{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:04:10Z","timestamp":1776791050143,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"268","license":[{"start":{"date-parts":[[2010,2,2]],"date-time":"2010-02-02T00:00:00Z","timestamp":1265068800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Explicit formulas for general order multivariate Pad\u00e9 approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function\n \n \n \n \n E<\/mml:mi>\n \n (<\/mml:mo>\n \n x<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munderover>\n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \u22ef\n \n <\/mml:mo>\n \n x<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \n \n (<\/mml:mo>\n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n !<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} E\\left (\\underline {x}\\right ) =\\sum _{j_{1},j_{2},\\ldots ,j_{m}=0}^{\\infty } \\frac {x_{1}^{j_{1}}x_{2}^{j_{2}}\\cdots x_{m}^{j_{m}}}{\\left ( j_{1}+j_{2}+\\cdots +j_{m}\\right ) !}, \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n the logarithm function\n \n \n \n \n L<\/mml:mi>\n (<\/mml:mo>\n \n x<\/mml:mi>\n \n _\n \n <\/mml:mo>\n <\/mml:munder>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munder>\n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \u22ef\n \n <\/mml:mo>\n \n x<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:mfrac>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} L(\\underline {x})=\\sum _{j_{1}+j_{2}+\\cdots +j_{m}\\geq 1}\\frac { x_{1}^{j_{1}}x_{2}^{j_{2}}\\cdots x_{m}^{j_{m}}}{j_{1}+j_{2}+\\cdots +j_{m}}, \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n the Lauricella function\n \n \n \n \n \n F<\/mml:mi>\n \n D<\/mml:mi>\n <\/mml:mrow>\n \n \n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ;<\/mml:mo>\n c<\/mml:mi>\n ;<\/mml:mo>\n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n j<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munderover>\n \n \n \n (<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n (<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mfrac>\n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n \n \u22ef\n \n <\/mml:mo>\n \n x<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n \n \n j<\/mml:mi>\n \n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msubsup>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} F_{D}^{\\left ( m\\right ) }\\left ( a,1,\\ldots ,1;c;x_{1},\\ldots ,x_{m}\\right ) =\\sum _{j_{1},j_{2},\\ldots ,j_{m}=0}^{\\infty }\\frac {\\left ( a\\right ) _{j_{1}+\\cdots +j_{m}}}{\\left ( c\\right ) _{j_{1}+\\cdots +j_{m}}} x_{1}^{j_{1}}\\cdots x_{m}^{j_{m}}, \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n and many more. We prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives. These properties do not hold in general for multivariate Pad\u00e9 approximants. A truncation error upperbound is also given.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02226-1","type":"journal-article","created":{"date-parts":[[2009,6,30]],"date-time":"2009-06-30T10:39:02Z","timestamp":1246358342000},"page":"2137-2155","source":"Crossref","is-referenced-by-count":2,"title":["General order multivariate Pad\u00e9 approximants for Pseudo-multivariate functions. II"],"prefix":"10.1090","volume":"78","author":[{"given":"Ping","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Annie","family":"Cuyt","sequence":"additional","affiliation":[]},{"given":"Jieqing","family":"Tan","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,2,2]]},"reference":[{"key":"1","unstructured":"P. Appell and J. Kamp\u00e9 de F\u00e9riet, Fonctions hyperg\u00e9om\u00e9triques et hypersph\u00e9riques: polyn\u00f4mes d\u2019Hermite, Paris: Gauthier-Villars, 1926."},{"key":"2","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Pad\\'{e} approximants. Part I","volume":"13","author":"Baker, George A., Jr.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0201135124"},{"issue":"1-2","key":"3","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/S0377-0427(99)00028-X","article-title":"How well can the concept of Pad\u00e9 approximant be generalized to the multivariate case?","volume":"105","author":"Cuyt, Annie","year":"1999","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/0377-0427(95)00044-5","article-title":"A direct approach to convergence of multivariate, nonhomogeneous, Pad\u00e9 approximants","volume":"69","author":"Cuyt, A.","year":"1996","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s002110050195","article-title":"Kronecker type theorems, normality and continuity of the multivariate Pad\u00e9 operator","volume":"73","author":"Cuyt, Annie","year":"1996","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"254","key":"6","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1090\/S0025-5718-06-01789-3","article-title":"General order multivariate Pad\u00e9 approximants for pseudo-multivariate functions","volume":"75","author":"Cuyt, Annie","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"G. Lauricella, Sulla funzioni ipergeometriche a pi\u00f9 variabili, Rend. Circ. Math. Palermo 7(1893), 111-158.","DOI":"10.1007\/BF03012437"},{"key":"8","isbn-type":"print","volume-title":"Constructive theory of multivariate functions","author":"Reimer, Manfred","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/341114601X"},{"key":"9","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2272-7","volume-title":"Introduction to numerical analysis","volume":"12","author":"Stoer, J.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/038797878X","edition":"2"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/s10444-004-1838-0","article-title":"On the finite sum representations of the Lauricella functions \ud835\udc39_{\ud835\udc37}","volume":"23","author":"Tan, Jieqing","year":"2005","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"11","first-page":"302","article-title":"More examples on general order multivariate Pad\u00e9 approximants for pseudo-multivariate functions","volume":"25","author":"Zhou, Ping","year":"2006","journal-title":"Electron. Trans. Numer. Anal."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02226-1\/S0025-5718-09-02226-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02226-1\/S0025-5718-09-02226-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:12:24Z","timestamp":1776787944000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-268\/S0025-5718-09-02226-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2,2]]},"references-count":11,"journal-issue":{"issue":"268","published-print":{"date-parts":[[2009,10]]}},"alternative-id":["S0025-5718-09-02226-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02226-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,2,2]]}}}