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On certain generalizations of q-hypergeometric functions of two variables

Thomas Ernst

Abstract


Based on a preprint by Per Karlsson 1976, the aim of the present study is to present transformation- and summation formulas for q-functions of 2n variables. We do expect such functions to possess more complicated parameter systems than the four q-Lauricella functions; on the other hand, they should not be so complicated that a practical notation becomes impossible; a reasonably high symmetry will be required. We made the decision that this study should comprise 43 functions. Loosely speaking, we may describe them as certain hypergeometric functions of 2n variables having parameters associated with 1, 2, n or 2n variables (but not any 2, nor any n);
for n = 1 they reduce to q-Appell, q-Horn, q-Humbert, and simpler q-hypergeometric functions. Relations involving Lauricella functions of 2n variables occasionally occur, and a few results appear which are particular cases of known results involving certain more general hypergeometric functions which have been studied in other contexts without particular attention to the case of 2n variables; omission of these results was found unacceptable in a reasonably systematic treatment. Our formulas will be more general than those of Karlsson-Srivastava 1999, since they are of two types, with vector index, and with absolute value index. Generalizing a paper from 2011, where the q-analougue of the 'triangle' operator was introduced; our formulas are of two types: reduction formulas with general terms and special cases, which use our 43 generic names. Once again, our formulas can be divided into four cases depending on the relation between the first and the second vector argument.

Keywords


q-hypergeometric formula, transformation- and summation formulas, 'triangle' operator, even number of variables, q-Lauricella function

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