Degenerate Miller-Paris transformations

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 Karlsson-Minton theorem.