,c+e]]

FullSimplify[sum-Binomial[a+c+d+e,a+c]Binomial[b+c+d+e,c+e],a>0&&b>0&&c>0&&d>0&&e>0]

{sum,Binomial[a+c+d+e,a+c]Binomial[b+c+d+e,c+e]}/.{a->RandomComplexDisk[],b->RandomComplexDisk[],c->RandomComplexDisk[],d->RandomComplexDisk[],e->RandomComplexDisk[]}

Subtract@@@Table[{sum,Binomial[a+c+d+e,a+c]Binomial[b+c+d+e,c+e]}/.{a->RandomReal[{0,5},WorkingPrecision->100],b->RandomReal[{0,5},WorkingPrecision->100],c->RandomReal[{0,5},WorkingPrecision->100],d->RandomReal[{0,5},WorkingPrecision->100],e->RandomReal[{0,5},WorkingPrecision->100]},{1}]//Chop//Union//Timing

Subtract@@@Table[{sum,Binomial[a+c+d+e,a+c]Binomial[b+c+d+e,c+e]}/.{a->RandomReal[{0,5},WorkingPrecision->50],b->RandomReal[{0,5},WorkingPrecision->50],c->RandomReal[{0,5},WorkingPrecision->50],d->RandomReal[{0,5},WorkingPrecision->50],e->RandomReal[{0,5},WorkingPrecision->50]},{2}]//Chop//Union//Timing

...very slow to numericize...