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$q$-extensions of Barnes’, Cauchy’s, and Euler’s beta integrals. (English) Zbl 0748.33012
Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 294-314 (1989).

[For the entire collection see Zbl 0721.00014.]

Various $q$-extensions of the beta-integral and the integrals of Barnes and Cauchy have been deduced in recent years. See, for example, R. Askey and R. Roy [Rocky Mt. J. Math. 16, 365-372 (1986; Zbl 0599.33002)].

Cauchy’s residue theorem and certain summation formulas for $q$- hypergeometric series are used in this study to provide further extensions of the same integrals. Furthermore, various $q$-contour integrals are evaluated and a transformation formula of general character involving $q$-hypergeometric series of one variable is obtained.