All the Coefficients whose "distance" from the constant term is at most, 2, in the q-Dyson product with , 6, variables By Shalosh B. Ekhad In this article we follow the brilliant method of Gyula Karolyi and Zoltan Lorant Nagy given in arXiv:1211.6484v1, to find (proved!) explicit expressions for all coefficients in the q-Dyson proudct F:= 6 ------------' ' | | | | | | | | | | | | i = 1 / 6 / a[i] - 1 \ / a[j] \\ | --------' | --------' / t \| | --------' / t \|| |' | | |' | | | q x[i]|| |' | | | q x[j]||| | | | | | | |1 - -------|| | | | |1 - -------||| | | | | | | \ x[j] /| | | | \ x[i] /|| | | | | | | | | | | || \j = i + 1 \ t = 0 / \ t = 1 // of monomials whose sum of positive powers is between 1 and, 2 Definition: Let Q(n) be the q-analog of n!: n --------' ' | | i Q(n) = | | (1 - q ) | | | | i = 1 Theorem: The coefficients of the various coefficients in the q-Dyson product F, given above divided by the q-multinomial coefficients Q(a[1] + a[2] + a[3] + a[4] + a[5] + a[6]) ----------------------------------------------- Q(a[1]) Q(a[2]) Q(a[3]) Q(a[4]) Q(a[5]) Q(a[6]) are as follows where, for i from 1 to , 6, we abbreviate: a[i] z[i] = q ------------------------------------------------------ When the sum of the positive exponents is, 1 there are, 1, possibe monomials: {[1, -1, 0, 0, 0, 0]} x[1] For the coeff., ---- x[2] we have: -1 + z[1] - ------------------------------ q z[2] z[3] z[4] z[5] z[6] - 1 and in Maple input notation: -(-1+z[1])/(q*z[2]*z[3]*z[4]*z[5]*z[6]-1) ------------------------------------------------------ When the sum of the positive exponents is, 2 there are, 4, possibe monomials: {[1, 1, -2, 0, 0, 0], [1, 1, -1, -1, 0, 0], [2, -2, 0, 0, 0, 0], [2, -1, -1, 0, 0, 0]} x[1] x[2] For the coeff., --------- 2 x[3] we have: (z[2] - 1) (-1 + z[1]) (q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] - z[2] - 1)/( (q z[3] z[4] z[5] z[6] - 1) (q z[2] z[3] z[4] z[5] z[6] - 1) (q z[1] z[3] z[4] z[5] z[6] - 1)) and in Maple input notation: (z[2]-1)*(-1+z[1])*(q*z[1]*z[2]*z[3]*z[4]*z[5]*z[6]+q*z[2]*z[3]*z[4]*z[5]*z[6]- z[2]-1)/(q*z[3]*z[4]*z[5]*z[6]-1)/(q*z[2]*z[3]*z[4]*z[5]*z[6]-1)/(q*z[1]*z[3]*z [4]*z[5]*z[6]-1) x[1] x[2] For the coeff., --------- x[3] x[4] we have: z[3] (z[2] - 1) (-1 + z[1]) (q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] - z[2] - 1)/( (q z[3] z[4] z[5] z[6] - 1) (q z[2] z[3] z[4] z[5] z[6] - 1) (q z[1] z[3] z[4] z[5] z[6] - 1)) and in Maple input notation: z[3]*(z[2]-1)*(-1+z[1])*(q*z[1]*z[2]*z[3]*z[4]*z[5]*z[6]+q*z[2]*z[3]*z[4]*z[5]* z[6]-z[2]-1)/(q*z[3]*z[4]*z[5]*z[6]-1)/(q*z[2]*z[3]*z[4]*z[5]*z[6]-1)/(q*z[1]*z [3]*z[4]*z[5]*z[6]-1) 2 x[1] For the coeff., ----- 2 x[2] we have: 5 4 5 4 4 4 - (-1 + z[1]) (q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 4 4 4 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 5 4 4 + q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 4 4 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 3 5 4 + q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 3 4 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 3 3 5 + q z[1] z[2] z[3] z[4] z[5] z[6] 5 4 3 3 3 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 4 4 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 4 3 4 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 4 3 3 4 5 4 3 3 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 4 4 3 3 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 3 3 4 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 3 3 3 4 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 4 4 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 4 3 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 3 3 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 3 2 4 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 3 3 4 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 2 4 4 4 3 4 3 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 4 3 3 2 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 3 2 2 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 3 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 3 2 4 4 3 3 3 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 4 3 2 4 3 3 4 3 3 2 2 3 - q z[2] z[3] z[4] z[5] z[6] + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 3 2 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 3 2 2 3 3 4 3 2 3 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 4 3 2 2 4 3 4 3 2 2 3 3 - q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 4 3 2 2 2 4 3 2 3 3 3 2 - q z[2] z[3] z[4] z[5] z[6] + q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 3 3 2 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 3 2 3 3 4 3 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 4 3 2 3 2 2 4 3 2 2 3 2 - q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 3 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 3 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 3 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 2 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 3 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 3 3 3 2 3 3 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 3 2 3 2 3 2 3 2 3 2 2 3 + q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 3 3 2 2 2 3 2 + q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 3 3 2 3 3 2 - q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 3 2 3 2 3 3 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 3 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 2 2 3 2 3 2 2 2 3 + q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 2 3 3 3 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 2 2 3 3 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 2 2 3 2 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] 3 2 2 2 3 2 2 2 + q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 3 2 2 2 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 2 - q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[2] z[4] z[5] z[6] 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 - q z[2] z[3] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 - q z[2] z[3] z[4] z[6] - q z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[6] + q z[1] z[2] z[3] z[5] z[6] + q z[1] z[2] z[4] z[5] z[6] / + q z[3] z[4] z[5] z[6] - z[1]) / ((q z[2] z[3] z[4] z[5] - 