Andrews-Schur Identity

Author

Eric W. Weisstein
February 1, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Andrews-SchurIdentity.nb.

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

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

Code

QBracket[n_,q_]:=((1-(q^n))/(1-q))

MakeBoxes[QBracket[n_,q_],TraditionalForm]:=SubscriptBox[RowBox[{"[",MakeBoxes[n,TraditionalForm],"]"}],MakeBoxes[q,TraditionalForm]]

Full Identity

(∑_(k=0)^n)(q^(a k+(k^2))) (QBinomial[a-k+2 n, k, q])==(∑_(k=-∞)^∞)(((q^((4 a-1) k+10 (k^2))) (Subscript[[2 a+10 k+2], q]) (QBinomial[2 a+2 n+2, n-5 k, q]))/(Subscript[[2 a+2 n+2], q]))/;a==0∨a==1

Sum[(q^((k^2)+a k))QBinomial[2n-k+a,k,q],{k,0,n}]//Timing

Sum[(q^(10(k^2)+(4a-1) k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-∞,∞}]//Timing

(∑_(k=0)^n)(q^(k^2)) (QBinomial[-k+2 n, k, q])==(∑_(k=-∞)^∞)(((Subscript[(q^(- k+10 (k^2))) [10 k+2], q]) (QBinomial[2 n+2, n-5 k, q]))/(Subscript[[2 n+2], q]))

With[{a=0},Sum[(q^((k^2)+a k))QBinomial[2n-k+a,k,q],{k,0,n}]]//Timing

With[{a=0},Sum[(q^((k^2)+a k))FunctionExpand[QBinomial[2n-k+a,k,q]],{k,-∞,∞}]]//Timing

Table[With[{a=0},Sum[(q^((k^2)+a k))FunctionExpand[QBinomial[2n-k+a,k,q]],{k,0,n}]],{n,0,3}]//Expand

With[{a=0},Sum[(q^(10(k^2)+(4a-1) k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-∞,∞}]]//Timing

With[{a=0},Sum[(q^(10(k^2)+(4a-1) k))FunctionExpand[QBinomial[2n+2a+2,n-5k,q]]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-∞,∞}]]//Timing

Table[With[{a=0},Sum[(q^(10(k^2)+(4a-1) k))FunctionExpand[QBinomial[2n+2a+2,n-5k,q]]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-10,10}]],{n,0,3}]//Expand//Factor

(∑_(k=0)^n)(q^(k+(k^2))) (QBinomial[1-k+2 n, k, q])==(∑_(k=-∞)^∞)(((Subscript[(q^(3k+10 (k^2))) [10 k+4], q]) (QBinomial[2 n+4, n-5 k, q]))/(Subscript[[2 n+4], q]))

With[{a=1},Sum[(q^((k^2)+a k))QBinomial[2n-k+a,k,q],{k,0,n}]]//Timing

With[{a=1},Sum[(q^((k^2)+a k))FunctionExpand[QBinomial[2n-k+a,k,q]],{k,0,n}]]//Timing

Table[With[{a=1},Sum[(q^((k^2)+a k))FunctionExpand[QBinomial[2n-k+a,k,q]],{k,0,n}]],{n,0,3}]//Expand

With[{a=1},Sum[(q^(10(k^2)+(4a-1) k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-∞,∞}]]//Timing

With[{a=1},Sum[(q^(10(k^2)+(4a-1) k))FunctionExpand[QBinomial[2n+2a+2,n-5k,q]]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-∞,∞}]]//Timing

Table[With[{a=1},Sum[(q^(10(k^2)+(4a-1) k))FunctionExpand[QBinomial[2n+2a+2,n-5k,q]]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-10,10}]],{n,0,3}]//Expand//Factor//Expand

Variant

(∑_(k=-(Floor[(a/2)]))^n)(q^(2ak+(k^2)))(QBinomial[a+k+n, n-k, q])==(∑_(k=-(Floor[(1/5) (2 a+n+2)]))^(Floor[(n/5)]))(((q^((6a+1)k+15(k^2)))(Subscript[[2a+10k+2], q])(QBinomial[2 a+2 n+2, n-5 k, q]))/(Subscript[[2a+2n+2], q]))/;a==0∨a==1

