Some Programs for Analyzing of the Conjectures of Borwein Type MARKO D. PETKOVIĆ & PREDRAG M. RAJKOVIĆ Department of Mathematics, University of Niš, Višegradska 33 SERBIA AND MONTENEGRO Abstract: - In our trials to do contributions to the conjectures about expanding some products,which Peter Borwein established, we made a lot of useful programs for their discussions. They enabled us to prove some properties od such expansions and more often to nominate new conjectures. We think that they can be useful in final proof. Key-words:- Polynomials, Basic Hypergeometric Functions 1 Introduction Let us denote by (a; q) n the next product n 1 (a; q ) 0 1, (a; q ) n (1 aq k ). k 0 Also, for a number m , let J m (n, q) be a function defined by m 1 n 1 m 1 j 1 k 0 j 1 J m (n, q) (q j ; q m ) n (1 q mk j ). The polynomial J m (n, q) has (1) degree dg ( J m ) (m 1)mn 2 / 2. We can expand it in the next way m 1 J m (n, q) A1(,mn ) (q m ) q k Ak( m,n) (q m ) (2) k 1 Next we will denote A1(,mn) (q) just as A1, n (q) when it is clear which value of m corresponds to noted polynomial. The program GenerateJA [ns_, ne_, m_, q_] is our program which generates the polynomials Ak , n (q m ) for ns n ne. It uses temporary function J [n_, m_, q_] which generates product J m (n, q) . J[n_, m_, q_] := Expand[Product[Product[1 – q^(m*i + j), {j, 1, m - 1}],{i, 0, n - 1}]]; GenerateJA[ns_, ne_, m_, q_] := Block[{T, t, A, h, j, n, k}, Clear[q]; k = m; n = ne - ns + 1; A = Table[Table[0, {i, m}], {j, n}]; For [j = 1, j <= n, j++, T = J[j + ns - 1, m, q]; t = Length[T]; For [h = 1, h <= t, h++, d = PolyDegree[T[[h]]]; A[[j, 1 + Mod[d, m]]] += T[[h]]; ]; Do[ A[[j, l]] = Expand[ReplaceAll[A[[j, 1]]/q^(l - 1), q -> q^(1/m)]], {l, 1 , m}]; Do [A[[j, l]] = -A[[j, l]], {l, 2, m}]; ]; Return[A]; ]; (* End Function *) We have used temporary MATHEMATICA function PolyDegree[p_] for computing degree of polynomial p ( q ) . At first, the function J[n_, k_, q_] expands product (1). Then loop of variable h collects addends of expanded product corresponding to q l m k where l and k are integers and 0 k m 1 is fixed. These members are addends of ql 1 Ak , n (q m ) collecting in the matrix A n, k . Finally two Do-loops forms Ak , n from ql 1 Ak , n (q m ) . In a personal communication in 1990., Peter Borwein had conjectured that all polynomials Ak , n (q) have nonnegative coefficients for m=3 and m=5. According to results obtained by our programs, we can establish two conjectures. Conjecture 1. (The first Borwein type conjecture) For any number m N , the polynomials A1, n (q) (n 1,2,) have nonnegative coefficients. m -even number and n -odd, besides A1, n (q) (n 1,2,) , we can notice one more For polynomial sequence with nonegative coefficients. Conjecture conjecture) 2. (The second Borwein type For any even number m N and Mod [(m 1)mn2 , m] 0 , the coefficients of the polynomials A 1, n (q) ( n N ) are nonegative. Example 1.1. For concrete values m 6 and n 1 , we have J 6 (1, q) 1 q q 2 q5 q 6 q 7 q8 10 13 14 on Make [n_, m_, q_] finds matrix F (m, n, q ) in the recurrence relation An F (m, n, q) An1, n N . m 