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In the relativistic and the nonrelativistic theoretical treatment of moderateand high-power laser-matter interaction, the generalized Bessel function occursnaturally when a Schr\"odinger-Volkov and Dirac-Volkov solution is expandedinto plane waves. For the evaluation of cross sections of quantumelectrodynamic processes in a linearly polarized laser field, it is oftennecessary to evaluate large arrays of generalized Bessel functions, ofarbitrary index but with fixed arguments. We show that the generalized Besselfunction can be evaluated, in a numerically stable way, by utilizing arecurrence relation and a normalization condition only, without having tocompute any initial value. We demonstrate the utility of the method byillustrating the quantum-classical correspondence of the Dirac-Volkov solutionsvia numerical calculations.

Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions