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17 q-Hypergeometric and Related FunctionsProperties

Β§17.13 Integrals

In this section, for the function Ξ“q see Β§5.18(ii).

17.13.1 βˆ«βˆ’cd(βˆ’q⁒x/c;q)∞⁒(q⁒x/d;q)∞(βˆ’a⁒x/c;q)∞⁒(b⁒x/d;q)∞⁒dqx=(1βˆ’q)⁒(q;q)∞⁒(a⁒b;q)∞⁒c⁒d⁒(βˆ’c/d;q)∞⁒(βˆ’d/c;q)∞(a;q)∞⁒(b;q)∞⁒(c+d)⁒(βˆ’b⁒c/d;q)∞⁒(βˆ’a⁒d/c;q)∞,

or, when 0<q<1,

17.13.2 βˆ«βˆ’cd(βˆ’q⁒x/c;q)∞⁒(q⁒x/d;q)∞(βˆ’x⁒qΞ±/c;q)∞⁒(x⁒qΞ²/d;q)∞⁒dqx=Ξ“q⁑(Ξ±)⁒Γq⁑(Ξ²)Ξ“q⁑(Ξ±+Ξ²)⁒c⁒dc+d⁒(βˆ’c/d;q)∞⁒(βˆ’d/c;q)∞(βˆ’qβ⁒c/d;q)∞⁒(βˆ’qα⁒d/c;q)∞.

Ramanujan’s Integrals

17.13.3 ∫0∞tΞ±βˆ’1⁒(βˆ’t⁒qΞ±+Ξ²;q)∞(βˆ’t;q)∞⁒dt=Γ⁑(Ξ±)⁒Γ⁑(1βˆ’Ξ±)⁒Γq⁑(Ξ²)Ξ“q⁑(1βˆ’Ξ±)⁒Γq⁑(Ξ±+Ξ²),
17.13.4 ∫0∞tΞ±βˆ’1⁒(βˆ’c⁒t⁒qΞ±+Ξ²;q)∞(βˆ’c⁒t;q)∞⁒dqt=Ξ“q⁑(Ξ±)⁒Γq⁑(Ξ²)⁒(βˆ’c⁒qΞ±;q)∞⁒(βˆ’q1βˆ’Ξ±/c;q)βˆžΞ“q⁑(Ξ±+Ξ²)⁒(βˆ’c;q)∞⁒(βˆ’q/c;q)∞.

Askey (1980) conjectured extensions of the foregoing integrals that are closely related to Macdonald (1982). These conjectures are proved independently in Habsieger (1988) and Kadell (1988).