{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:19:25Z","timestamp":1776788365615,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"264","license":[{"start":{"date-parts":[[2009,5,14]],"date-time":"2009-05-14T00:00:00Z","timestamp":1242259200000},"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                    We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 1 Baseline upper F 1 left-parenthesis a plus epsilon 1 n semicolon c plus epsilon 2 n semicolon z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi\/>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">_1F_1(a+\\epsilon _1 n;c+\\epsilon _2 n;z)<\/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 U left-parenthesis a plus epsilon 1 n comma c plus epsilon 2 n comma z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>U<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">U(a+\\epsilon _1 n,c+\\epsilon _2 n,z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon Subscript i Baseline equals 0 comma plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03f5\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\epsilon _i=0,\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (not both equal to 0). The results are obtained by applying Perron\u2019s theorem, together with uniform asymptotic estimates derived by T.\u00a0M.\u00a0Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c\">\n                        <mml:semantics>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">c<\/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=\"z\">\n                        <mml:semantics>\n                          <mml:mi>z<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue arg z EndAbsoluteValue greater-than pi\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>arg<\/mml:mi>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|\\arg \\,z|&gt;\\pi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02122-4","type":"journal-article","created":{"date-parts":[[2008,7,25]],"date-time":"2008-07-25T12:55:42Z","timestamp":1216990542000},"page":"2277-2293","source":"Crossref","is-referenced-by-count":7,"title":["Identifying minimal and dominant solutions for Kummer recursions"],"prefix":"10.1090","volume":"77","author":[{"given":"Alfredo","family":"Dea\u00f1o","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Javier","family":"Segura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nico","family":"Temme","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,5,14]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1137\/0520052","article-title":"Uniform asymptotic expansions for Whittaker\u2019s confluent hypergeometric functions","volume":"20","author":"Dunster, T. 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J.","year":"1960"},{"key":"8","series-title":"A Wiley-Interscience Publication","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032572","volume-title":"Special functions","author":"Temme, Nico M.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0471113131"},{"issue":"1-4","key":"9","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1524\/anly.1983.3.14.221","article-title":"Uniform asymptotic expansions of Laplace integrals","volume":"3","author":"Temme, Nico M.","year":"1983","journal-title":"Analysis","ISSN":"https:\/\/id.crossref.org\/issn\/0174-4747","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02122-4\/S0025-5718-08-02122-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02122-4\/S0025-5718-08-02122-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:29:49Z","timestamp":1776785389000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02122-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,5,14]]},"references-count":9,"journal-issue":{"issue":"264","published-print":{"date-parts":[[2008,10]]}},"alternative-id":["S0025-5718-08-02122-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02122-4","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":[[2008,5,14]]}}}