TY - CONF

T1 - Disturbing the Dyson Conjecture (in a Generally GOOD Way)

AU - Sills, Andrew V.

N1 - Any physical experience provides only a small number of results, whereas a large number of parameters may vary. What is the "value" of the information obtained this way, if, for instance, we have 300 measures, where the whole experiment may involve 50 parameters, thus leading to possible states, if each parameter can take 10 values?

PY - 2005/10/27

Y1 - 2005/10/27

N2 - Let F n := F n ( x 1 , x 2 , . . . , x n ; a 1 , a 2 , . . . , a n ) := ∏ 1≤i≠j≤n (1-x i / x j ) aj . In 1962, Freeman Dyson conjectured that the constant term in the expansion of F n is the multinomial coefficient ( a 1 + a 2 + . . . + a n )!/( a 1 ! a 2 ! . . . a n !). In 1975, George Andrews extended Dyson's conjecture to a q -analog. A particularly elegant proof of Dyson's conjecture was given by I. J. Good in 1970. Good's proof does not extend to the q -analog, however, and the q -Dyson conjecture was not settled until 1985 when Zeilberger and Bressoud proved it combinatorially. Last March in the Experimental Mathematics Seminar, I demonstrated how a Maple package that I developed with Professor Zeilberger could be used to automatically conjecture and prove closed form expressions for coefficients in the expansion of Fn besides the constant term, for fixed n . The automated proofs are based on a generalization of Good's proof. In this lecture, I will discuss more recent work, where the "disturbed Dyson conjectures" and their proofs are extended to symbolic n , and corresponding q -analogs are conjectured.

AB - Let F n := F n ( x 1 , x 2 , . . . , x n ; a 1 , a 2 , . . . , a n ) := ∏ 1≤i≠j≤n (1-x i / x j ) aj . In 1962, Freeman Dyson conjectured that the constant term in the expansion of F n is the multinomial coefficient ( a 1 + a 2 + . . . + a n )!/( a 1 ! a 2 ! . . . a n !). In 1975, George Andrews extended Dyson's conjecture to a q -analog. A particularly elegant proof of Dyson's conjecture was given by I. J. Good in 1970. Good's proof does not extend to the q -analog, however, and the q -Dyson conjecture was not settled until 1985 when Zeilberger and Bressoud proved it combinatorially. Last March in the Experimental Mathematics Seminar, I demonstrated how a Maple package that I developed with Professor Zeilberger could be used to automatically conjecture and prove closed form expressions for coefficients in the expansion of Fn besides the constant term, for fixed n . The automated proofs are based on a generalization of Good's proof. In this lecture, I will discuss more recent work, where the "disturbed Dyson conjectures" and their proofs are extended to symbolic n , and corresponding q -analogs are conjectured.

KW - Dyson Conjecture

UR - http://sites.math.rutgers.edu/~bte14/expmath/archive05.html

M3 - Presentation

T2 - Rutgers Experimental Mathematics Seminar

Y2 - 27 October 2005

ER -