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
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
((-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.
The following versions are available: