On bounds for ratios of contiguous hypergeometric functions

We review recent results on analytical properties (monotonicity and bounds) for ratios of contiguous functions of hypergeometric type. The cases of parabolic cylinder functions and modified Bessel functions have been discussed with considerable detail in the literature, and we give a brief account of these results, completing some aspects in the case of parabolic cylinder functions. Different techniques for obtaining these bounds are considered. They are all based on simple qualitative descriptions of the solutions of associated ODEs (mainly Riccati equations, but not only Riccati). In spite of their simplicity, they provide the most accurate global bounds known so far. We also provide examples of application of these ideas to the more general cases of the Kummer confluent function and the Gauss hypergeometric function. The function ratios described in this paper are important functions appearing in a large number of applications, in which simple approximations are very often required.