Abstract

A multiple generalization of the Euler transformation formula for basic hypergeometric series 2 φ 1 is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff–Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U( n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised 10 φ 9 and 8 φ 7 series, balanced 4 φ 3 series and 3 φ 2 series are also given.

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