All the Coefficients whose "distance" from the constant term is at most, 2, in the q-Dyson product with , 5, 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:= 5 ------------' ' | | | | | | | | | | | | i = 1 / 5 / 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]) --------------------------------------- Q(a[1]) Q(a[2]) Q(a[3]) Q(a[4]) Q(a[5]) are as follows where, for i from 1 to , 5, 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]} x[1] For the coeff., ---- x[2] we have: -1 + z[1] - ------------------------- q z[2] z[3] z[4] z[5] - 1 and in Maple input notation: -(-1+z[1])/(q*z[2]*z[3]*z[4]*z[5]-1) ------------------------------------------------------ When the sum of the positive exponents is, 2 there are, 4, possibe monomials: {[1, 1, -2, 0, 0], [1, 1, -1, -1, 0], [2, -2, 0, 0, 0], [2, -1, -1, 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] + q z[2] z[3] z[4] z[5] - z[2] - 1)/( (q z[3] z[4] z[5] - 1) (q z[2] z[3] z[4] z[5] - 1) (q z[1] z[3] z[4] z[5] - 1)) and in Maple input notation: (z[2]-1)*(-1+z[1])*(q*z[1]*z[2]*z[3]*z[4]*z[5]+q*z[2]*z[3]*z[4]*z[5]-z[2]-1)/(q *z[3]*z[4]*z[5]-1)/(q*z[2]*z[3]*z[4]*z[5]-1)/(q*z[1]*z[3]*z[4]*z[5]-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] + q z[2] z[3] z[4] z[5] - z[2] - 1)/( (q z[3] z[4] z[5] - 1) (q z[2] z[3] z[4] z[5] - 1) (q z[1] z[3] z[4] z[5] - 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]+q*z[2]*z[3]*z[4]*z[5]-z[2]-\ 1)/(q*z[3]*z[4]*z[5]-1)/(q*z[2]*z[3]*z[4]*z[5]-1)/(q*z[1]*z[3]*z[4]*z[5]-1) 2 x[1] For the coeff., ----- 2 x[2] we have: 4 3 4 3 3 - (-1 + z[1]) (q z[1] z[2] z[3] z[4] z[5] 4 3 3 3 3 4 3 2 4 3 - q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 4 3 2 3 3 4 3 2 2 4 - q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 4 3 2 2 3 3 2 3 3 2 - q z[1] z[2] z[3] z[4] z[5] - q z[1] z[2] z[3] z[4] z[5] 3 2 3 2 3 4 3 2 2 2 - q z[1] z[2] z[3] z[4] z[5] + q z[2] z[3] z[4] z[5] 3 3 2 2 2 3 2 2 2 3 - q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 3 2 3 3 3 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 3 2 2 3 3 2 3 2 2 - q z[1] z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 3 2 2 2 3 2 2 2 + q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 3 2 2 2 2 3 2 3 2 + q z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 3 2 2 2 3 2 3 + q z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 2 2 2 2 3 2 2 + q z[1] z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 3 2 2 2 2 2 - q z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 2 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 2 2 2 2 2 2 + q z[2] z[3] z[4] z[5] + q z[2] z[3] z[4] z[5] 2 2 2 - q z[1] z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 2 2 2 2 2 2 + q z[2] z[4] z[5] - q z[2] z[3] z[4] z[5] + q z[2] z[3] z[5] 2 + q z[2] z[3] z[4] - q z[1] z[2] z[3] z[4] - q z[1] z[2] z[3] z[5] / - q z[1] z[2] z[4] z[5] - q z[3] z[4] z[5] + z[1]) / ( / (q z[2] z[3] z[4] - 1) (q z[2] z[3] z[5] - 1) (q z[2] z[4] z[5] - 1) 2 (q z[2] z[3] z[4] z[5] - 1) (q z[2] z[3] z[4] z[5] - 1)) and in Maple input notation: -(-1+z[1])*(q^4*z[1]*z[2]^3*z[3]^4*z[4]^3*z[5]^3-q^4*z[1]*z[2]^3*z[3]^3*z[4]^3* z[5]^3+q^4*z[1]*z[2]^3*z[3]^2*z[4]^4*z[5]^3-q^4*z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^ 3+q^4*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^4-q^4*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^3-q^3 *z[1]*z[2]^2*z[3]^3*z[4]^3*z[5]^2-q^3*z[1]*z[2]^2*z[3]^3*z[4]^2*z[5]^3+q^4*z[2] ^3*z[3]^2*z[4]^2*z[5]^2-q^3*z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^2+q^3*z[1]*z[2]^2*z[ 3]^2*z[4]^2*z[5]^3-q^3*z[1]*z[2]^2*z[3]*z[4]^3*z[5]^3+q^3*z[1]*z[2]^2*z[3]^2*z[ 4]^2*z[5]^2-q^3*z[1]*z[2]^2*z[3]^2*z[4]*z[5]^3-q^3*z[2]^2*z[3]^3*z[4]^2*z[5]^2+ q^3*z[1]*z[2]^2*z[3]^2*z[4]*z[5]^2+q^3*z[1]*z[2]^2*z[3]*z[4]^2*z[5]^2+q^3*z[2]^ 2*z[3]^2*z[4]^2*z[5]^2-q^3*z[2]^2*z[3]*z[4]^3*z[5]^2+q^3*z[2]^2*z[3]*z[4]^2*z[5 ]^2-q^3*z[2]^2*z[3]*z[4]*z[5]^3+q^2*z[1]*z[2]*z[3]^2*z[4]^2*z[5]^2-q^3*z[2]^2*z [3]^2*z[4]*z[5]-q^3*z[2]^2*z[3]*z[4]^2*z[5]+q^2*z[1]*z[2]^2*z[3]^2*z[4]*z[5]+q^ 2*z[1]*z[2]^2*z[3]*z[4]^2*z[5]+q^2*z[1]*z[2]^2*z[3]*z[4]*z[5]^2+q^2*z[2]*z[3]^2 *z[4]^2*z[5]+q^2*z[2]*z[3]^2*z[4]*z[5]^2-q^2*z[1]*z[2]*z[3]*z[4]*z[5]-q^2*z[2]* z[3]*z[4]*z[5]^2+q^2*z[2]*z[4]^2*z[5]^2-q^2*z[2]*z[3]*z[4]*z[5]+q^2*z[2]*z[3]*z [5]^2+q^2*z[2]*z[3]*z[4]-q*z[1]*z[2]*z[3]*z[4]-q*z[1]*z[2]*z[3]*z[5]-q*z[1]*z[2 ]*z[4]*z[5]-q*z[3]*z[4]*z[5]+z[1])/(q*z[2]*z[3]*z[4]-1)/(q*z[2]*z[3]*z[5]-1)/(q *z[2]*z[4]*z[5]-1)/(q^2*z[2]*z[3]*z[4]*z[5]-1)/(q*z[2]*z[3]*z[4]*z[5]-1) 2 x[1] For the coeff., --------- x[2] x[3] we have: 3 3 3 2 2 3 3 2 3 2 (q z[1] z[2] z[3] z[4] z[5] - q z[1] z[2] z[3] z[4] z[5] 3 3 2 2 2 3 3 2 3 + q z[1] z[2] z[3] z[4] z[5] - q z[1] z[2] z[3] z[4] z[5] 3 3 2 2 3 3 2 + q z[1] z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 2 3 2 2 2 2 2 + q z[1] z[2] z[3] z[4] z[5] - q z[1] z[2] z[3] z[4] z[5] 2 2 2 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] + q z[1] z[2] z[3] z[4] z[5] 2 2 2 2 2 - q z[1] z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] 2 2 2 2 2 2 2 2 + q z[2] z[3] z[4] z[5] - q z[2] z[3] z[4] z[5] + q z[2] z[3] z[5] 2 2 2 2 + q z[2] z[3] z[4] - q z[1] z[2] z[3] z[4] - q z[1] z[2] z[3] z[5] + q z[1] z[2] z[3] z[4] z[5] + q z[2] z[3] z[4] + q z[2] z[3] z[5] / - q z[2] z[4] z[5] + z[1] z[2] - 1) (-1 + z[1]) / ( / 2 (q z[2] z[3] z[5] - 1) (q z[2] z[3] z[4] - 1) (q z[2] z[3] z[4] z[5] - 1) (q z[2] z[3] z[4] z[5] - 1)) and in Maple input notation: (q^3*z[1]*z[2]^3*z[3]^3*z[4]^2*z[5]^2-q^3*z[1]*z[2]^3*z[3]^2*z[4]^3*z[5]^2+q^3* z[1]*z[2]^3*z[3]^2*z[4]^2*z[5]^2-q^3*z[1]*z[2]^3*z[3]^2*z[4]*z[5]^3+q^3*z[1]*z[ 2]^3*z[3]^2*z[4]*z[5]^2-q^3*z[2]^3*z[3]^2*z[4]*z[5]+q^2*z[1]*z[2]^3*z[3]^2*z[4] *z[5]-q^2*z[1]*z[2]^2*z[3]^2*z[4]^2*z[5]-q^2*z[1]*z[2]^2*z[3]^2*z[4]*z[5]^2+q^2 *z[1]*z[2]^2*z[3]*z[4]^2*z[5]^2-q^2*z[1]*z[2]^2*z[3]*z[4]*z[5]-q^2*z[2]^2*z[3]^ 2*z[4]*z[5]+q^2*z[2]^2*z[3]*z[4]^2*z[5]-q^2*z[2]^2*z[3]*z[4]*z[5]+q^2*z[2]^2*z[ 3]*z[5]^2+q^2*z[2]^2*z[3]*z[4]-q*z[1]*z[2]^2*z[3]*z[4]-q*z[1]*z[2]^2*z[3]*z[5]+ q*z[1]*z[2]*z[3]*z[4]*z[5]+q*z[2]*z[3]*z[4]+q*z[2]*z[3]*z[5]-q*z[2]*z[4]*z[5]+z [1]*z[2]-1)*(-1+z[1])/(q*z[2]*z[3]*z[5]-1)/(q*z[2]*z[3]*z[4]-1)/(q^2*z[2]*z[3]* z[4]*z[5]-1)/(q*z[2]*z[3]*z[4]*z[5]-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, 3.664, seconds to generate. 3.692