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Keywords:
Jackson's $_2\phi _2$ transformation; reverse the order of summation; $_1\phi _1$ summation
Jackson's $_2\phi _2$ transformation; reverse the order of summation; $_1\phi _1$ summation
Summary:
We derive twenty different multivariable generalizations of Jackson's $_2\phi _2$ transformation by applying elementary proofs. Then, by reversing the order of summation, we obtain eight multivariable extensions of transformations of terminating $_2\phi _1$ series. As limiting cases of our main results, we obtain three $A_n$ $_1\phi _1$ summation formulae.
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