Abstract
Important new transformations for the generalized hypergeometric functions with integral parameter differences have been discovered some years ago by Miller and Paris and studied in detail in a series of papers by a number of authors. These transformations fail if the free bottom parameter is greater than a free top parameter by a small positive integer. In this paper we fill this gap in the theory of Miller–Paris transformations by computing the limit cases of these transformations in such previously prohibited situations. This leads to a number of new transformation and summation formulas including extensions of the Karlsson–Minton theorem.
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Acknowledgements
We express our gratitude to Professor Richard Paris for sharing his insights on the topic of this work. We are also indebted to anonymous referees for a number of useful remarks that helped to improve the paper.
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Karp, D.B., Prilepkina, E.G. Degenerate Miller–Paris Transformations. Results Math 74, 94 (2019). https://doi.org/10.1007/s00025-019-1017-8
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DOI: https://doi.org/10.1007/s00025-019-1017-8
Keywords
- Generalized hypergeometric function
- Miller–Paris transformation
- summation formula
- Karlsson–Minton formula