Sum[(q^((k^2)+2a k))QBinomial[n+k+a,n-k,q],{k,-Floor[a/2],n}]//Timing

Sum[(q^(15(k^2)+(6a+1)k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]//Timing

(∑_(k=0)^n)(q^(k^2)) (QBinomial[k+n, n-k, q])==(∑_(k=-(Floor[(1/5) (n+2)]))^(Floor[(n/5)]))(((Subscript[(q^(k+15 (k^2))) [10 k+2], q]) (QBinomial[2 n+2, n-5 k, q]))/(Subscript[[2 n+2], q]))

With[{a=0},Sum[(q^((k^2)+2a k))QBinomial[n+k+a,n-k,q],{k,-Floor[a/2],n}]]//Timing

With[{a=0},Table[Sum[(q^((k^2)+2a k))QBinomial[n+k+a,n-k,q],{k,-Floor[a/2],n}],{n,3}]]//FunctionExpand//Expand

With[{a=0},Sum[(q^(15(k^2)+(6a+1)k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]]//Timing

With[{a=0},Table[Sum[(q^(15(k^2)+(6a+1)k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-Floor[(n+2a+2)/5],Floor[n/5]}],{n,3}]]//FunctionExpand//Factor//Expand

(∑_(k=0)^n)(q^(2 k+(k^2))) (QBinomial[1+k+n, n-k, q])==(∑_(k=-(Floor[(1/5) (n+4)]))^(Floor[(n/5)]))(((Subscript[(q^(7k+15 (k^2))) [10 k+4], q]) (QBinomial[2 n+4, n-5 k, q]))/(Subscript[[2 n+4], q]))

With[{a=1},Sum[(q^((k^2)+2a k))QBinomial[n+k+a,n-k,q],{k,-Floor[a/2],n}]]//Timing

With[{a=1},Table[Sum[(q^((k^2)+2a k))QBinomial[n+k+a,n-k,q],{k,-Floor[a/2],n}],{n,3}]]//FunctionExpand//Expand

With[{a=1},Sum[(q^(15(k^2)+(6a+1)k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]]//Timing

With[{a=1},Table[Sum[(q^(15(k^2)+(6a+1)k))QBinomial[2n+2a+2,n-5k,q]((QBracket[10k+2a+2,q])/(QBracket[2n+2a+2,q])),{k,-Floor[(n+2a+2)/5],Floor[n/5]}],{n,3}]]//FunctionExpand//Factor//Expand

Reciprocal Identity

(∑_(k=0)^∞)((q^(2 a k+(k^2)))/(QPochhammer[q, q, a+2 k]))==(∏_(j=0)^∞)(1/((1-(q^(2 j+1))) (1-(q^(4 a+20 j+4))) (1-(q^(-4 a+20 j+16)))))==(1/((QPochhammer2[q, (q^2)]) (QPochhammer2[(q^(16-4 a)), (q^20)]) (QPochhammer2[(q^(4 a+4)), (q^20)])))/;a==0∨a==1

Sum[((q^((k^2)+2a k))/(QPochhammer[q,2k+a])),{k,0,∞}]//Timing

Product[(1/((1-(q^(2j+1)))(1-(q^(20j+4a+4)))(1-(q^(20j-4a+16))))),{j,0,∞}]//Timing

(∑_(k=0)^∞)((q^(k^2))/(QPochhammer[q, q, 2 k]))==(∏_(j=0)^∞)(1/((1-(q^(2 j+1))) (1-(q^(20 j+4))) (1-(q^(20 j+16)))))

With[{a=0},Sum[((q^((k^2)+2a k))/(QPochhammer[q,q,2k+a])),{k,0,∞}]]//Timing

Series[With[{a=0},Sum[((q^((k^2)+2a k))/(QPochhammer[q,q,2k+a])),{k,0,20}]],{q,0,10}]

