In this article, we study some new characterizations of primitive recursive functions based on restricted forms of primitive recursion, improving the pioneering work of R. M. Robinson and M. D. Gladstone. We reduce certain recursion schemes (mixed/pure iteration without parameters) and we characterize one-argument primitive recursive functions as the closure under substitution and iteration of certain optimal sets.
"Unary primitive recursive functions." J. Symbolic Logic 73 (4) 1122 - 1138, December 2008. https://doi.org/10.2178/jsl/1230396909