A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914536531951616 |
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| author | Pandey, Puja Vellaisamy, Palaniappan |
| author_facet | Pandey, Puja Vellaisamy, Palaniappan |
| contents | The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A fundamental quantity of interest is the probability that a simple symmetric random walk returns to the origin after 2n steps. In this paper, we develop a unified probabilistic approach that connects the return probabilities in arbitrary dimensions with moment representations. Using this framework, we provide probabilistic proofs of several combinatorial identities involving beta and gamma functions, and derive new combinatorial identities in general dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05420 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks Pandey, Puja Vellaisamy, Palaniappan Probability 60G50, 60E05, 05A19 The study of random walks has increasingly been popular across diverse disciplines such as statistics, mathematics, quantum physics, where they are used to model paths consisting of successive random steps in a mathematical space. A fundamental quantity of interest is the probability that a simple symmetric random walk returns to the origin after 2n steps. In this paper, we develop a unified probabilistic approach that connects the return probabilities in arbitrary dimensions with moment representations. Using this framework, we provide probabilistic proofs of several combinatorial identities involving beta and gamma functions, and derive new combinatorial identities in general dimensions. |
| title | A Unified Approach to Beta Moments, Combinatorial Identities, and Random Walks |
| topic | Probability 60G50, 60E05, 05A19 |
| url | https://arxiv.org/abs/2605.05420 |