{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:02:57Z","timestamp":1760148177782,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T00:00:00Z","timestamp":1681084800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University","award":["2268"],"award-info":[{"award-number":["2268"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider birth and death processes with different sequences of transition rates and find the bound for the extreme zeros of orthogonal polynomials related to the three term recurrence relations and birth and death processes. Furthermore, we find the related chain sequences. Using these chain sequences, we find the transition probabilities for the corresponding process. As a consequence, transition probabilities related to G-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.<\/jats:p>","DOI":"10.3390\/sym15040890","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T03:49:29Z","timestamp":1681098569000},"page":"890","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4540-1601","authenticated-orcid":false,"given":"Saiful R.","family":"Mondal","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al Hasa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2270-8109","authenticated-orcid":false,"given":"Sourav","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur 831014, Jharkhand, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,10]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Characterization theorems for orthogonal polynomials","volume":"Volume 294","year":"1990","journal-title":"Orthogonal Polynomials (Columbus, OH, 1989)"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nikiforov, A.F., Uvarov, V.B., and Suslov, S.K. 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