[math-fun] tiny 2x2 simultaneously computes both Rogers-Ramanujan sums
(For R-R, special-case a:=q) (c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]); [ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] (c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6); [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] (d45)/T/ [ ] = [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] [ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] [ ] [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ] This came from the eensy path-invariant 3x3s {{k, {{0, q^n, 0}, {q^(k + n), q^n, 1 - q^n}, {0, 0, 1}}}, {n, {{q^(k + 2 n)/(1 - q^(1 + n)), 0, q^n}, {0, q^(k + 2 n)/(1 - q^(1 + n)), 1}, {0, 0, 1}}}}. --rwg with the usual big help from Corey&Julian
Nice. Could you supply an non-garbled version (image?) of the things below (c45)? These may be of interest: {Johann Cigler: {A new class of $q$-Fibonacci polynomials}, The Electronic Journal of Combinatorics, vol.10, no.1, (2003). URL: \url{http://www.combinatorics.org/Volume_10/Abstracts/v10i1r19.html}.} {Johann Cigler: {$q$-Fibonacci Polynomials and the Rogers-Ramanujan Identities}, Annals of Combinatorics, vol.8, no.3, pp.269-285, (September-2004). URL: \url{http://homepage.univie.ac.at/johann.cigler/preprints/fibon.pdf}.} (I can email the final version of second, it's pay-walled). Possibly more pertinent papers at http://homepage.univie.ac.at/johann.cigler/electr.html Regards, jj * Bill Gosper <billgosper@gmail.com> [Oct 08. 2012 08:02]:
(For R-R, special-case a:=q)
(c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]);
[ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
(c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6);
[GARBLED]
* Bill Gosper <billgosper@gmail.com> [Oct 08. 2012 08:02]:
(For R-R, special-case a:=q)
[...]
[ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
[...]
I toyed around a bit (note my start k=1 of the product): [ inf 2 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (UR = H(q) = A003106) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 1 ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n + n ] [ \ q ] [ 0 > -------------- ] (LR= G(q) = A003114) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] I.e., R-R proper. I am much tempted to put that into both A003106 and A003114 (obviously with attribution to RWG). OK, Neil ? [ inf 2 ] [ ==== n - n ] [ \ q ] [ 0 q > -------------- ] (UR=A006141) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 q ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (LR= H(q) = A003106) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] inf /===\ | | [ 0 q^n ] [ 0 D(q) ] (D=A000009) (dxx) | | [ ] = [ ] | | [ k ] [ 0 D(q) ] (D=A000009) k = 1 [ 1+q 1 ] inf /===\ | | [ 0 -q^n ] [ 0 E(q) ] (E=A010815 = 1/A000041) (dxx) | | [ ] = [ ] | | [ k ] [ 0 E(q) ] (E=A010815 = 1/A000041) k = 1 [1-q 1 ] ASCII for president!
Joerg, Yes, certainly! Thank you! Neil On Mon, Oct 8, 2012 at 10:25 AM, Joerg Arndt <arndt@jjj.de> wrote:
* Bill Gosper <billgosper@gmail.com> [Oct 08. 2012 08:02]:
(For R-R, special-case a:=q)
[...]
[ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
[...]
I toyed around a bit (note my start k=1 of the product):
[ inf 2 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (UR = H(q) = A003106) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 1 ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n + n ] [ \ q ] [ 0 > -------------- ] (LR= G(q) = A003114) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
I.e., R-R proper. I am much tempted to put that into both A003106 and A003114 (obviously with attribution to RWG). OK, Neil ?
[ inf 2 ] [ ==== n - n ] [ \ q ] [ 0 q > -------------- ] (UR=A006141) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 q ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (LR= H(q) = A003106) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
inf /===\ | | [ 0 q^n ] [ 0 D(q) ] (D=A000009) (dxx) | | [ ] = [ ] | | [ k ] [ 0 D(q) ] (D=A000009) k = 1 [ 1+q 1 ]
inf /===\ | | [ 0 -q^n ] [ 0 E(q) ] (E=A010815 = 1/A000041) (dxx) | | [ ] = [ ] | | [ k ] [ 0 E(q) ] (E=A010815 = 1/A000041) k = 1 [1-q 1 ]
ASCII for president!
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
* meg <njasloane@gmail.com> [Oct 09. 2012 07:01]:
Joerg, Yes, certainly! Thank you! Neil
[...]
