{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T04:29:13Z","timestamp":1782966553617,"version":"3.54.5"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"318","license":[{"start":{"date-parts":[[2019,11,8]],"date-time":"2019-11-08T00:00:00Z","timestamp":1573171200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000161","name":"National Institute of Standards and Technology","doi-asserted-by":"publisher","award":["GRANT11863412\/ 70NANB15H221"],"award-info":[{"award-number":["GRANT11863412\/ 70NANB15H221"]}],"id":[{"id":"10.13039\/100000161","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000161","name":"National Institute of Standards and Technology","doi-asserted-by":"publisher","award":["GRANT11863412\/70NANB15H221"],"award-info":[{"award-number":["GRANT11863412\/70NANB15H221"]}],"id":[{"id":"10.13039\/100000161","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000161","name":"National Institute of Standards and Technology","doi-asserted-by":"publisher","award":["GRANT11863412\/ 70NANB15H221"],"award-info":[{"award-number":["GRANT11863412\/ 70NANB15H221"]}],"id":[{"id":"10.13039\/100000161","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000161","name":"National Institute of Standards and Technology","doi-asserted-by":"publisher","award":["GRANT11863412\/70NANB15H221"],"award-info":[{"award-number":["GRANT11863412\/70NANB15H221"]}],"id":[{"id":"10.13039\/100000161","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We construct asymptotic expansions for the normalised incomplete gamma function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q left-parenthesis a comma z right-parenthesis equals normal upper Gamma left-parenthesis a comma z right-parenthesis slash normal upper Gamma left-parenthesis a right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Q<\/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:mo>=<\/mml:mo>\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 class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\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 stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q(a,z)=\\Gamma (a,z)\/\\Gamma (a)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that are valid in the transition regions, including the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"z almost-equals a\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2248\n                              \n                            <\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">z\\approx a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a right-arrow negative normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a\\to -\\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the unique negative zero of the regularised incomplete gamma function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma Superscript asterisk Baseline left-parenthesis a comma x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma ^*(a,x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3391","type":"journal-article","created":{"date-parts":[[2018,7,19]],"date-time":"2018-07-19T08:27:09Z","timestamp":1531988829000},"page":"1805-1827","source":"Crossref","is-referenced-by-count":13,"title":["Asymptotic expansions for the incomplete gamma function in the transition regions"],"prefix":"10.1090","volume":"88","author":[{"given":"Gerg\u0151","family":"Nemes","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adri","family":"Olde Daalhuis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,11,8]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s11139-010-9237-2","article-title":"The asymptotic expansion for \ud835\udc5b! and the Lagrange inversion formula","volume":"24","author":"Brassesco, Stella","year":"2011","journal-title":"Ramanujan J.","ISSN":"https:\/\/id.crossref.org\/issn\/1382-4090","issn-type":"print"},{"key":"2","volume-title":"Asymptotic expansions: their derivation and interpretation","author":"Dingle, R. 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