THE KARLIN–MCGREGOR FORMULA FOR PATHS CONNECTED WITH
A CLIQUE
Abstract: The Karlin–McGregor formula, a well-known integral expression of the -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 one-mode interacting Fock spaces is newly introduced and an integral
expression for the -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.
2000 AMS Mathematics Subject Classification: Primary: 42C05, 47B36; Secondary:
46L53, 60J10, 81S25.
Keywords and phrases: Jacobi matrix, Karlin–McGregor formula, Kesten
distribution, one-mode interacting Fock space, orthogonal polynomials, tridiagonal
matrix.