1 Tn (q ) (1 q m ( n 1) i ). i 1 When we replace J m (n, q) and J m (n 1, q) with (2), and expand right side by collecting addends corresponding to same ql (0 l m 1) , we obtain (4) The matrix M i , j ( q ) is equal to F (m, n, q ) . The Let An A1, n (q), A2, n (q),, Am, n (q) . . Our functi- where j 1 J1,1 (q) q 1, J 4,1 (q) q 2 q. J m (n, q ) J m (n 1, q ) Tn (q) n 15 hence 2 The matrix form The main idea is to represent (1) in a form Ai , n (q m ) M i , j (q)Aj , n 1 (q m ) . q q q q q 9 For [j = 1, j <= t, j++, Tm=ReplaceAll[Tmp[[j]], n -> (m+1)]; tm = Mod[PolyDegree[Tm], m]; Tm1 = ExpandAll[ReplaceAll[ Tmp[[j]]/q^tm, q -> q^(1/m)]]; If[tm > 0, Tm1 = -Tm1]; A[[tm + 1, h + 1]] += Tm1; ]; ]; Return[A]; ]; (3) Make[n_, m_, q_] := Block[{Var, h, i, j, l, T, t, Anm1, An, A, Tmp, Tm, tm, Tm1}, Clear[Var]; A =0* IdentityMatrix[m]; T = ExpandAll[Product[1 - q^(m*(n - 1) + j),{j, 1, m - 1}]]; Anm1 = ExpandAll[Var[0] – Sum [Var[i]*q^i, {i, m - 1}]]; An = ExpandAll[Anm1*T]; t = Length[An]; For [h = 0, h <= m - 1, h++, Tmp = Coefficient[An, Var[h], 1]; t = Length[Tmp]; product J m (n 1, q) is represented by Anm1 in our program and right side of equation (4) is computing with An = ExpandAll[Anm1*T]; where T represents Tn (q ) . The next loop (of variable h ) forms polynomials M i , j ( q ) . Polynomials Aj , n 1 (q) are represented via array of temporary symbolic variables Var[j]. First we take addends with common Var[h] in variable Tmp. Then, we are grouping addends corresponding to ql m k and finally obtaining A[[tm+1,h+1]], elements of matrix F (m, n, q ) . Note that input parameters n and q are symbolic variables and m is integer. The resulting matrix A (representing F (m, n, q ) ) is a function of n and q . Example 2.1. For m 3 , we have 1 q 2 n 1 qn qn F (3, n, q) 1 q 2 n 1 1 q 2 n 1 1 q 2 n 1 . q n 1 q n 1 1 q 2 n 1 Relation (3) can be used for computing The next algorithm computes An by using the relation (3) for given matrix F (n, m, q ) which can be computed by the function Make [n_, m_, q_]. MatrixGenerator[nn_, k_, q_, F_] := Block[{AaA, i, j}, Clear[AaA]; AaA[0] = Insert[Table[0, {j, k - 1}], 1, 1]; Do[AaA[i + 1] = ExpandAll[ ReplaceAll[F, n -> (i + 1)]. AaA[i]]; , {i, 0, nn - 1}]; Return[Table[AaA[i], {i, 1, nn}]]; ]; Here input matrix F is functional matrix of symbolic variables q and n . Output is list consists of elements An for 1 n nn . Print["Simplifying system....."]; Sys = Simplify[Sys]; Print["Solving....."]; sol = Solve[Sys, Table[Var[i], {i, 2, d}]]; Print["Simplifying solution....."]; sol = Simplify[sol, TimeConstraint -> 600]; J = ReplaceAll[A[[d, 1, 1]], sol[[1]]]; Print["Simplifying J....."]; J = Simplify[J]; J = Collect[J, Union[{Var[1]}, Table[Var[1, i], {i, 1, d}]]]; Return[J]; ]; Main idea is to make a system of ( m 1) equations 3 Fundamental recurrence relation It seems to be of great importance to find separate nN , recurrence relations for the sequences A1,n (q ) i 1, with Var 2 , variables ,Var m (k 2, An , k , m) and to replace its solution to m th equation ( A d ,1,1 0 ). Using vector equation (3), it can be easily obtained An1 