Zeilberger-Bressoud Therem

Author

Eric W. Weisstein
January 10, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/Zeilberger-BressoudTherem.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Product

ConstantTerm[n_]:=Series[Product[If[i==j,1,(((1-((x[i])/(x[j]))))^(a[i]))],{i,n},{j,n}],Sequence@@({x[#],0,0}&/@Range[n])]

ConstantTerm[5]

Sums

(∑_(k=-∞)^∞)(((-1))^k) (q^(k (3 k+1)/2)) (QBinomial[a+b, b+k, q]) (QBinomial[a+c, a+k, q]) (QBinomial[b+c, c+k, q])==((QPochhammer[q, q, a+b+c])/((QPochhammer[q, q, a]) (QPochhammer[q, q, b]) (QPochhammer[q, q, c])))

Sum[(((-1))^k)(q^(k(3k+1)/2))QBinomial[b+c,c+k,q]QBinomial[c+a,a+k,q]QBinomial[a+b,b+k,q],{k,-∞,∞}]//Timing

{Sum[(((-1))^k)(q^(k(3k+1)/2))QBinomial[b+c,c+k,q]QBinomial[c+a,a+k,q]QBinomial[a+b,b+k,q],{k,-50,50}],((QPochhammer[q,q,a+b+c])/(QPochhammer[q,q,a]QPochhammer[q,q,b]QPochhammer[q,q,c]))}/.{a->1,b->2,c->3,q->.3}

(∑_(k=0)^(2 p))(((-1))^k) ((Binomial[2 p, k])^3)==(((((-1))^p) (3 p)!)/(((p!))^3))

Table[Sum[(((-1))^k)((Binomial[2p,k])^3),{k,0,2p}],{p,0,5}]

V7.0.0

Sum[(((-1))^k)((Binomial[2p,k])^3),{k,0,2p}]

...wrong...

Table[%,{p,0,5}]

V7.0.1

Sum[(((-1))^k)((Binomial[2p,k])^3),{k,0,2p}]

Table[(((((-1))^p) (3 p)!)/(((p!))^3)),{p,0,5}]

((QPochhammer[q,q,a+b+c])/(QPochhammer[q,q,a]QPochhammer[q,q,b]QPochhammer[q,q,c]))/.{a->p,b->p,c->p}//FunctionExpand

Limit[((QPochhammer[q,q,3 p])/((QPochhammer[q,q,p])^3)),q->1]

expr=Sum[(((-1))^k)Binomial[2p,p+k]Binomial[2p,p+k]Binomial[2p,p+k],{k,-∞,∞}]

Table[Sum[(((-1))^k)Binomial[2p,p+k]Binomial[2p,p+k]Binomial[2p,p+k],{k,-∞,∞}],{p,5}]

Table[expr,{p,0,5}]

Table[(((3 p)!)/(((p!))^3)),{p,0,5}]

FullSimplify[expr,p∈Integers&&p>0]===expr

FullSimplify[expr-(((3 p)!)/(((p!))^3)),p∈Integers&&p>0]

Table[expr-(((3 p)!)/(((p!))^3)),{p,0,10}]//Union