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17 q-Hypergeometric and Related FunctionsProperties

Β§17.14 Constant Term Identities

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

17.14.1 (q;q)a1+a2+β‹―+an(q;q)a1⁒(q;q)a2⁒⋯⁒(q;q)an=Β coeff. of ⁒x10⁒x20⁒⋯⁒xn0⁒ in ⁒∏1≀j<k≀n(xjxk;q)aj⁒(q⁒xkxj;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 βˆ‘n=0∞qn⁒(n+1)(q2;q2)n⁒(βˆ’q;q2)n+1=Β coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(zβˆ’1⁒q2;q2)∞⁒(βˆ’q;q2)∞⁒(zβˆ’1⁒q;q2)∞=1(βˆ’q;q2)∞⁒ coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(zβˆ’1⁒q;q)∞=H⁑(q)(βˆ’q;q2)∞,
17.14.3 βˆ‘n=0∞qn⁒(n+1)(q2;q2)n⁒(βˆ’q;q2)n+1=Β coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(zβˆ’1;q2)∞⁒(βˆ’q;q2)∞⁒(zβˆ’1⁒q;q2)∞=1(βˆ’q;q2)∞⁒ coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(zβˆ’1;q)∞=G⁑(q)(βˆ’q;q2)∞,
17.14.4 βˆ‘n=0∞qn2(q2;q2)n⁒(q;q2)n=Β coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(βˆ’zβˆ’1;q2)∞⁒(q;q2)∞⁒(zβˆ’1;q2)∞=1(q;q2)∞⁒ coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(zβˆ’2;q4)∞=G⁑(q4)(q;q2)∞,
17.14.5 βˆ‘n=0∞qn2+2⁒n(q2;q2)n⁒(q;q2)n+1=Β coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(βˆ’q2⁒zβˆ’1;q2)∞⁒(q;q2)∞⁒(zβˆ’1⁒q2;q2)∞=1(q;q2)∞⁒ coeff. of ⁒z0⁒ in ⁒(βˆ’z⁒q;q2)∞⁒(βˆ’zβˆ’1⁒q;q2)∞⁒(q2;q2)∞(q4⁒zβˆ’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).