Séminaire Lotharingien de Combinatoire, B87c (2023), 16 pp.

On the Problem of Random Flights in Odd Dimensions

Alexander Kovačec and Pedro Barata de Tovar Sá

Abstract. In the first part of this paper we give a procedure to compute the exact probability for a particle starting from the origin of an odd-dimensional Euclidean space to be encountered within a distance r from the start after n random jumps of unit length. In the second part we use a combinatorial identity to deduce for integers m≥0 and a certain large family of integers l≥0, detailed information concerning the primitives ∫ xl-2m ((-1+x+s)(1-x+s)(1+x-s)(1+x+s))m dx. This will imply that the density function associated with this random flight problem is piecewise polynomial. The approach is significantly different from the one chosen by García-Pelayo [J. Math. Phys. 53 (2012), 103504] who used advanced analytical tools.

Received: September 12, 2022. Revised: April 4, 2023. Accepted: May 19, 2023.

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