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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

The singular value decomposition going with many problems in medical imaging, non-destructive testing, geophysics, etc. is of central importance. The singular functions are characterized as the eigenfunctions of a compact, selfadjoint integral operator with (as in the case of limited angle tomography) a certain number of significantly nonzero eigenvalues. A stable reconstruction algorithm should aim at finding the projection of the unknown object on the linear span of the singular functions going with only these eigenvalues. If this linear span is not satisfactory for the spatial resolution that one wants to achieve, one needs to measure over a larger range of angles and compute the new singular value decomposition from scratch. Unfortunately the effective numerical determination of the eigenfunctions of these integral operators, i.e. the singular functions in question, is a very ill-posed problem. The best known remedy to this problem goes back the work of Slepian, Landau and Pollak, Bell Labs 1960-1965. They found a mathematical miracle: one can produce a differential operator that shares these eigenfunctions and has a very spread out spectrum, resulting in a numerically stable problem. The search for other situations where this (albeit exceptional) miracle holds is the motivation of this paper. We show that the master symmetries of the Korteweg-de Vries equation give a way to extend the remarkable result of David Slepian in connection with the Bessel integral kernel and the existence of a differential operator that commutes with it. The original result of Slepian has already played an important role in signal processing as well as in Random matrix theory.