In 1954, M.Kac gave a very simple proof of the Strong Szego Limit Theorem (SSLT) for the unit circle. In the same article the expected value of the maximum of a random walk on the real line is computed. The proofs of these two results in analysis rely on a certain purely combinatorial identity due to G.A.Hunt and F.J.Dyson. In 1997, V.Guillemin and K.Okikiolu proved a generalization of SSLT for a sphere of any dimension. They suggested an elegant abstract scheme which reduces the proof to a use of the Hunt-Dyson formula in the spirit of the proof by M.Kac. It turns out that the latter scheme can be used to compute further asymptotic terms. One thing needed on the way is a generalization of the Hunt-Dyson combinatorial identity to an arbitrary natural power. We have found such a generalization and calculated the third term in the Szego asymptotics. In 1956, F.Spitzer fully described the maximum of a random walk with the help of a remarkable combinatorial theorem of H.F.Bohnenblust. It turns out that the generalized Hunt-Dyson formula is an equivalent version of Bohnenblust-Spitzer theorem. Other related results are due to G.Baxter anf J.Wendel. We mention here that in 1969 G.-C.Rota has introduced a notion of a Baxter algebra and proved that the standard Baxter algebra is free. This fact is another form of Bohnenblust-Spitzer theorem. G.P.Thomas has used the Baxter operators in a certain generalization of the Schensted-Robinson construction.