A quantum calculus formulation of dynamic programming and ordered derivatives

Much recent research activity has focused on the theory and application of quantum calculus. This branch of mathematics continues to find new and useful applications and there is much promise left for investigation into this field. We present a formulation of dynamic programming grounded in the quantum calculus. Our results include the standard dynamic programming induction algorithm which can be interpreted as the Hamilton-Jacobi-Bellman equation in the quantum calculus. Furthermore, we show that approximate dynamic programming in quantum calculus is tenable by laying the groundwork for the backpropagation algorithm common in neural network training. In particular, we prove that the chain rule for ordered derivatives, fundamental to backpropagation, is valid in quantum calculus. In doing this we have connected two major fields of research.