{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:50:32Z","timestamp":1760151032084,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electronics"],"abstract":"<jats:p>In this contribution, we consider the sequence {Hn(x;q)}n\u22650 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of {Hn(x;q)}n\u22650 in a watermarking problem. Several differences are found in this watermarking application for the non-standard cases (when j&gt;0) with respect to the standard classical Krawtchouk case \u03bb=\u03bc=0.<\/jats:p>","DOI":"10.3390\/electronics11030500","type":"journal-article","created":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T01:03:40Z","timestamp":1644368620000},"page":"500","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Watermarking Applications of Krawtchouk\u2013Sobolev Type Orthogonal Moments"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6802-3303","authenticated-orcid":false,"given":"Edmundo J.","family":"Huertas","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1ticas, Facultad de Ciencias, Universidad de Alcal\u00e1, 28805 Alcal\u00e1 de Henares, Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4012-6471","authenticated-orcid":false,"given":"Alberto","family":"Lastra","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1ticas, Facultad de Ciencias, Universidad de Alcal\u00e1, 28805 Alcal\u00e1 de Henares, Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3488-3094","authenticated-orcid":false,"given":"Anier","family":"Soria-Lorente","sequence":"additional","affiliation":[{"name":"Departamento de Tecnolog\u00eda, Universidad de Granma, Km. 17,5 de la Carretera de Bayamo-Manzanillo, Bayamo 85100, Cuba"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"key":"ref_1","unstructured":"Costas-Santos, R.S., and Soria-Lorente, A. 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