{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:56:58Z","timestamp":1753883818518,"version":"3.41.2"},"reference-count":10,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,8,9]],"date-time":"2012-08-09T00:00:00Z","timestamp":1344470400000},"content-version":"vor","delay-in-days":221,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Let  and <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) be the ultraspherical polynomials with respect to <jats:italic>w<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>). Then, we denote the Stieltjes polynomials with respect to <jats:italic>w<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) by <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) satisfying , 0 \u2264 <jats:italic>m<\/jats:italic> &lt; <jats:italic>n<\/jats:italic> + 1, , <jats:italic>m<\/jats:italic> = <jats:italic>n<\/jats:italic> + 1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) and the product <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>)<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>). Especially, we estimate the even\u2010order derivative values of <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) and <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>)<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) at the zeros of <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) and the product <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>)<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>), respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) and <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>)<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) at the zeros of <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>) and <jats:italic>E<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic>+1<\/jats:sub>(<jats:italic>x<\/jats:italic>)<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03bb<\/jats:italic>,<jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>) on a closed subset of (\u22121, 1), respectively. These estimates will play important roles in investigating convergence and divergence of the higher\u2010order Hermite\u2010Fej\u00e9r interpolation polynomials.<\/jats:p>","DOI":"10.1155\/2012\/482935","type":"journal-article","created":{"date-parts":[[2012,8,9]],"date-time":"2012-08-09T21:03:06Z","timestamp":1344546186000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Asymptotic Properties of Derivatives of the Stieltjes Polynomials"],"prefix":"10.1155","volume":"2012","author":[{"given":"Hee Sun","family":"Jung","sequence":"first","affiliation":[]},{"given":"Ryozi","family":"Sakai","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,8,9]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01448041"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1006\/jath.1995.1079"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00808-9"},{"key":"e_1_2_6_4_2","first-page":"187","volume-title":"Approximation Theory and Function Series","author":"Ehrich S.","year":"1996"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2008588"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2004.02.004"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01795-3"},{"key":"e_1_2_6_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(92)90083-Z"},{"key":"e_1_2_6_9_2","series-title":"Colloquium Publications (Amer Mathematical Soc)","volume-title":"Orthogonal Polynomials","author":"Szeg\u00f6 G.","year":"1975"},{"key":"e_1_2_6_10_2","doi-asserted-by":"publisher","DOI":"10.1137\/1024039"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/482935.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/482935.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/482935","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T07:16:06Z","timestamp":1718090166000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/482935"}},"subtitle":[],"editor":[{"given":"Jin L.","family":"Kuang","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/482935"],"URL":"https:\/\/doi.org\/10.1155\/2012\/482935","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-03-16","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-05-24","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-08-09","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"482935"}}