A basis in orthogonal polynomials for Sierpinski like triangles

[Roger Bagula]

I have been working in the area of generalized Pascal triangles in
Integer mathematics and
fractals. You may be familiar with the q-Boson statistics and the q-Fermion.
I have discovered six new such product based sequences that give
Integer triangles of q-combinations from q-factorial.
The most important are the ones based on orthogonal polynomials:
The A_n Cartan Matrix characteristic polynomial products
and the Shabat Chebyshev polynomial products
as q sequences.
Try these links:
http://www.research.att.com/~njas/sequences/?q=A_n+bagula+q-combinations&language=english&go=Search
http://www.research.att.com/~njas/sequences/?q=bagula+q-combinations&sort=0&fmt=0&language=english&go=Search



q combinations from Cartan A_n with a Fibonacci connection

-------- Original Message --------
Subject: q combinations from Cartan A_n with a Fibonacci connection
Date: Wed, 11 Feb 2009 05:02:52 -0800
From: Roger Bagula <***@sbcglobal.net>
Reply-To: ***@yahoo.com
To: true number theory <***@yahoogroups.com>,
quant-***@yahoogroups.com


A connection between Pascal-Sierpinski
and quantum algebra in Cartan sets using
Cartan A_n characteristics polynomials to form a
q-like general factorial
and then,
making combinations of the result.
Since the first level above the Pascal is:
(A034801 Triangle of Fibonomial coefficients (k=2).)
{1},
{1, 1},
{1, -3, 1},
{1, 8, 8,1},
{1, -21, 56, -21, 1},
{1, 55, 385, 385, 55, 1},
{1, -144, 2640, -6930, 2640, -144, 1},
{1, 377, 18096, 124410, 124410, 18096, 377, 1},
{1, -987, 124033, -2232594, 5847270, -2232594, 124033, -987, 1},
{1, 2584, 850136, 40062659, 274715376, 274715376, 40062659, 850136,
2584, 1},
{1, -6765, 5826920, -718896255, 12905899435, -33789991248, 12905899435,
-718896255, 5826920, -6765, 1}

a(n, k) = product(fibonacci(2*(n-j)),

j=0..k-1)/product(fibonacci(2*j), j=1..k)


It seems to pull at lot together. I tried to get the Indeterminant
levels out, but only got rational
combinations for my trouble. This set seems to be the only way I can get
it to work.



%I A156602
%S A156602
1,1,1,1,7,1,1,48,48,1,1,329,2256,329,1,1,2255,105985,105985,2255,1,
%T A156602 1,15456,4979040,34127170,4979040,15456,1,1,105937,233908896,
%U A156602 10988845010,10988845010,233908896,105937,1,1,726103,10988739073
%V A156602
1,1,1,1,-7,1,1,48,48,1,1,-329,2256,-329,1,1,2255,105985,105985,2255,1,
%W A156602 1,-15456,4979040,-34127170,4979040,-15456,1,1,105937,233908896,
%X A156602 10988845010,10988845010,233908896,105937,1,1,-726103,10988739073
%N A156602 A q-combination triangle sequence built of Cartan A_n
polynomials: m=8;q=9; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%C A156602 Row sums are:
%C A156602 {1, 2, -5, 98, 1600, 216482, -24200000, 22445719688,
17197609000000,
%C A156602 109329711296575112, -574146991795520000000,...}.
%C A156602 I get as m levels:
%C A156602 m=0;Binomial
%C A156602 m=1;Indeterminant
%C A156602 m=2;Indeterminant
%C A156602 m=3;signed Binomial at {1.-2,1}
%C A156602 m=4;A034801 at {1,-3,1}
%C A156602 m=5;A156599 at {1,-4,1}
%C A156602 m=6;A156600 at {1,-5,1}
%C A156602 m=7;A156601 at {1,-6,1}
%C A156602 m=8; this sequence at {1,-7,1}.
%C A156602 These sequences are important because they relate Cartan
quantum orthogonal
%C A156602 group theory ( Lie algebra)to a set of combinatorial triangle
sequences.
%C A156602 As A_1 or SU(2) is associated to weak and electromagnetic theory
%C A156602 and A_2 or SU(3) to strong nuclear force and A_4 or SU(5) to
the standard model of physics,
%C A156602 a q particle modeled on these combinations would be very
important to physics.
%F A156602 m=8;q=9; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%e A156602 {1},
%e A156602 {1, 1},
%e A156602 {1, -7, 1},
%e A156602 {1, 48, 48, 1},
%e A156602 {1, -329, 2256, -329, 1},
%e A156602 {1, 2255, 105985, 105985, 2255, 1},
%e A156602 {1, -15456, 4979040, -34127170, 4979040, -15456, 1},
%e A156602 {1, 105937, 233908896, 10988845010, 10988845010, 233908896,
105937, 1},
%e A156602 {1, -726103, 10988739073, -3538373981506, 24252380937070,
-3538373981506, 10988739073, -726103, 1},
%e A156602 {1, 4976784, 516236827536, 1139345433305859,
53524993973177376, 53524993973177376, 1139345433305859, 516236827536,
4976784, 1},
%e A156602 {1, -34111385, 24252142155120, -366865691151231195,
118129637457410270835, -809672583832254166752, 118129637457410270835,
-366865691151231195, 24252142155120, -34111385, 1}
%t A156602 Clear[t, n, m, i, k, a, b, T, M, p];
%t A156602 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1,
-1, 0]];
%t A156602 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A156602 p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];
%t A156602 a0 = Table[p[x, n], {n, 0, 20}] /. x -> m + 1;
%t A156602 t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];
%t A156602 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156602 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0,
10}]], {m, 0, 15}]
%Y A156602 A034801,A156599,A156600,A156601
%K A156602 sign,tabl
%O A156602 0,5
%A A156602 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2009