Now in https://oeis.org/draft/A003106 :
From _Joerg Arndt_, Oct 10 2012: (Start) R. W. Gosper gives (message to the math-fun mailing list, Oct 07 2012) prod(k>=0, [0 , a; q^k, 1]) = [0, X(a,q); 0, Y(a,q)] where X(a,q) = a * sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ) and Y(a,q) = sum(n>=0, a^n*q^(n^2-n) / prod(k=1..n, 1-q^n) ). Set a=q to obtain prod(k>=0, [0 , a; q^k, 1]) = [0, q*H(q); 0, G(q)] where H(q) is the g.f. of A003106 and G(q) is the g.f. of A003114. (End)
I will now look at the following message. Regards, jj
Warren, you have GMail. Neil just taught me to click on▼, then "Message text garbled?" That should fix everything but the overwrapped lines, which you can fix by copying and pasting into an editor and deleting the bogus linebreaks. I too hate ASCII art. Why can't math-fun allow html with in-line images?
From the path-invariant 3x3s {{j, {{0, q^(j + 2*n), 1 - q^n}, {1, q^n, 0}, {0, 0, 1}}}, {n, {{q^(j + 2*n)/(1 - q^(1 + n)), 0, 1 - q^n}, {0, q^(1 + j + 2*n)/(1 - q^(1 + n)),1}, {0, 0, 1}}}}
Neil and I just found [ inf ] [ ==== n n (n + 1) ] [ \ a q ] [ ? a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ j ] [ n = 0 ] | | [ 0 a q ] = [ ] | | [ ] [ inf 2 ] j = 0 [ 1 1 ] [ ==== n n ] [ \ a q ] [ ? > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] Maybe Corey&Julian can help me determine the "?"s. --rwg I only recently learned that a lot of broken links to gosper,org are due to a disk crash. If you hit one, send me mail and I'll try to restore it. On Sun, Oct 7, 2012 at 4:02 PM, Bill Gosper <billgosper@gmail.com> wrote:
(For R-R, special-case a:=q)
(c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]);
[ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
(c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6);
[ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] (d45)/T/ [ ] = [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ]
[ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ]
[ ] [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ]
This came from the eensy path-invariant 3x3s {{k, {{0, q^n, 0}, {q^(k + n), q^n, 1 - q^n}, {0, 0, 1}}}, {n, {{q^(k + 2 n)/(1 - q^(1 + n)), 0, q^n}, {0, q^(k + 2 n)/(1 - q^(1 + n)), 1}, {0, 0, 1}}}}. --rwg with the usual big help from Corey&Julian
* Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Oct 08. 2012 08:02]:> (For R-R, special-case a:=q)> > [...]> > [ inf 2 ]> [ ==== n n ]> [ \ a q ]> [ 0 a > -------------- ]> inf [ / qpoch(q, q, n) ]> /===\ [ ==== ]> | | [ 0 a ] [ n = 0 ]> (d44) | | [ ] = [ ]> | | [ k ] [ inf 2 ]> k = 0 [ q 1 ] [ ==== n n - n ]> [ \ a q ]> [ 0 > -------------- ]> [ / qpoch(q, q, n) ]> [ ==== ]> [ n = 0 ]> > [...] Jörg>I toyed around a bit (note my start k=1 of the product): [ inf 2 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (UR = H(q) = A003106) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 1 ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n + n ] [ \ q ] [ 0 > -------------- ] (LR= G(q) = A003114) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] I.e., R-R proper. I am much tempted to put that into both A003106 and A003114 (obviously with attribution to RWG). OK, Neil ? [ inf 2 ] [ ==== n - n ] [ \ q ] [ 0 q > -------------- ] (UR=A006141) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 q ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (LR= H(q) = A003106) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] inf /===\ | | [ 0 q^n ] [ 0 D(q) ] (D=A000009) (dxx) | | [ ] = [ ] | | [ k ] [ 0 D(q) ] (D=A000009) k = 1 [ 1+q 1 ] inf /===\ | | [ 0 -q^n ] [ 0 E(q) ] (E=A010815 = 1/A000041) (dxx) | | [ ] = [ ] | | [ k ] [ 0 E(q) ] (E=A010815 = 1/A000041) k = 1 [1-q 1 ] ASCII for president!