Fi* m, n, q An , m . Program Fundamental [F] finds funda- mental recurrence relation for matrix computed by function Make [n_, m_, q_]. where Fundamental[F_] := Block[{Sys, J, J1, h, i, n, d, M, A, CF}, Clear[Var]; {d, d1} = Dimensions[F]; A = Table[0, {d + 1}]; M = IdentityMatrix[d]; A[[1]] = Var[1, 1] {First[ReplaceAll[F, n -> n + 1]]}. Table[{Var[i]}, {i, d}]; Do [ M = ExpandAll[ReplaceAll[F, n -> n + h - 1].M]; A[[h]] = Collect[ExpandAll[{{Var[1, h]}} - {First[ ReplaceAll[F, n -> n + h]]}.M. Table[{Var[i]}, {i, d}]], Table[Var[i], {i, d}]]; , {h, 2, d + 1} ]; (* End Do h*) Sys = Table[A[[h, 1, 1]] == 0, {h, 1, d - 1}]; Also, it holds Fi* m, n, q F m, n i, q m F m, n 1, q . Ani ,1 Fi* m, n, q j 1 1j Aj ,1 . (5) Previous equations, for i 1, , m 1 , form system of independent linear equations (parameter An i ,1 is contained only in i th equation) of variables An , j , j 2, , m . When we replace solution of this system into ( m 1) th equation we obtain equation G An ,1 , , An m 1,1 0 which is fundamental recurrence equation of sequence An ,1 . In the first Do - loop, we are forming equations (5). Product Fi* m, n, q An is computing iteratively and is represented by M . Equations are storing in variable A and are being computed by definition in body of loop. Next, we are solving formed system, replacing solution to ( m 1) th (in code, we used variable d for m , so equation is d+1-th) equation and obtaining fundamental relation. In all code we use MATHEMATICA function Simplify in order to reduce expressions we obtain. The next theorems deal with general recurrence relation for concrete values m 2 and m 3 . n 1 m 1 2 2 1 q n m( m 1) 2 J m (n,1 q) q n m( m 1) 2 1 mi k q i 0 k 1 n 1 m 1 i 0 k 1 Theorem 3.1. For m 2 , the sequences Ai ,n (q ) , (i 1, 2) satisfy the same recurrence relation (1)n ( m 1) J m (n, q) m 2 q n m( m 1) 2 A1,n 1 q m q ( k 1) Ak ,n 1 q m k 2 with only difference in initial values. 1 Theorem 3.2. For m 2 , the sequences Ai ,n (q ) , (i 1, 2,3) satisfy the same recurrence relation F n 1q(1 q q )(1 q q n ( m 1) m m A q q k 1 Ak ,n q m 1,n k 2 ) F n q 3 (1 q 3n 1 )(1 q 3n 2 ) 1 with only difference in initial values: F n A1,n : F 0 1, F 1 1 q, m q n m( m 1) 2 A1,n 1 q m q n m( m 1) 2( k 1) Ak ,n 1 q m 2 4n 4 i.e. F n 3 F n 2 (1 q q 2 )(1 q 2 n 3 ) 2n2 we have F n 2 (1 q) F n 1 (q q 2 n 2 ) F n 0 2 q mi k 1 2 k 2 m A1,n q m 1 n ( m 1) n ( m 1) k 2 q k 1 Ak ,n q m wherefrom conclusions follow. F 2 1 q 2q q q , 2 3 4 F n A2,n : F 0 1, F 1 1, F 2 1 q q3 , F n A3,n : F 0 1, F 1 1, F 2 1 q2 q3. We have generated recurrence equations for m 4,5, 6, 7 , but these relations are rather complicated, and that is why they are not presented here. Also for m 6 and m 7 our program needs large amount of time to produce the result, because polynomial and rational expressions obtaining by our program are so large. Also, the matrices F (m, n, q ) have large polynomials as its elements. 