We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N + 1 − β. We combine both lattices to obtain the bi-lattice N ∪ (N + 1 − β) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.

SUMMARYThe resonance frequencies of the 250-m-high arch dam of Mauvoisin are obtained by way of ambient vibration tests. It is observed that the resonance frequencies initially increase with rising water level and then decrease with a further rise. This is linked to the two competing features of increasing entrained mass of water (reduction of the resonance frequencies) and of dam sti!ening due to closing of the vertical construction joints (augmentation of the resonance frequencies). The ambient vibration test results are complemented by those obtained during earthquakes at an array of 12 accelerographs.

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We present an asymmetric q-Painlevé equation. We will derive this using q-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this q-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution, which gives the recurrence coefficients. We establish a connection with a-q-P V .

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.