A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions

In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the $G_{n}^{(1)}$ transformation and Slevinsky-Safouhi formula for differentiation. In the present contribution, we improve this existing algorithm for incomplete Bessel functions by developing a recurrence relation for the numerator sequence and the denominator sequence whose ratio forms the sequence of approximations. By finding this recurrence relation, we reduce the complexity from ${\cal O}(n^4)$ to ${\cal O}(n)$. We plot relative error showing that the algorithm is capable of extremely high accuracy for incomplete Bessel functions.