{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:49:52Z","timestamp":1776836992411,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2007,2,1]],"date-time":"2007-02-01T00:00:00Z","timestamp":1170288000000},"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":"<p>\n                    Although general order multivariate Pad\u00e9 approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Pad\u00e9 approximants to the class of so-called pseudo-multivariate functions, using the Pad\u00e9 approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Pad\u00e9 approximants. Examples include the multivariate forms of the exponential and the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -exponential functions\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction x Superscript i Baseline y Superscript j Baseline Over left-parenthesis i plus j right-parenthesis factorial EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:msup>\n                                <mml:msup>\n                                  <mml:mi>y<\/mml:mi>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mrow>\n                                  <mml:mo>(<\/mml:mo>\n                                  <mml:mi>i<\/mml:mi>\n                                  <mml:mo>+<\/mml:mo>\n                                  <mml:mi>j<\/mml:mi>\n                                  <mml:mo>)<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mo>!<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E\\left ( x,y\\right ) =\\sum _{i,j=0}^\\infty \\frac {x^iy^j}{\\left ( i+j\\right ) !}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    and\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E Subscript q Baseline left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction x Superscript i Baseline y Superscript j Baseline Over left-bracket i plus j right-bracket Subscript q Baseline factorial EndFraction comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>E<\/mml:mi>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:msup>\n                                <mml:msup>\n                                  <mml:mi>y<\/mml:mi>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mo stretchy=\"false\">[<\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:msub>\n                                  <mml:mo stretchy=\"false\">]<\/mml:mo>\n                                  <mml:mi>q<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mo>!<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E_q\\left ( x,y\\right ) =\\sum _{i,j=0}^\\infty \\frac {x^iy^j}{[i+j]_q!},<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    as well as the Appell function\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F 1 left-parenthesis a comma 1 comma 1 semicolon c semicolon x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction left-parenthesis a right-parenthesis Subscript i plus j Baseline x Superscript i Baseline y Superscript j Baseline Over left-parenthesis c right-parenthesis Subscript i plus j Baseline EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>;<\/mml:mo>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mo>;<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mrow>\n                                    <mml:mo>(<\/mml:mo>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mo>)<\/mml:mo>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>i<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mi>j<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:msup>\n                                <mml:msup>\n                                  <mml:mi>y<\/mml:mi>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                              <mml:msub>\n                                <mml:mrow>\n                                  <mml:mo>(<\/mml:mo>\n                                  <mml:mi>c<\/mml:mi>\n                                  <mml:mo>)<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi>i<\/mml:mi>\n                                  <mml:mo>+<\/mml:mo>\n                                  <mml:mi>j<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:msub>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">F_1\\left ( a,1,1;c;x,y\\right ) =\\sum _{i,j=0}^\\infty \\frac {\\left ( a\\right ) _{i+j}x^iy^j}{\\left ( c\\right ) _{i+j}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    and the multivariate form of the partial theta function\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts q Superscript minus left-parenthesis i plus j right-parenthesis squared slash 2 Baseline x Superscript i Baseline y Superscript j Baseline period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:msup>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:msup>\n                                  <mml:mrow>\n                                    <mml:mo>(<\/mml:mo>\n                                    <mml:mi>i<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mi>j<\/mml:mi>\n                                    <mml:mo>)<\/mml:mo>\n                                  <\/mml:mrow>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:msup>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mi>j<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">F\\left ( x,y\\right ) =\\sum _{i,j=0}^\\infty q^{-\\left ( i+j\\right ) ^2\/2}x^iy^j.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01789-3","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"727-741","source":"Crossref","is-referenced-by-count":6,"title":["General order multivariate Pad\u00e9 approximants for pseudo-multivariate functions"],"prefix":"10.1090","volume":"75","author":[{"given":"Annie","family":"Cuyt","sequence":"first","affiliation":[]},{"given":"Jieqing","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Ping","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,2,1]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF01208907","article-title":"On the structure of a table of multivariate rational interpolants","volume":"8","author":"Allouche, H.","year":"1992","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"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":"4","key":"3","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/BF02075469","article-title":"Pad\u00e9 approximants for the \ud835\udc5e-elementary functions","volume":"4","author":"Borwein, Peter B.","year":"1988","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"4","unstructured":"P. B. Borwein, A. Cuyt and P. Zhou, Explicit construction of general multivariate Pad\u00e9 approximants to an Appell function, Adv. in Comp. Math., to appear."},{"issue":"1-2","key":"5","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":"6","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":"7","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":"1","key":"8","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/A:1018918429917","article-title":"Exploring multivariate Pad\u00e9 approximants for multiple hypergeometric series","volume":"10","author":"Cuyt, Annie","year":"1999","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"9","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","volume-title":"Basic hypergeometric series","volume":"35","author":"Gasper, George","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521350492"},{"issue":"1","key":"10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/imamat\/18.1.1","article-title":"General order Pad\u00e9-type rational approximants defined from double power series","volume":"18","author":"Levin, D.","year":"1976","journal-title":"J. Inst. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-2932","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/BF01890574","article-title":"Convergence of Pad\u00e9 approximants of partial theta functions and the Rogers-Szeg\u0151 polynomials","volume":"3","author":"Lubinsky, D. S.","year":"1987","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"12","unstructured":"H. Pad\u00e9, Recherches sur la convergence des d\u00e9"},{"key":"13","volume-title":"Generalized hypergeometric functions","author":"Slater, Lucy Joan","year":"1966"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1142\/9789812798886_0033","article-title":"Some explicit formulas for Pad\u00e9 approximants of ratios of hypergeometric functions","author":"Wimp, Jet","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/9810214375"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0377-0427(96)00095-7","article-title":"Explicit construction of multivariate Pad\u00e9 approximants","volume":"79","author":"Zhou, Ping","year":"1997","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1006\/jath.1997.3158","article-title":"Multivariate Pad\u00e9 approximants associated with functional relations","volume":"93","author":"Zhou, Ping","year":"1998","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1006\/jath.1999.3409","article-title":"Explicit construction of multivariate Pad\u00e9 approximants for a \ud835\udc5e-logarithm function","volume":"103","author":"Zhou, Ping","year":"2000","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01789-3\/S0025-5718-06-01789-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01789-3\/S0025-5718-06-01789-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:34:46Z","timestamp":1776782086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01789-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,2,1]]},"references-count":17,"journal-issue":{"issue":"254","published-print":{"date-parts":[[2006,4]]}},"alternative-id":["S0025-5718-06-01789-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01789-3","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":[[2006,2,1]]}}}