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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
∫ *x*^{l-2m}
((-1+*x*+*s*)(1-*x*+*s*)(1+*x*-*s*)(1+*x*+*s*))^{m} *d**x*. 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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