The enumeration of restricted random walks by Sheffer polynomials with applications to statistics |
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| Affiliation: | 1. Department of Materials Science and Engineering, National Taiwan University of Science and Technology, Taipei, 106, Taiwan;2. Department of Radiology, Tri-Service General Hospital, National Defense Medical Center, 325, Sec. 2, Cheng-Gong Rd, Nei-Hu, Taipei, 114, Taiwan;1. College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China;2. State Key Laboratory of Mining Disaster Prevention and Control Co-found by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, Shandong 266590, China;3. College of Chemical and Environmental Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China;4. College of Resources and Environmental Engineering, Binzhou University, Binzhou, Shandong 256603, China;1. Department of Mathematics and Statistics, Smith College, United States of America;2. Department of Physics and Astronomy, University of Rochester, United States of America;3. Department of Mathematics, Dartmouth College, United States of America;4. Department of Mathematics, MIT, United States of America;5. Department of Mathematics, University of Miami, United States of America;1. Department of Chemical Engineering, Faculty of Industrial Technology, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia;2. Center for Catalysis and Reaction Engineering, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia;3. Division of Inorganic and Physical Chemistry, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia;4. Department of Chemistry, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia;5. Research Center for Nanosciences and Nanotechnology, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung 40132, Indonesia;1. Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel;2. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, Athens 15784, Greece;3. Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA |
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| Abstract: | ![]()
Sheffer polynomials are solutions of certain systems of operator equations. Difference equations, which frequently occur in path enumeration, belong in that class. To find representations of the solutions, the restriction on the paths has to be in the form of boundaries. Such problems have applications in two-sample tests. We also consider paths with more than two step vectors. The gambler's ruin problem illustrates the method. If paths with a given area underneath are counted, q-binomial coefficients come into play. Eulerian Sheffer sequence solve some of such problems. |
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