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The Karlin-McGregor formula for paths connected with a clique

Obata N.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

451--466

The Karlin-McGregor formula, a well-known integral expression of the m-step transition probability for a nearest-neighbor random walk on the non-negative integers (an infinite path graph), is reformulated in terms of one-mode interacting Fock spaces. A truncated direct sum of onemode interacting Fock spaces is newly introduced and an integral expression for the m-th moment of the associated operator is derived. This integral expression gives rise to an extension of the Karlin-McGregor formula to the graph of paths connected with a clique.