A canonical two-scale Sonine fractional calculus induced by the Tricomi function

We introduce a Tricomi-type generalized fractional calculus in the Sonine kernel framework.

The key result is that the Tricomi branch is a Stieltjes function in the admissible parameter range, so its reciprocal is a complete Bernstein function.

This fact induces a Sonine fractional calculus together with the canonical Tricomi integral and the associated Riemann--Liouville-type and Caputo-type derivatives.

We also prove that, within the Kummer class, the Tricomi branch is the unique Stieltjes representative, once the natural asymptotic normalization is fixed, and we derive the corresponding Lévy--Khintchine representation, Volterra formulation, and scalar Cauchy problem.

A distinctive feature of the resulting operators is the emergence of two independent asymptotic orders.