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17 q-Hypergeometric and Related FunctionsProperties

§17.14 Constant Term Identities

Zeilberger–Bressoud Theorem (Andrews’ q-Dyson Conjecture)

17.14.1 (q;q)a1+a2++an(q;q)a1(q;q)a2(q;q)an= coeff. of x10x20xn0 in 1j<kn(xjxk;q)aj(qxkxj;q)ak.

Rogers–Ramanujan Constant Term Identities

In the following, G(q) and H(q) denote the left-hand sides of (17.2.49) and (17.2.50), respectively.

17.14.2 n=0qn(n+1)(q2;q2)n(-q;q2)n+1= coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(z-1q2;q2)(-q;q2)(z-1q;q2)=1(-q;q2) coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(z-1q;q)=H(q)(-q;q2),
17.14.3 n=0qn(n+1)(q2;q2)n(-q;q2)n+1= coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(z-1;q2)(-q;q2)(z-1q;q2)=1(-q;q2) coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(z-1;q)=G(q)(-q;q2),
17.14.4 n=0qn2(q2;q2)n(q;q2)n= coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(-z-1;q2)(q;q2)(z-1;q2)=1(q;q2) coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(z-2;q4)=G(q4)(q;q2),
17.14.5 n=0qn2+2n(q2;q2)n(q;q2)n+1= coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(-q2z-1;q2)(q;q2)(z-1q2;q2)=1(q;q2) coeff. of z0 in (-zq;q2)(-z-1q;q2)(q2;q2)(q4z-2;q4)=H(q4)(q;q2).

Macdonald (1982) includes extensive conjectures on generalizations of (17.14.1) to root systems. These conjectures were proved in Cherednik (1995), Habsieger (1986), and Kadell (1994); see also Macdonald (1998). For additional results of the type (17.14.2)–(17.14.5) see Andrews (1986, Chapter 4).