{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:38:46Z","timestamp":1776724726540,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"222","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d\">\n                        <mml:semantics>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">d<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    - and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D\">\n                        <mml:semantics>\n                          <mml:mi>D<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transformations. The\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">D_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"d 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">d_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00955-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"695-714","source":"Crossref","is-referenced-by-count":1,"title":["The \ud835\udc51\u2082-transformation for infinite double series and the \ud835\udc37\u2082-transformation for infinite double integrals"],"prefix":"10.1090","volume":"67","author":[{"given":"Chen","family":"Greif","sequence":"first","affiliation":[]},{"given":"David","family":"Levin","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"841","DOI":"10.2307\/2005519","article-title":"Rational approximants defined from double power series","volume":"27","author":"Chisholm, J. S. R.","year":"1973","journal-title":"Math. 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Math. 32 (1990), no. 1-2, 47\u201357; MR1091774 (92b:65015)]","volume":"41","author":"Cuyt, A.","year":"1992","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"5","unstructured":"C. Greif, 1994 Singularity detection and bivariate generalization of the \ud835\udc51 and \ud835\udc37 transformations, M.Sc. Thesis, Tel Aviv Univ."},{"key":"6","unstructured":"D. Levin, 1971 Development of non-linear transformations of series and sequences to increase rate and domain of convergence, and their use for computing results from formal solutions to applied math. problems, M.Sc. Thesis, Tel Aviv Univ."},{"key":"7","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1080\/00207167308803075","article-title":"Development of non-linear transformations of improving convergence of sequences","volume":"3","author":"Levin, David","year":"1973","journal-title":"Internat. J. Comput. 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Sidi, 1977 Exponential function approximation to Laplace transform inversion and development of non-linear methods for accelerating the convergence of infinite integrals and series, Ph.D. Thesis, Tel Aviv Univ."},{"issue":"145","key":"14","doi-asserted-by":"publisher","first-page":"315","DOI":"10.2307\/2006044","article-title":"Convergence properties of some nonlinear sequence transformations","volume":"33","author":"Sidi, Avram","year":"1979","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/imamat\/24.3.327","article-title":"Some properties of a generalization of the Richardson extrapolation process","volume":"24","author":"Sidi, A.","year":"1979","journal-title":"J. Inst. Math. 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