Symmetric Sheffer sequences and their applications to lattice path counting |
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| Institution: | 1. Catalan Health Institute, Catsalut, Generalitat of Catalonia, Spain;2. Girona Research Support Unit, Jordi Gol Primary Care Research Institute, Spain;3. Nursing Department, Research Group on Health and Health Care, University of Girona, Spain;4. Nursing Department, Research Group on the Development of the Nursing Profession, University of Girona, Spain |
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| Abstract: | A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks. |
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