1) / (q z[2] z[3] z[4] z[6] - 1) (q z[2] z[3] z[5] z[6] - 1) (q z[2] z[4] z[5] z[6] - 1) (q z[2] z[3] z[4] z[5] z[6] - 1) 2 (q z[2] z[3] z[4] z[5] z[6] - 1)) and in Maple input notation: -(-1+z[1])*(q^5*z[1]*z[2]^4*z[3]^5*z[4]^4*z[5]^4*z[6]^4-q^5*z[1]*z[2]^4*z[3]^4* z[4]^4*z[5]^4*z[6]^4+q^5*z[1]*z[2]^4*z[3]^3*z[4]^5*z[5]^4*z[6]^4-q^5*z[1]*z[2]^ 4*z[3]^3*z[4]^4*z[5]^4*z[6]^4+q^5*z[1]*z[2]^4*z[3]^3*z[4]^3*z[5]^5*z[6]^4-q^5*z [1]*z[2]^4*z[3]^3*z[4]^3*z[5]^4*z[6]^4+q^5*z[1]*z[2]^4*z[3]^3*z[4]^3*z[5]^3*z[6 ]^5-q^5*z[1]*z[2]^4*z[3]^3*z[4]^3*z[5]^3*z[6]^4-q^4*z[1]*z[2]^3*z[3]^4*z[4]^4*z [5]^3*z[6]^3-q^4*z[1]*z[2]^3*z[3]^4*z[4]^3*z[5]^4*z[6]^3-q^4*z[1]*z[2]^3*z[3]^4 *z[4]^3*z[5]^3*z[6]^4+q^5*z[2]^4*z[3]^3*z[4]^3*z[5]^3*z[6]^3-q^4*z[1]*z[2]^4*z[ 3]^3*z[4]^3*z[5]^3*z[6]^3+q^4*z[1]*z[2]^3*z[3]^3*z[4]^3*z[5]^4*z[6]^3+q^4*z[1]* z[2]^3*z[3]^3*z[4]^3*z[5]^3*z[6]^4-q^4*z[1]*z[2]^3*z[3]^2*z[4]^4*z[5]^4*z[6]^3- q^4*z[1]*z[2]^3*z[3]^2*z[4]^4*z[5]^3*z[6]^4+q^4*z[1]*z[2]^3*z[3]^3*z[4]^3*z[5]^ 3*z[6]^3-q^4*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^4*z[6]^3+q^4*z[1]*z[2]^3*z[3]^2*z[4 ]^3*z[5]^3*z[6]^4-q^4*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^4*z[6]^4-q^4*z[2]^3*z[3]^4 *z[4]^3*z[5]^3*z[6]^3+q^4*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^3*z[6]^3-q^4*z[1]*z[2] ^3*z[3]^3*z[4]^2*z[5]^2*z[6]^4+q^4*z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^3*z[6]^3-q^4* z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^2*z[6]^4+q^4*z[2]^3*z[3]^3*z[4]^3*z[5]^3*z[6]^3- q^4*z[2]^3*z[3]^2*z[4]^4*z[5]^3*z[6]^3+q^4*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^2*z[6 ]^3+q^4*z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^2*z[6]^3+q^4*z[1]*z[2]^3*z[3]^2*z[4]^2*z [5]^3*z[6]^3+q^4*z[2]^3*z[3]^2*z[4]^3*z[5]^3*z[6]^3-q^4*z[2]^3*z[3]^2*z[4]^2*z[ 5]^4*z[6]^3+q^4*z[2]^3*z[3]^2*z[4]^2*z[5]^3*z[6]^3-q^4*z[2]^3*z[3]^2*z[4]^2*z[5 ]^2*z[6]^4+q^3*z[1]*z[2]^2*z[3]^3*z[4]^3*z[5]^3*z[6]^2+q^3*z[1]*z[2]^2*z[3]^3*z [4]^3*z[5]^2*z[6]^3+q^3*z[1]*z[2]^2*z[3]^3*z[4]^2*z[5]^3*z[6]^3-q^4*z[2]^3*z[3] ^3*z[4]^2*z[5]^2*z[6]^2-q^4*z[2]^3*z[3]^2*z[4]^3*z[5]^2*z[6]^2-q^4*z[2]^3*z[3]^ 2*z[4]^2*z[5]^3*z[6]^2+q^3*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^2*z[6]^2+q^3*z[1]*z[2 ]^3*z[3]^2*z[4]^3*z[5]^2*z[6]^2+q^3*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^3*z[6]^2+q^3 *z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^2*z[6]^3-q^3*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]^3*z [6]^3+q^3*z[1]*z[2]^2*z[3]*z[4]^3*z[5]^3*z[6]^3-q^3*z[1]*z[2]^2*z[3]^2*z[4]^2*z [5]^2*z[6]^3+q^3*z[1]*z[2]^2*z[3]^2*z[4]*z[5]^3*z[6]^3+q^3*z[2]^2*z[3]^3*z[4]^3 *z[5]^2*z[6]^2+q^3*z[2]^2*z[3]^3*z[4]^2*z[5]^3*z[6]^2+q^3*z[2]^2*z[3]^3*z[4]^2* z[5]^2*z[6]^3-q^3*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]^2*z[6]^2+q^3*z[1]*z[2]^2*z[3]^ 2*z[4]^2*z[5]*z[6]^3-q^3*z[2]^2*z[3]^2*z[4]^2*z[5]^3*z[6]^2-q^3*z[2]^2*z[3]^2*z [4]^2*z[5]^2*z[6]^3+q^3*z[2]^2*z[3]*z[4]^3*z[5]^3*z[6]^2+q^3*z[2]^2*z[3]*z[4]^3 *z[5]^2*z[6]^3-q^3*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]*z[6]^2-q^3*z[1]*z[2]^2*z[3]^2 *z[4]*z[5]^2*z[6]^2-q^3*z[1]*z[2]^2*z[3]*z[4]^2*z[5]^2*z[6]^2-q^3*z[2]^2*z[3]^2 *z[4]^2*z[5]^2*z[6]^2+q^3*z[2]^2*z[3]^2*z[4]*z[5]^3*z[6]^2-q^3*z[2]^2*z[3]*z[4] ^2*z[5]^2*z[6]^3+q^3*z[2]^2*z[3]*z[4]*z[5]^3*z[6]^3-q^3*z[2]^2*z[3]^2*z[4]*z[5] ^2*z[6]^2+q^3*z[2]^2*z[3]^2*z[4]*z[5]*z[6]^3-q^3*z[2]^2*z[3]*z[4]^2*z[5]^2*z[6] ^2+q^3*z[2]^2*z[3]*z[4]^2*z[5]*z[6]^3-q^2*z[1]*z[2]*z[3]^2*z[4]^2*z[5]^2*z[6]^2 +q^3*z[2]^2*z[3]^2*z[4]^2*z[5]*z[6]+q^3*z[2]^2*z[3]^2*z[4]*z[5]^2*z[6]+q^3*z[2] ^2*z[3]*z[4]^2*z[5]^2*z[6]-q^2*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]*z[6]-q^2*z[1]*z[2 ]^2*z[3]^2*z[4]*z[5]^2*z[6]-q^2*z[1]*z[2]^2*z[3]^2*z[4]*z[5]*z[6]^2-q^2*z[1]*z[ 2]^2*z[3]*z[4]^2*z[5]^2*z[6]-q^2*z[1]*z[2]^2*z[3]*z[4]^2*z[5]*z[6]^2-q^2*z[1]*z [2]^2*z[3]*z[4]*z[5]^2*z[6]^2-q^2*z[2]*z[3]^2*z[4]^2*z[5]^2*z[6]-q^2*z[2]*z[3]^ 2*z[4]^2*z[5]*z[6]^2-q^2*z[2]*z[3]^2*z[4]*z[5]^2*z[6]^2+q^2*z[2]*z[3]*z[4]*z[5] ^2*z[6]^2-q^2*z[2]*z[4]^2*z[5]^2*z[6]^2+q^2*z[1]*z[2]*z[3]*z[4]*z[5]*z[6]+q^2*z [2]*z[3]*z[4]*z[5]*z[6]^2-q^2*z[2]*z[3]*z[5]^2*z[6]^2+q^2*z[2]*z[3]*z[4]*z[5]*z [6]-q^2*z[2]*z[3]*z[4]*z[6]^2-q^2*z[2]*z[3]*z[4]*z[5]+q*z[1]*z[2]*z[3]*z[4]*z[5 ]+q*z[1]*z[2]*z[3]*z[4]*z[6]+q*z[1]*z[2]*z[3]*z[5]*z[6]+q*z[1]*z[2]*z[4]*z[5]*z [6]+q*z[3]*z[4]*z[5]*z[6]-z[1])/(q*z[2]*z[3]*z[4]*z[5]-1)/(q*z[2]*z[3]*z[4]*z[6 ]-1)/(q*z[2]*z[3]*z[5]*z[6]-1)/(q*z[2]*z[4]*z[5]*z[6]-1)/(q*z[2]*z[3]*z[4]*z[5] *z[6]-1)/(q^2*z[2]*z[3]*z[4]*z[5]*z[6]-1) 2 x[1] For the coeff., --------- x[2] x[3] we have: 4 4 4 3 3 3 (-1 + z[1]) (q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 4 3 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 3 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 2 4 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 2 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 2 2 4 - q z[1] z[2] z[3] z[4] z[5] z[6] 4 4 3 2 2 3 4 4 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 4 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 3 3 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 3 2 3 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 3 2 2 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 3 3 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 3 2 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 2 3 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 3 3 + q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 3 3 3 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 3 2 3 2 2 3 3 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] 3 3 2 2 2 3 3 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 3 2 3 2 3 3 2 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 3 3 2 3 3 3 2 2 + q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 3 3 2 2 2 3 2 2 + q z[2] z[3] z[4] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] 2 3 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 3 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 2 + q z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 2 2 - q z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 2 - q z[2] z[3] z[5] z[6] + q z[2] z[3] z[4] z[5] z[6] 2 2 2 2 2 - q z[2] z[3] z[4] z[6] - q z[2] z[3] z[4] z[5] 2 2 + q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[6] 