%I A156599
%S A156599 1,1,1,1,4,1,1,15,15,1,1,56,210,56,1,1,209,2926,2926,209,1,1,780,
%T A156599
40755,152152,40755,780,1,1,2911,567645,7909187,7909187,567645,2911,1,
%U A156599 1,10864,7906276,411126352,1534382278,411126352,7906276,10864,1,1
%V A156599
1,1,1,1,-4,1,1,15,15,1,1,-56,210,-56,1,1,209,2926,2926,209,1,1,-780,
%W A156599
40755,-152152,40755,-780,1,1,2911,567645,7909187,7909187,567645,2911,1,
%X A156599
1,-10864,7906276,-411126352,1534382278,-411126352,7906276,-10864,1,1
%N A156599 A q-combination triangle sequence built of Cartan A_n
polynomials: m=5;q=6; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%C A156599 Row sums are:
%C A156599 {1, 2, -2, 32, 100, 6272, -72200, 16959488, 727920400,
638287290368, -102236420180000,...}.
%C A156599 I get as m levels:
%C A156599 m=0;Binomial
%C A156599 m=1;Indeterminant
%C A156599 m=2;Indeterminant
%C A156599 m=3;signed Binomial at {1.-2,1}
%C A156599 m=4;A034801 at {1,-3,1}
%C A156599 m=5; this sequence at {1,-4,1}
%F A156599 m=5;q=6; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%e A156599 {1},
%e A156599 {1, 1},
%e A156599 {1, -4, 1},
%e A156599 {1, 15, 15, 1},
%e A156599 {1, -56, 210, -56, 1},
%e A156599 {1, 209, 2926, 2926, 209, 1},
%e A156599 {1, -780, 40755, -152152, 40755, -780, 1},
%e A156599 {1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1},
%e A156599 {1, -10864, 7906276, -411126352, 1534382278, -411126352,
7906276, -10864, 1},
%e A156599 {1, 40545, 110120220, 21370664028, 297662820390,
297662820390, 21370664028, 110120220, 40545, 1},
%e A156599 {1, -151316, 1533776805, -1110863413968, 57745060679658,
-215507881962360, 57745060679658, -1110863413968, 1533776805, -151316, 1}
%t A156599 Clear[t, n, m, i, k, a, b, T, M, p];
%t A156599 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1,
-1, 0]];
%t A156599 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A156599 p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];
%t A156599 a0 = Table[p[x, n], {n, 0, 20}] /. x -&gt; m + 1;
%t A156599 t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];
%t A156599 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156599 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0,
10}]], {m, 0, 15}]
%Y A156599 A034801
%K A156599 sign,tabl
%O A156599 0,5
%A A156599 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2009


