{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:58:17Z","timestamp":1776866297009,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"318","license":[{"start":{"date-parts":[[2019,11,5]],"date-time":"2019-11-05T00:00:00Z","timestamp":1572912000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["MTM2014-53178-P"],"award-info":[{"award-number":["MTM2014-53178-P"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider the confluent hypergeometric function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M left-parenthesis a comma b semicolon z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/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\">M(a,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z element-of double-struck upper C\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">z\\in \\mathbb {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=\"normal fraktur upper R b greater-than normal fraktur upper R a greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re b&gt;\\Re a&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and the confluent hypergeometric function\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 comma b semicolon 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:mi>b<\/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,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b element-of double-struck upper C\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">C<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b\\in \\mathbb {C}<\/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=\"normal fraktur upper R a greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re a&gt;0<\/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=\"normal fraktur upper R z greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re z&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We derive two convergent expansions of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M left-parenthesis a comma b semicolon z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/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\">M(a,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; one of them in terms of incomplete gamma functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma left-parenthesis a comma z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/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\">\\gamma (a,z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and another one in terms of rational functions of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e Superscript z\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">e^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=\"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                    . We also derive a convergent expansion of\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 comma b semicolon 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:mi>b<\/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,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in terms of incomplete gamma functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma left-parenthesis a comma z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/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\">\\gamma (a,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=\"normal upper Gamma left-parenthesis a comma z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0393\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/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\">\\Gamma (a,z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The expansions of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M left-parenthesis a comma b semicolon z right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/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\">M(a,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    hold uniformly in either\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal fraktur upper R z greater-than-or-equal-to 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re z\\ge 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    or\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal fraktur upper R z less-than-or-equal-to 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re z\\le 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; the expansion of\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 comma b semicolon 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:mi>b<\/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,b;z)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    holds uniformly in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal fraktur upper R z greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u211c\n                              \n                            <\/mml:mi>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Re z&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The accuracy of the approximations is illustrated by means of some numerical experiments.\n                  <\/p>","DOI":"10.1090\/mcom\/3389","type":"journal-article","created":{"date-parts":[[2018,7,25]],"date-time":"2018-07-25T09:28:34Z","timestamp":1532510914000},"page":"1773-1789","source":"Crossref","is-referenced-by-count":6,"title":["Convergent expansions of the confluent hypergeometric functions in terms of elementary functions"],"prefix":"10.1090","volume":"88","author":[{"given":"Blanca","family":"Bujanda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9","family":"L\u00f3pez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro","family":"Pagola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,11,5]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.amc.2014.02.099","article-title":"New series expansions for the confluent hypergeometric function \ud835\udc40(\ud835\udc4e,\ud835\udc4f,\ud835\udc67)","volume":"235","author":"L\u00f3pez, Jos\u00e9 L.","year":"2014","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1142\/S0219530517500099","article-title":"Convergent expansions of the incomplete gamma functions in terms of elementary functions","volume":"16","author":"Bujanda, Blanca","year":"2018","journal-title":"Anal. Appl. (Singap.)","ISSN":"https:\/\/id.crossref.org\/issn\/0219-5305","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10444-017-9543-y","article-title":"Convergent expansions of the Bessel functions in terms of elementary functions","volume":"44","author":"L\u00f3pez, Jos\u00e9 L.","year":"2018","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s002110100285","article-title":"Computing the confluent hypergeometric function, \ud835\udc40(\ud835\udc4e,\ud835\udc4f,\ud835\udc65)","volume":"90","author":"Muller, Keith E.","year":"2001","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"5","unstructured":"R. B. Paris, Incomplete Gamma functions, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010, pp. 173\u2013192 (Chapter 8)."},{"key":"6","isbn-type":"print","first-page":"321","article-title":"Confluent hypergeometric functions","author":"Olde Daalhuis, A. B.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521140638"},{"key":"7","isbn-type":"print","first-page":"383","article-title":"Hypergeometric function","author":"Olde Daalhuis, A. B.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521140638"},{"key":"8","isbn-type":"print","first-page":"403","article-title":"Generalized hypergeometric functions and Meijer \ud835\udc3a-function","author":"Askey, R. A.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521140638"},{"issue":"1-2","key":"9","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1016\/S0377-0427(02)00627-1","article-title":"Large parameter cases of the Gauss hypergeometric function","volume":"153","author":"Temme, Nico M.","year":"2003","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"10","series-title":"Computer Science and Scientific Computing","isbn-type":"print","volume-title":"Asymptotic approximations of integrals","author":"Wong, R.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0127625356"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-318\/S0025-5718-2018-03389-0\/mcom3389_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2019-88-318\/S0025-5718-2018-03389-0\/S0025-5718-2018-03389-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-318\/S0025-5718-2018-03389-0\/S0025-5718-2018-03389-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:09:43Z","timestamp":1776802183000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-318\/S0025-5718-2018-03389-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,5]]},"references-count":10,"journal-issue":{"issue":"318","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["S0025-5718-2018-03389-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3389","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":[[2018,11,5]]}}}