{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:50:45Z","timestamp":1776797445975,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"294","license":[{"start":{"date-parts":[[2015,12,3]],"date-time":"2015-12-03T00:00:00Z","timestamp":1449100800000},"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":"

\n Numerical evaluation of the Gauss hypergeometric function\n \n \n \n \n \n \n 2<\/mml:mn>\n <\/mml:msub>\n \n F<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ;<\/mml:mo>\n c<\/mml:mi>\n ;<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n {}_2F_1(a,b;c;z)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with complex parameters\n \n \n \n \n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n <\/mml:mrow>\n a,b,c<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and complex argument\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is notoriously difficult. Carrying out the summation that defines the function may fail, even for moderate values of\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Formulae are available to transform the effective argument in the series, potentially leading to a numerically successful summation. Unfortunately, these transformations have a singularity when\n \n \n \n \n b<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n a<\/mml:mi>\n <\/mml:mrow>\n b-a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or\n \n \n \n \n c<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n a<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n b<\/mml:mi>\n <\/mml:mrow>\n c-a-b<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is an integer, and suffer numerical instability near that. This singularity has to be removed analytically after collecting powers in\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n The contributions in this paper are fourfold. First, analytical expressions are provided that remove the singularity from B\u00fchring\u2019s\n \n \n \n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n z<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n z<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n 1\/(z-z_0)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n transformation. This is more difficult, because the singularity occurs twice, and it is necessary to collect powers of\n \n \n \n \n z<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n z_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , as well as\n \n \n \n z<\/mml:mi>\n z<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The resulting expression has a three-term recursion, like the original. Next, improved expressions are derived for the cases that have been addressed before. We study a transformation that converges outside\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n z<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 0.32<\/mml:mn>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 0.32<\/mml:mn>\n <\/mml:mrow>\n |z-0.32| > 0.32<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n z<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n {\\mathcal {R}}z>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which is tighter than the\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n z<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 0.5<\/mml:mn>\n \n |<\/mml:mo>\n <\/mml:mrow>\n ><\/mml:mo>\n 0.5<\/mml:mn>\n <\/mml:mrow>\n |z-0.5| > 0.5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is normally considered. Finally, we derive an improved algorithm for the numerical evaluation of\n \n \n \n \n \n \n 2<\/mml:mn>\n <\/mml:msub>\n \n F<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n {}_2F_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2014-02905-0","type":"journal-article","created":{"date-parts":[[2014,12,3]],"date-time":"2014-12-03T11:40:03Z","timestamp":1417606803000},"page":"1813-1833","source":"Crossref","is-referenced-by-count":8,"title":["Numerical evaluation of the Gauss hypergeometric function by power summations"],"prefix":"10.1090","volume":"84","author":[{"given":"Jurgen","family":"Doornik","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,12,3]]},"reference":[{"key":"1","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1970."},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"884","DOI":"10.1137\/0518066","article-title":"An analytic continuation of the hypergeometric series","volume":"18","author":"B\u00fchring, Wolfgang","year":"1987","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"J. A. Doornik, Conversion of high-period random numbers to floating point, ACM Transactions on Modeling and Computer Simulation 17 (2007).","DOI":"10.1145\/1189756.1189759"},{"key":"4","unstructured":"\\bysame, Object-oriented Matrix Programming using Ox, 7th ed., Timberlake Consultants Press, London, 2013."},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1006\/jcph.1997.5794","article-title":"Computing the hypergeometric function","volume":"137","author":"Forrey, Robert C.","year":"1997","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1-2","key":"6","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/j.cam.2005.01.041","article-title":"The ABC of hyper recursions","volume":"190","author":"Gil, Amparo","year":"2006","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"259","key":"7","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.1090\/S0025-5718-07-01918-7","article-title":"Numerically satisfactory solutions of hypergeometric recursions","volume":"76","author":"Gil, Amparo","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s10444-012-9283-y","article-title":"New series expansions of the Gauss hypergeometric function","volume":"39","author":"L\u00f3pez, Jos\u00e9 L.","year":"2013","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"7","key":"9","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1016\/j.cpc.2007.11.007","article-title":"Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the P\u00f6schl-Teller-Ginocchio potential wave functions","volume":"178","author":"Michel, N.","year":"2008","journal-title":"Comput. Phys. Comm.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-4655","issn-type":"print"},{"key":"10","unstructured":"A. B. Olde Daalhuis, Hypergeometric function, NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010."},{"issue":"12","key":"11","first-page":"1808","article-title":"A regularization method for computing the hypergeometric function \ud835\udc39(\ud835\udc4e,\ud835\udc4f;\ud835\udc50;\ud835\udc67) in a neighborhood of the singular points \ud835\udc67=1 and \ud835\udc67=\u221e","volume":"41","author":"Skorokhodov, S. L.","year":"2001","journal-title":"Zh. Vychisl. Mat. Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1017\/S0962492906330012","article-title":"Numerical aspects of special functions","volume":"16","author":"Temme, Nico M.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521877435","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02905-0\/S0025-5718-2014-02905-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02905-0\/S0025-5718-2014-02905-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:24:11Z","timestamp":1776795851000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02905-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,3]]},"references-count":12,"journal-issue":{"issue":"294","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["S0025-5718-2014-02905-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02905-0","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":[[2014,12,3]]}}}