2 + q z[1] z[2] z[3] z[5] z[6] - q z[1] z[2] z[3] z[4] z[5] z[6] - q z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[6] - q z[2] z[3] z[5] z[6] / + q z[2] z[4] z[5] z[6] - z[1] z[2] + 1) / ((q z[2] z[3] z[4] z[6] - 1) / (q z[2] z[3] z[5] z[6] - 1) (q z[2] z[3] z[4] z[5] - 1) 2 (q z[2] z[3] z[4] z[5] z[6] - 1) (q z[2] z[3] z[4] z[5] z[6] - 1)) and in Maple input notation: (-1+z[1])*(q^4*z[1]*z[2]^4*z[3]^4*z[4]^3*z[5]^3*z[6]^3-q^4*z[1]*z[2]^4*z[3]^3*z [4]^4*z[5]^3*z[6]^3+q^4*z[1]*z[2]^4*z[3]^3*z[4]^3*z[5]^3*z[6]^3-q^4*z[1]*z[2]^4 *z[3]^3*z[4]^2*z[5]^4*z[6]^3+q^4*z[1]*z[2]^4*z[3]^3*z[4]^2*z[5]^3*z[6]^3-q^4*z[ 1]*z[2]^4*z[3]^3*z[4]^2*z[5]^2*z[6]^4+q^4*z[1]*z[2]^4*z[3]^3*z[4]^2*z[5]^2*z[6] ^3-q^4*z[2]^4*z[3]^3*z[4]^2*z[5]^2*z[6]^2+q^3*z[1]*z[2]^4*z[3]^3*z[4]^2*z[5]^2* z[6]^2-q^3*z[1]*z[2]^3*z[3]^3*z[4]^3*z[5]^2*z[6]^2-q^3*z[1]*z[2]^3*z[3]^3*z[4]^ 2*z[5]^3*z[6]^2-q^3*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^2*z[6]^3+q^3*z[1]*z[2]^3*z[3 ]^2*z[4]^3*z[5]^3*z[6]^2+q^3*z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^2*z[6]^3-q^3*z[1]*z [2]^3*z[3]^2*z[4]^2*z[5]^2*z[6]^3+q^3*z[1]*z[2]^3*z[3]^2*z[4]*z[5]^3*z[6]^3-q^3 *z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^2*z[6]^2+q^3*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]*z[6 ]^3-q^3*z[2]^3*z[3]^3*z[4]^2*z[5]^2*z[6]^2+q^3*z[2]^3*z[3]^2*z[4]^3*z[5]^2*z[6] ^2-q^3*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]*z[6]^2-q^3*z[1]*z[2]^3*z[3]^2*z[4]*z[5]^2 *z[6]^2-q^3*z[2]^3*z[3]^2*z[4]^2*z[5]^2*z[6]^2+q^3*z[2]^3*z[3]^2*z[4]*z[5]^3*z[ 6]^2-q^3*z[2]^3*z[3]^2*z[4]*z[5]^2*z[6]^2+q^3*z[2]^3*z[3]^2*z[4]*z[5]*z[6]^3+q^ 3*z[2]^3*z[3]^2*z[4]^2*z[5]*z[6]+q^3*z[2]^3*z[3]^2*z[4]*z[5]^2*z[6]-q^2*z[1]*z[ 2]^3*z[3]^2*z[4]^2*z[5]*z[6]-q^2*z[1]*z[2]^3*z[3]^2*z[4]*z[5]^2*z[6]-q^2*z[1]*z [2]^3*z[3]^2*z[4]*z[5]*z[6]^2+q^2*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]^2*z[6]+q^2*z[1 ]*z[2]^2*z[3]^2*z[4]^2*z[5]*z[6]^2+q^2*z[1]*z[2]^2*z[3]^2*z[4]*z[5]^2*z[6]^2-q^ 2*z[1]*z[2]^2*z[3]*z[4]^2*z[5]^2*z[6]^2+q^2*z[2]^2*z[3]^2*z[4]^2*z[5]*z[6]+q^2* z[2]^2*z[3]^2*z[4]*z[5]^2*z[6]+q^2*z[2]^2*z[3]^2*z[4]*z[5]*z[6]^2-q^2*z[2]^2*z[ 3]*z[4]^2*z[5]^2*z[6]-q^2*z[2]^2*z[3]*z[4]^2*z[5]*z[6]^2+q^2*z[1]*z[2]^2*z[3]*z [4]*z[5]*z[6]+q^2*z[2]^2*z[3]*z[4]*z[5]*z[6]^2-q^2*z[2]^2*z[3]*z[5]^2*z[6]^2+q^ 2*z[2]^2*z[3]*z[4]*z[5]*z[6]-q^2*z[2]^2*z[3]*z[4]*z[6]^2-q^2*z[2]^2*z[3]*z[4]*z [5]+q*z[1]*z[2]^2*z[3]*z[4]*z[5]+q*z[1]*z[2]^2*z[3]*z[4]*z[6]+q*z[1]*z[2]^2*z[3 ]*z[5]*z[6]-q*z[1]*z[2]*z[3]*z[4]*z[5]*z[6]-q*z[2]*z[3]*z[4]*z[5]-q*z[2]*z[3]*z [4]*z[6]-q*z[2]*z[3]*z[5]*z[6]+q*z[2]*z[4]*z[5]*z[6]-z[1]*z[2]+1)/(q*z[2]*z[3]* z[4]*z[6]-1)/(q*z[2]*z[3]*z[5]*z[6]-1)/(q*z[2]*z[3]*z[4]*z[5]-1)/(q^2*z[2]*z[3] *z[4]*z[5]*z[6]-1)/(q*z[2]*z[3]*z[4]*z[5]*z[6]-1) Note that all these statements are proved (internally) using the Karolyi-Nagy method, as extended by my beloved master, Doron Zeilberger. as explained in his brilliant article "Variations on Gyula Karolyi and Zoltan Lorant Nagy's BRILLIANT proof of the Zeilberger-Bressoud q-Dyson Theorem" published in our famous Personal Journal. This book took, 16.161, seconds to generate. 16.190