%I A156600
%S A156600 1,1,1,1,5,1,1,24,24,1,1,115,552,115,1,1,551,12673,12673,551,1,1,
%T A156600 2640,290928,1394030,290928,2640,1,1,12649,6678672,153331178,
%U A156600
153331178,6678672,12649,1,1,60605,153318529,16865038190,80805530806
%V A156600
1,1,1,1,-5,1,1,24,24,1,1,-115,552,-115,1,1,551,12673,12673,551,1,1,
%W A156600 -2640,290928,-1394030,290928,-2640,1,1,12649,6678672,153331178,
%X A156600
153331178,6678672,12649,1,1,-60605,153318529,-16865038190,80805530806
%N A156600 A q-combination triangle sequence built of Cartan A_n
polynomials: m=6;q=7; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%C A156600 Row sums are:
%C A156600 {1, 2, -3, 50, 324, 26450, -817452, 320045000, 47381970276,
88885777805000,
%C A156600 -63049671623224368,...}.
%C A156600 I get as m levels:
%C A156600 m=0;Binomial
%C A156600 m=1;Indeterminant
%C A156600 m=2;Indeterminant
%C A156600 m=3;signed Binomial at {1.-2,1}
%C A156600 m=4;A034801 at {1,-3,1}
%C A156600 m=5;A156599 at {1,-4,1}
%C A156600 m=6; this sequence at {1,-5,1}
%F A156600 m=6;q=7; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%e A156600 {1},
%e A156600 {1, 1},
%e A156600 {1, -5, 1},
%e A156600 {1, 24, 24, 1},
%e A156600 {1, -115, 552, -115, 1},
%e A156600 {1, 551, 12673, 12673, 551, 1},
%e A156600 {1, -2640, 290928, -1394030, 290928, -2640, 1},
%e A156600 {1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1},
%e A156600 {1, -60605, 153318529, -16865038190, 80805530806,
-16865038190, 153318529, -60605, 1},
%e A156600 {1, 290376, 3519647496, 1855000882371, 42584368082256,
42584368082256, 1855000882371, 3519647496, 290376, 1},
%e A156600 {1, -1391275, 80798573880, -204033232083225,
22441881327136635, -107525529407696400, 22441881327136635,
-204033232083225, 80798573880, -1391275, 1}
%t A156600 Clear[t, n, m, i, k, a, b, T, M, p];
%t A156600 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1,
-1, 0]];
%t A156600 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A156600 p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];
%t A156600 a0 = Table[p[x, n], {n, 0, 20}] /. x -&gt; m + 1;
%t A156600 t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];
%t A156600 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156600 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0,
10}]], {m, 0, 15}]
%Y A156600 A034801,A156599
%K A156600 sign,tabl
%O A156600 0,5
%A A156600 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2009

%I A156601
%S A156601
1,1,1,1,6,1,1,35,35,1,1,204,1190,204,1,1,1189,40426,40426,1189,1,1,
%T A156601 6930,1373295,8004348,1373295,6930,1,1,40391,46651605,1584821667,
%U A156601 1584821667,46651605,40391,1,1,235416,1584781276,313786692648
%V A156601
1,1,1,1,-6,1,1,35,35,1,1,-204,1190,-204,1,1,1189,40426,40426,1189,1,1,
%W A156601
-6930,1373295,-8004348,1373295,-6930,1,1,40391,46651605,1584821667,
%X A156601 1584821667,46651605,40391,1,1,-235416,1584781276,-313786692648
%N A156601 A q-combination triangle sequence built of Cartan A_n
polynomials: m=7;q=8; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%C A156601 Row sums are:
%C A156601 {1, 2, -4, 72, 784, 83232, -5271616, 3263027328, 1204479910144,
%C A156601 4345425701134848, -9348927348636058624,...}.
%C A156601 I get as m levels:
%C A156601 m=0;Binomial
%C A156601 m=1;Indeterminant
%C A156601 m=2;Indeterminant
%C A156601 m=3;signed Binomial at {1.-2,1}
%C A156601 m=4;A034801 at {1,-3,1}
%C A156601 m=5;A156599 at {1,-4,1}
%C A156601 m=6;A156600 at {1,-5,1}
%C A156601 m=7; this sequence at {1,-6,1}
%F A156601 m=7;q=8; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!,
Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k,
m]*t[n - k, m])].
%e A156601 {1},
%e A156601 {1, 1},
%e A156601 {1, -6, 1},
%e A156601 {1, 35, 35, 1},
%e A156601 {1, -204, 1190, -204, 1},
%e A156601 {1, 1189, 40426, 40426, 1189, 1},
%e A156601 {1, -6930, 1373295, -8004348, 1373295, -6930, 1},
%e A156601 {1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1},
%e A156601 {1, -235416, 1584781276, -313786692648, 1828884203718,
-313786692648, 1584781276, -235416, 1},
%e A156601 {1, 1372105, 53835911780, 62128180363028, 2110530832920510,
2110530832920510, 62128180363028, 53835911780, 1372105, 1},
%e A156601 {1, -7997214, 1828836219245, -12301065925422312,
2435550753890846098, -14195430382223350260, 2435550753890846098,
-12301065925422312, 1828836219245, -7997214, 1}
%t A156601 Clear[t, n, m, i, k, a, b, T, M, p];
%t A156601 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1,
-1, 0]];
%t A156601 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A156601 p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];
%t A156601 a0 = Table[p[x, n], {n, 0, 20}] /. x -&gt; m + 1;
%t A156601 t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];
%t A156601 b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
%t A156601 Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0,
10}]], {m, 0, 15}]
%Y A156601 A034801,A156599,A156600
%K A156601 sign,tabl
%O A156601 0,5
%A A156601 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2009

