{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:04:09Z","timestamp":1776834249514,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"334","license":[{"start":{"date-parts":[[2022,11,5]],"date-time":"2022-11-05T00:00:00Z","timestamp":1667606400000},"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                    In this paper we present lower and upper bounds for Kummer\u2019s function ratios of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction upper M left-parenthesis a comma b comma z right-parenthesis prime Over upper M left-parenthesis a comma b comma z right-parenthesis EndFraction\">\n                        <mml:semantics>\n                          <mml:mfrac>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\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:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\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:mfrac>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {{M(a, b, z)}\u2019}{M(a, b, z)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than a greater-than b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0&gt;a&gt;b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The derived bounds are asymptotically precise, theoretically well-defined, numerically accurate, and easy to compute. Moreover, we show how the bounds can be used as starting values for monotonically convergent sequences to approximate the ratio with even higher precision while avoiding the anomalous convergence discussed by Gautschi [Math. Comp. 31 (1977), pp. 994\u2013999]. This allows to apply the results in multiple areas, as for example the estimation of Watson distributions in statistical modelling. Furthermore, we extend the convergence results provided by Gautschi and the list of known bounds for the inverse of Kummer\u2019s function ratio given by Sra and Karp [J. Multivariate Anal. 114 (2013), pp. 256\u2013269]. In addition, the derived starting bounds are compared and connected to other results from the literature.\n                  <\/p>","DOI":"10.1090\/mcom\/3690","type":"journal-article","created":{"date-parts":[[2021,9,15]],"date-time":"2021-09-15T09:57:14Z","timestamp":1631699834000},"page":"887-907","source":"Crossref","is-referenced-by-count":6,"title":["On bounds for Kummer\u2019s function ratio"],"prefix":"10.1090","volume":"91","author":[{"given":"Lukas","family":"Sablica","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kurt","family":"Hornik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2021,11,5]]},"reference":[{"key":"1","unstructured":"[AM16] R\u00e9my Abergel and Lionel Moisan. 2016. 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In Marina Meila and Xiaotong Shen, eds, AISTATS 2007: Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, volume 2 of JMLR Workshop and Conference Proceedings, pp. 35\u201342, San Juan, Puerto Rico, \\url{http:\/\/jmlr.org\/proceedings\/papers\/v2\/bijral07a\/bijral07a.pdf}."},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1145\/1145768.1145782","article-title":"Application of high-precision computing for pricing arithmetic Asian options","author":"Boyle, Phelim","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/1595932763"},{"issue":"318","key":"5","doi-asserted-by":"publisher","first-page":"1773","DOI":"10.1090\/mcom\/3389","article-title":"Convergent expansions of the confluent hypergeometric functions in terms of elementary functions","volume":"88","author":"Bujanda, Blanca","year":"2019","journal-title":"Math. 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