{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:42:30Z","timestamp":1776793350066,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,2,11]],"date-time":"2012-02-11T00:00:00Z","timestamp":1328918400000},"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                    The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 3 Baseline upper F 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mspace width=\"thickmathspace\"\/>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\;_3F_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials:\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 3 Baseline upper F 2 left-parenthesis negative n comma tau n plus a comma b semicolon tau n plus c comma negative n plus d semicolon z right-parenthesis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mspace width=\"thickmathspace\"\/>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/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>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>d<\/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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\;_3F_2(-n, \\tau n+a, b;\\tau n+c, -n+d;z),<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a nonnegative integer. Some families of the hypergeometric\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 3 Baseline upper F 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi\/>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">_3F_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    functions, which are connected (by means of a hypergeometric reduction formula) with the Gauss hypergeometric polynomials of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 2 Baseline upper F 1 left-parenthesis negative n comma k n plus l plus 1 semicolon k n plus l plus 2 semicolon z right-parenthesis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mspace width=\"thickmathspace\"\/>\n                              <mml:mn>2<\/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:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\;_2F_1(-n,kn+l+1;kn+l+2;z),<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    are also investigated. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by\n                    <italic>Mathematica<\/italic>\n                    (Version 4.0).\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02409-9","type":"journal-article","created":{"date-parts":[[2011,2,11]],"date-time":"2011-02-11T08:40:57Z","timestamp":1297413657000},"page":"1769-1784","source":"Crossref","is-referenced-by-count":9,"title":["Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials"],"prefix":"10.1090","volume":"80","author":[{"given":"H.","family":"Srivastava","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian-Rong","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi-Gang","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,2,11]]},"reference":[{"key":"1","series-title":"Cambridge Tracts in Mathematics and Mathematical Physics, No. 32","volume-title":"Generalized hypergeometric series","author":"Bailey, W. N.","year":"1964"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/BF03320990","article-title":"Zeros of hypergeometric functions","volume":"1","author":"Boggs, Kathryn","year":"2001","journal-title":"Comput. Methods Funct. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1617-9447","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"D. Dominici, K. Driver and K. Jordaan, Polynomial solutions of differential-difference equations, J. Approx. Theory (2009) [doi:10.1016\/j.jat.2009.05.010].","DOI":"10.1016\/j.jat.2009.05.010"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/S0019-3577(00)88572-9","article-title":"Zeros of the hypergeometric polynomials \ud835\udc39(-\ud835\udc5b,\ud835\udc4f;2\ud835\udc4f;\ud835\udc67)","volume":"11","author":"Driver, Kathy","year":"2000","journal-title":"Indag. Math. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0019-3577","issn-type":"print"},{"issue":"3-4","key":"5","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1023\/A:1020126822435","article-title":"Zeros of \u2083\ud835\udc39\u2082(\\smallmatrix-\ud835\udc5b,\ud835\udc4f,\ud835\udc50\ud835\udc51,\ud835\udc52\\endsmallmatrix;\ud835\udc67) polynomials","volume":"30","author":"Driver, K.","year":"2002","journal-title":"Numer. Algorithms","ISSN":"https:\/\/id.crossref.org\/issn\/1017-1398","issn-type":"print"},{"issue":"3-4","key":"6","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/S0019-3577(03)90049-8","article-title":"Asymptotic zero distribution of a class of \u2083\ud835\udc39\u2082 hypergeometric functions","volume":"14","author":"Driver, K.","year":"2003","journal-title":"Indag. Math. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0019-3577","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.cam.2005.05.039","article-title":"Separation theorems for the zeros of certain hypergeometric polynomials","volume":"199","author":"Driver, K.","year":"2007","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"1045","DOI":"10.1016\/j.jmaa.2006.10.080","article-title":"P\u00f3lya frequency sequences and real zeros of some \u2083\ud835\udc39\u2082 polynomials","volume":"332","author":"Driver, K.","year":"2007","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"10","key":"9","doi-asserted-by":"publisher","first-page":"2424","DOI":"10.1016\/j.apnum.2009.04.007","article-title":"Interlacing of zeros of linear combinations of classical orthogonal polynomials from different sequences","volume":"59","author":"Driver, Kathy","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1006\/jath.2000.3548","article-title":"Zeros of the hypergeometric polynomials \ud835\udc39(-\ud835\udc5b,\ud835\udc4f;-2\ud835\udc5b;\ud835\udc67)","volume":"110","author":"Driver, Kathy","year":"2001","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1006\/jath.2001.3580","article-title":"Asymptotic properties of zeros of hypergeometric polynomials","volume":"111","author":"Duren, Peter L.","year":"2001","journal-title":"J. Approx. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9045","issn-type":"print"},{"key":"12","series-title":"Introductions to Higher Mathematics","volume-title":"Analytic function theory. Vol. II","author":"Hille, Einar","year":"1962"},{"key":"13","first-page":"1","article-title":"Orthogonality of Jacobi polynomials with general parameters","volume":"19","author":"Kuijlaars, A. B. J.","year":"2005","journal-title":"Electron. Trans. Numer. Anal."},{"key":"14","series-title":"Mathematical Surveys, No. 3","volume-title":"Geometry of polynomials","author":"Marden, Morris","year":"1966","edition":"2"},{"key":"15","doi-asserted-by":"crossref","unstructured":"P. Mart\u00ednez-Gonz\u00e1lez and A. Zarzo, Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials, J. Comput. Appl. Math. 233 (2010), 1577\u20131583.","DOI":"10.1016\/j.cam.2009.02.088"},{"key":"16","isbn-type":"print","volume-title":"Integrals and series. Vol. 3","author":"Prudnikov, A. P.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/2881246826"},{"key":"17","series-title":"Ellis Horwood Series: Mathematics and its Applications","isbn-type":"print","volume-title":"Multiple Gaussian hypergeometric series","author":"Srivastava, H. M.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/085312602X"},{"key":"18","series-title":"Ellis Horwood Series: Mathematics and its Applications","isbn-type":"print","volume-title":"A treatise on generating functions","author":"Srivastava, H. M.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0853125082"},{"key":"19","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"},{"issue":"1-2","key":"20","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"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1111\/j.1467-9590.2007.00373.x","article-title":"An infinite asymptotic expansion for the extreme zeros of the Pollaczek polynomials","volume":"118","author":"Zhou, Jian-Rong","year":"2007","journal-title":"Stud. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2526","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02409-9\/S0025-5718-2011-02409-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02409-9\/S0025-5718-2011-02409-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:53:18Z","timestamp":1776790398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-275\/S0025-5718-2011-02409-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,11]]},"references-count":21,"journal-issue":{"issue":"275","published-print":{"date-parts":[[2011,7]]}},"alternative-id":["S0025-5718-2011-02409-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02409-9","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":[[2011,2,11]]}}}