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Let D, $$D'$$ be arbitrary domains in $${\mathbb C}^n$$ and $${\mathbb C}^N$$ respectively, $$1

Let D , D ′ be arbitrary domains in C n and C N respectively, 1 n ⩽ N , both possibly unbounded and let M ⊂ ∂ D , M ′ ⊂ ∂ D ′ be open pieces of the boundaries. Suppose that ∂ D is smooth real-analytic and minimal in an open neighborhood of M ¯ and ∂ D ′ is smooth real-algebraic and minimal in an open neighborhood of M ¯ ′ . Let f : D → D ′ be a holomorphic mapping. Assume that the cluster set cl f ( M ) does not intersect D ′ . It is proved that if the cluster set cl f ( p ) of a point p ∈ M contains some point q ∈ M ′ and the graph of f extends as an analytic set to a neighborhood of ( p , q ) ∈ C n × C N , then f extends as a holomorphic map near p .

In this paper, we give a reduction formula for a specific q-integral. Our formula is expressed as a three term recurrence relations for basic hypergeometric ${_{3}\phi_{2}}$ series. This is a q-analog of work by Watson and by Bailey of 1953.