* Bill Gosper <billgosper@gmail.com> [Oct 10. 2012 10:13]:
Warren, you have GMail. Neil just taught me to click on▼, then "Message text garbled?" That should fix everything but the overwrapped lines, which you can fix by copying and pasting into an editor and deleting the bogus linebreaks.
I too hate ASCII art. Why can't math-fun allow html with in-line images?
From the path-invariant 3x3s {{j, {{0, q^(j + 2*n), 1 - q^n}, {1, q^n, 0}, {0, 0, 1}}}, {n, {{q^(j + 2*n)/(1 - q^(1 + n)), 0, 1 - q^n}, {0, q^(1 + j + 2*n)/(1 - q^(1 + n)),1}, {0, 0, 1}}}}
Neil and I just found [ inf ] [ ==== n n (n + 1) ] [ \ a q ] [ ? a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ j ] [ n = 0 ] | | [ 0 a q ] = [ ] | | [ ] [ inf 2 ] j = 0 [ 1 1 ] [ ==== n n ] [ \ a q ] [ ? > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] Maybe Corey&Julian can help me determine the "?"s. --rwg
[...]
Let G(q)=sum(n=0,S,q^(n^2)/prod(k=1,n,1-q^k)); /* g.f. of A003114 */ \\Vec(G(q)) \\ A003114 partitions of n into parts 5k+1 or 5k+4. H(q)=sum(n=0,S,q^(n^2+n)/prod(k=1,n,1-q^k)); /* g.f. of A003106 */ \\Vec(H(q)) \\ A003106 partitions of n into parts 5k+2 or 5k+3. P(q) = sum(n=0,S,q^(n^2-n)/prod(k=1,n,1-q^k)); /* g.f. of A003113 */ \\Vec(P(q)) \\ A003113 Coefficients in expansion of permanent ... B(q) = q * sum(n=0,S,q^(n^2+2*n)/prod(k=1,n,1-q^k)); /* g.f. of A006141 */ \\Vec(B(q)) \\ A006141 balanced partitions: smallest part equals the number of parts. A specialization (a==1, starting product with k=1): inf /===\ | | [ k ] [ B B ] | | [ 0 q ] = [ ] | | [ ] [ H H ] k = 1 [ 1 1 ] And (a==1, starting with k=0): inf /===\ | | [ k ] [ H H ] [ H H ] | | [ 0 q ] = [ ] = [ ] | | [ ] [ G G ] [ H+B H+B ] k = 0 [ 1 1 ] So I added ( https://oeis.org/draft/A003106 ): R. W. Gosper and _N. J. A. Sloane_ give (message to math-fun, Oct 10 2012) prod(k>=0, [0 , a*q^k; 1, 1]) = [?, U(a,q); ?, V(a,q)] where U(a,q) = a * sum(n>=0, a^n*q^(n^2+n) / prod(k=1..n, 1-q^n) ) and V(a,q) = sum(n>=0, a^n*q^(n^2) / prod(k=1..n, 1-q^n) ). Set a=1 to obtain prod(k>=0, [0 , q^k; 1, 1]) = [H(q), H(q); G(q), G(q)]. Now we really like to know the two '?' ...
* Joerg Arndt <arndt@jjj.de> [Oct 10. 2012 13:02]:
[...]
Now we really like to know the two '?' ...
Writing R1 and R2 for the '?'s and starting the product at index one, I get: inf /===\ | | [ k ] [ X X ] | | [ 0 a*q ] = [ ] | | [ ] [ Y Y ] k = 1 [ 1 1 ] Now we can recognize your matrix product [R1, U; R2, V]: [0,1;1,1]^(-1) * [ X, X; Y, Y ] \\ == [ R1, U; R2, V ] \\ == [ -X+Y, -X+Y; X, X ] That is inf /===\ | | [ k ] [ U U ] | | [ 0 a*q ] = [ ] | | [ ] [ V V ] k = 0 [ 1 1 ] where (as before) U(a,q) = a * sum(n=0,N, a^n*q^(n^2+n) / prod(k=1,n, 1-q^k) ); V(a,q) = sum(n=0,N, a^n*q^(n^2) / prod(k=1,n, 1-q^k) ); I'll now put this in.
participants (3)
-
Bill Gosper -
Joerg Arndt -
Neil Sloane