With[{a=0},Product[(1/((1-(q^(2j+1)))(1-(q^(20j+4a+4)))(1-(q^(20j-4a+16))))),{j,0,∞}]]//Timing

Series[With[{a=0},Product[(1/((1-(q^(2j+1)))(1-(q^(20j+4a+4)))(1-(q^(20j-4a+16))))),{j,0,10}]],{q,0,10}]

(∑_(k=0)^∞)((q^(2 k+(k^2)))/(QPochhammer[q, q, 1+2 k]))==(∏_(j=0)^∞)(1/((1-(q^(2 j+1))) (1-(q^(20 j+8))) (1-(q^(20 j+12)))))

With[{a=1},Sum[((q^((k^2)+2 a k))/(QPochhammer[q,q,2k+a])),{k,0,∞}]]//Timing

Series[With[{a=1},Sum[((q^((k^2)+2 a k))/(QPochhammer[q,q,2k+a])),{k,0,20}]],{q,0,10}]

With[{a=1},Product[(1/((1-(q^(2j+1)))(1-(q^(20j+4a+4)))(1-(q^(20j-4a+16))))),{j,0,∞}]]//Timing

Series[With[{a=1},Product[(1/((1-(q^(2j+1)))(1-(q^(20j+4a+4)))(1-(q^(20j-4a+16))))),{j,0,20}]],{q,0,10}]

q==1

(∑_(k=-(Floor[(a/2)]))^n)(Binomial[a+k+n, n-k])==(∑_(k=-(Floor[(1/5) (2 a+n+2)]))^(Floor[(n/5)]))(((a+5 k+1) (Binomial[2 a+2 n+2, n-5 k]))/(a+n+1))/;a==0∨a==1

Sum[Binomial[n+k+a,n-k],{k,-Floor[a/2],n}]//Timing

Sum[Binomial[2n+2a+2,n-5k]((5k+a+1)/(n+a+1)),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]//Timing

FullSimplify[Last[%],n∈Integers&&n>0]//Timing

(∑_(k=-(Floor[(a/2)]))^n)(Binomial[k+n, n-k])==(∑_(k=-(Floor[((n+2)/5)]))^(Floor[(n/5)]))(((5 k+1) (Binomial[2 n+2, n-5 k]))/(n+1))

With[{a=0},Sum[Binomial[n+k+a,n-k],{k,-Floor[a/2],n}]]

Table[Fibonacci[1+2 n],{n,10}]

Table[With[{a=0},Sum[Binomial[n+k+a,n-k],{k,-Floor[a/2],n}]],{n,10}]

With[{a=0},Sum[Binomial[2n+2a+2,n-5k]((5k+a+1)/(n+a+1)),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]]//Timing

Table[Last[%],{n,10}]

Table[With[{a=0},Sum[Binomial[2n+2a+2,n-5k]((5k+a+1)/(n+a+1)),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]],{n,10}]

(∑_(k=-(Floor[(a/2)]))^n)(Binomial[k+n+1, n-k])==(∑_(k=-(Floor[((n+4)/5)]))^(Floor[(n/5)]))(((5 k+2) (Binomial[2 n+4, n-5 k]))/(n+2))

With[{a=1},Sum[Binomial[n+k+a,n-k],{k,-Floor[a/2],n}]]

Table[With[{a=1},Sum[Binomial[n+k+a,n-k],{k,-Floor[a/2],n}]],{n,10}]

With[{a=1},Sum[Binomial[2n+2a+2,n-5k]((5k+a+1)/(n+a+1)),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]]//Timing

FullSimplify[Last[%],n∈Integers&&n>0]//Timing

Table[Last[%],{n,10}]

Table[With[{a=1},Sum[Binomial[2n+2a+2,n-5k]((5k+a+1)/(n+a+1)),{k,-Floor[(n+2a+2)/5],Floor[n/5]}]],{n,10}]