Literature on Kummer and related three-term recurrence relations is sparse. Segura and Temme outline the contours of the field, fill in some unoccupied spaces, and extend the field by introducing two theorems regarding satisfactory solutions to families of Kummer relations.

The authors begin with a succinct discussion of Kummer recurrence relations and their interconnections with Hermite and Laguerre polynomials, and Bessel and Coulomb functions. They continue by noting that Perron’s theorem and Whittaker functions are central to their main result--the finding of minimal and dominant solutions in the complex plane. The introduction shows that these functions and theorems form a lattice, which the authors augment. Then, the authors present their main contribution: two theorems regarding satisfactory solutions to Kummer relations. One theorem treats cases where the second input parameter to a family of Kummer functions is equal to zero, and the other theorem applies when it is not. An appendix provides expansions of Kummer relations, when the second parameter approaches infinity.

This crisply written paper advances the area under discussion with focused precision. The paper would be more useful to nonspecialists if it suggested how the relations discussed could be used to solve problems.