5 The zeros Let us remind that a polynomial P q qk 1 k has the zeros 2 qk , j exp i j 1 k j 1, 2, ,k i 1 which have the property k , mod l , k 0 j 1 0, mod l , k 0 We will denote by sl the sum of l the powers of k q l k, j 4 Reciprocal polynomials Let us notice some reciprocal polynomials. all zeros of J n, q , i.e. n 1 m 1 mk 1 Theorem 4.1. The polynomial A1, n ( q ) is reciprocal l sl qmk j ,i . itself and the polynomial Ak 1,n ( q ) is reciprocal to the polynomial Am k 1,n ( q ) , i.e., A1,n q 1 Ak 1,n (q) (1) n ( m 1) n ( m 1) (k 1, 2, Proof. Starting from q qn 2 ( m 1) 2 n 2 ( m 1) 2 k 0 j 1 i 1 Hence A1,n 1 q Am k 1,n (1 q) , m 1) . n 1 m 1 sl mk j k ',0 k 'mod l ,mk j k 0 j 1 where k , j is Kronecker delta. Obviously, sl 1 . If l is a prime number greater than m n 1 m 1, then sl 1 . Now, the coefficients of the polynomial 1 ( m 1) n J n, q q mn d n ,1q mn 2 2 1 Note that values of sl are very small compared with d n , k . For n 20 and m 3 , maximal value d n ,mn2 1q d n ,mn2 and the sums sl are connected by Newton's formulae sl d n,1sl 1 d n,l 1s1 ld n,l 0 (l 1, 2, 6 Conclusion , mn2 ). or d n ,l 1 sl dn,1sl 1 dn,l 1s1 l (l 1, 2, , mn2 ). sl is 190 and maximal d 20,k is 21645518 . (6) wherefrom we get much faster algorithm for evaluating of the coefficients d n , k . Next is a code of function GenerateNewtonAi[n_, m_, q_] which generates vector An . GenerateNewtonAi[n_, m_, q_] := Module[{dgJ, h, j, k, l, m, n, s, a, list, mm}, dgJ = (m - 1)*m/2*n^2; Do[s[nn] = Sum[(m*k + j)* KroneckerDelta[ Mod[nn, m*k + j], 0], {j, 1, m - 1}, {k, 0, n - 1}]; , {n, 1, dgJ}]; a[0] = 1; a[1] = -s[1]; Do[ a[mm] = -(s[mm] + Sum[a[j]*s[mm - j], {j, 1, mm - 1}])/mm; , {mm, 2, dgJ}]; list = Table[0, {i, 1, m}]; list[[1]] = Sum[a[m*k]*q^k, {k, 0, dgJ/m}]; Do[ list[[i]] = -Sum[a[m*k + i - 1]*q^k, {k, 0, dgJ/m - 1}]; , {i, 2, m}]; Return[list]; ]; In this paper, we presented some algorithms and programs for analyzing conjectures of Borwein type. There are presented 3 algorithms for computation of polynomials Ak , n . From the computed polynomials, we conjectured two generalizations of Borwein conjecture. Next we made a programs for computing the matrix F n, k , q in recurrence relation (3), and separate (fundamental) recurrence relation for the sequence A 1,n n . Also we presented some theoretical results obtained by our programs. For further research, it seems to be interesting to consider sequence sl and relation (6) because values of sl are positive and small, and relation (6) makes a nice recurrent connection between coefficients of polynomials J m (n, q) and also Ak , n . References: [1] G. E. Andrews, On a Conjecture of Peter Borwein, J. Symbolic Computation, Vol. 20, 1995, pp. 487-501. [2] D. M. Bressoud, The Borwein Conjecture and Partitions with Prescribed Hook Differences, Electronic J. Combinatorics, Vol. 3, No. 2, 1996, pp. 1-14. [3] R. Koekoek and R. F. Swarttouw, Askeyscheme of hypergeometric orthogonal polynomials and its q-analogue, Report No. 98-17, Delft University of Technology, 1998. [4] S. Ole Warnaar, The generalized Borwein conjecture. I. The Burge transform, to appear in Contemporary Mathematics, arXiv:mathCO/001220, 2000. [5] S. Ole Warnaar, The generalized Borwein conjecture. II. Refined q-trinomial coefficients, arXiv:mathCO/0110307, 2001.