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17 q-Hypergeometric and Related FunctionsProperties

§17.13 Integrals

In this section, for the function Γq see §5.18(ii).

17.13.1 -cd(-qx/c;q)(qx/d;q)(-ax/c;q)(bx/d;q)qx=(1-q)(q;q)(ab;q)cd(-c/d;q)(-d/c;q)(a;q)(b;q)(c+d)(-bc/d;q)(-ad/c;q),

or, when 0<q<1,

17.13.2 -cd(-qx/c;q)(qx/d;q)(-xqα/c;q)(xqβ/d;q)qx=Γq(α)Γq(β)Γq(α+β)cdc+d(-c/d;q)(-d/c;q)(-qβc/d;q)(-qαd/c;q).

Ramanujan’s Integrals

17.13.3 0tα-1(-tqα+β;q)(-t;q)t=Γ(α)Γ(1-α)Γq(β)Γq(1-α)Γq(α+β),
17.13.4 0tα-1(-ctqα+β;q)(-ct;q)qt=Γq(α)Γq(β)(-cqα;q)(-q1-α/c;q)Γq(α+β)(-c;q)(-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).