{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:50:47Z","timestamp":1740124247474,"version":"3.37.3"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007088","name":"Jagiellonian University in Krakow","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007088","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2021,9]]},"abstract":"Abstract<\/jats:title>LetK<\/jats:italic>be a field and put$${\\mathcal {A}}:=\\{(i,j,k,m)\\in \\mathbb {N}^{4}:\\;i\\le j\\;\\text{ and }\\;m\\le k\\}$$<\/jats:tex-math>A<\/mml:mi>:<\/mml:mo>=<\/mml:mo>{<\/mml:mo>(<\/mml:mo>i<\/mml:mi>,<\/mml:mo>j<\/mml:mi>,<\/mml:mo>k<\/mml:mi>,<\/mml:mo>m<\/mml:mi>)<\/mml:mo><\/mml:mrow>\u2208<\/mml:mo>N<\/mml:mi><\/mml:mrow>4<\/mml:mn><\/mml:msup>:<\/mml:mo>i<\/mml:mi>\u2264<\/mml:mo>j<\/mml:mi>and<\/mml:mtext>m<\/mml:mi>\u2264<\/mml:mo>k<\/mml:mi>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For any given$$A\\in {\\mathcal {A}}$$<\/jats:tex-math>A<\/mml:mi>\u2208<\/mml:mo>A<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>we consider the sequence of polynomials$$(r_{A,n}(x))_{n\\in \\mathbb {N}}$$<\/jats:tex-math>(<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow>n<\/mml:mi>\u2208<\/mml:mo>N<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>defined by the recurrence$$\\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\\;n\\ge 2, \\end{aligned}$$<\/jats:tex-math>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>=<\/mml:mo>f<\/mml:mi>n<\/mml:mi><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>-<\/mml:mo>v<\/mml:mi>n<\/mml:mi><\/mml:msub>x<\/mml:mi>m<\/mml:mi><\/mml:msup>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi>-<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>,<\/mml:mo>n<\/mml:mi>\u2265<\/mml:mo>2<\/mml:mn>,<\/mml:mo><\/mml:mrow><\/mml:mtd><\/mml:mtr><\/mml:mtable><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:disp-formula>where the initial polynomials$$r_{A,0}, r_{A,1}\\in K[x]$$<\/jats:tex-math>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>0<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msub>\u2208<\/mml:mo>K<\/mml:mi>[<\/mml:mo>x<\/mml:mi>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>are of degreei<\/jats:italic>,\u00a0j<\/jats:italic>respectively and$$f_{n}\\in K[x], n\\ge 2$$<\/jats:tex-math>f<\/mml:mi>n<\/mml:mi><\/mml:msub>\u2208<\/mml:mo>K<\/mml:mi>[<\/mml:mo>x<\/mml:mi>]<\/mml:mo><\/mml:mrow>,<\/mml:mo>n<\/mml:mi>\u2265<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, is of degreek<\/jats:italic>with variable coefficients. The aim of the paper is to prove the formula for the resultant$${\\text {Res}}(r_{A,n}(x),r_{A,n-1}(x))$$<\/jats:tex-math>Res<\/mml:mtext>(<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>,<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msub>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Our result is an extension of the classical Schur formula which is obtained for$$A=(0,1,1,0)$$<\/jats:tex-math>A<\/mml:mi>=<\/mml:mo>(<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>1<\/mml:mn>,<\/mml:mo>0<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. As an application we get the formula for the resultant$${\\text {Res}}(r_{A,n},r_{A,n-2})$$<\/jats:tex-math>Res<\/mml:mtext>(<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:msub>,<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi>-<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where the sequence$$(r_{A,n})_{n\\in \\mathbb {N}}$$<\/jats:tex-math>(<\/mml:mo>r<\/mml:mi>A<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:mrow>n<\/mml:mi>\u2208<\/mml:mo>N<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric.<\/jats:p>","DOI":"10.1007\/s10998-020-00369-4","type":"journal-article","created":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T04:26:00Z","timestamp":1606105560000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On a generalization of Schur theorem concerning resultants"],"prefix":"10.1007","volume":"83","author":[{"given":"Maciej","family":"Ulas","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,11,23]]},"reference":[{"key":"369_CR1","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1090\/S0002-9939-1970-0251010-X","volume":"24","author":"TM Apostol","year":"1970","unstructured":"T.M. 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