%I A156603
%S A156603 1,1,1,1,1,2,1,1,0,6,1,1,1,0,24,1,1,2,0,0,120,1,1,3,6,0,0,720,1,1,
%T A156603 4,24,24,0,0,5040,1,1,5,60,504,120,0,0,40320,1,1,6,120,3360,27720,
%U A156603 720,0,0,362880,1,1,7,210,13800,702240,3991680,5040,0,0,3628800
%V A156603
1,1,1,1,1,2,1,1,0,6,1,1,-1,0,24,1,1,-2,0,0,120,1,1,-3,-6,0,0,720,1,1,
%W A156603
-4,-24,24,0,0,5040,1,1,-5,-60,504,120,0,0,40320,1,1,-6,-120,3360,27720,
%X A156603
-720,0,0,362880,1,1,-7,-210,13800,702240,-3991680,-5040,0,0,3628800
%N A156603 A q-factorial triangle sequence built of Cartan A_n
polynomials as anti-diagonals: p(x,n)=CartanAn(x,n): t(n,k)=If[m == 0,
n!, Product[p(m+1),k), {k, 1, n}]]; %C A156603 Row sums are:
%C A156603 {1, 2, 4, 8, 25, 120, 713, 5038, 40881, 393116, 347905,...}.
%F A156603 p(x,n)=CartanAn(x,n):
%F A156603 t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]];
%F A156603 Out_(n,m)=anti-diagonal(t(n,m)). %e A156603 {1}, %e A156603
{1, 1}, %e A156603 {1, 1, 2}, %e A156603 {1, 1, 0, 6}, %e A156603 {1, 1,
-1, 0, 24}, %e A156603 {1, 1, -2, 0, 0, 120}, %e A156603 {1, 1, -3, -6,
0, 0, 720}, %e A156603 {1, 1, -4, -24, 24, 0, 0, 5040}, %e A156603 {1,
1, -5, -60, 504, 120, 0, 0, 40320}, %e A156603 {1, 1, -6, -120, 3360,
27720, -720, 0, 0, 362880}, %e A156603 {1, 1, -7, -210, 13800, 702240,
-3991680, -5040, 0, 0, 3628800} %t A156603 Clear[t, n, m, i, k, a, b, T,
M, p];
%t A156603 T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1,
-1, 0]];
%t A156603 M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
%t A156603 p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];
%t A156603 a0 = Table[p[x, n], {n, 0, 20}] /. x -&gt; m + 1;
%t A156603 t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];
%t A156603 a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
%t A156603 b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1,
Length[a]}]'
%t A156603 Flatten[%] %Y A156603 A034801,A156599,A156600,A156601,A156602
%K A156603 sign,tabl
%O A156603 0,6
%A A156603 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 11 2009

[Roger Bagula]

The Shabat Chebyshev's behave well as
q-combinations.
These q-combinations will each have their own exponential like constant and
associated Bernoulli ( boson)
and Euler ( Fermion) numbers and statistical mechanical distributions.

The cyclotomic like q-factorial combinations have been around
long enough the Dr. Sloane calls them Gauss:
f[n_,q_]=Product[Sum[q^i,{i,0,k-1}],{k,1,n}]=Product[(1-q^n)/(1-q),{k,1,n}]
but nobody has carried it further than the Fermion types:
f[n_,q_]=Product[Sum[((-q)^i,{i,0,k-1}],{k,1,n}]=Product[(1-(-q)^n)/(1+q),{k,1,n}]
The Paul Hanna general sequence one hadn't even been connected to a
q-factorial from a polynomial
until I found it was.
I've found three new ones in q-combinations:
1) the Stirling 1st polynomial
2) the A_n polynomials
3) the Shabat polynomials
The last two are quantum orthogonal based polynomials.
The seem to alternate even ( Sierpinski -Pascal like) and odd
in their fractal patterns.

What seems important is that they would connect up to Bosons and
Fermions in an
n dimensional way to give a basic fractal connection to
statistical mechanics and quantum mechanics.
The product represents:
U(1)*SU(2)*SU(3)*SU(4)*SU(5)*....*SU(n)
or in Cartan terms:
Product[A_i,{i,0,n-1}]
I don't even have any idea what that might imply.