{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T16:08:21Z","timestamp":1786118901407,"version":"build-2736575974"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"362","license":[{"start":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T00:00:00Z","timestamp":1784764800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n For\n \n \n \n \n q<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n q \\in (0, 1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the deformed exponential function\n \n \n \n \n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n n<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munder>\n \n x<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msup>\n \n q<\/mml:mi>\n \n n<\/mml:mi>\n (<\/mml:mo>\n n<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n n<\/mml:mi>\n !<\/mml:mo>\n <\/mml:mrow>\n f(x) = \\sum _{n \\geq 0} x^n q^{n(n-1)\/2}\/n!<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is known to have infinitely many simple and negative zeros\n \n \n \n \n {<\/mml:mo>\n \n x<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n \n }<\/mml:mo>\n \n k<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \\{x_k(q)\\}_{k \\geq 1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In this paper, we analyze the series expansions of\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n \n x<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n k<\/mml:mi>\n <\/mml:mrow>\n -x_k(q)\/k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n k<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n x<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n k\/x_k(q)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in powers of\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove that the coefficients of these expansions are rational functions of the form\n \n \n \n \n \n P<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n P_n(k)\/Q_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \n P<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widehat {P}_n(k)\/Q_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \n Q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n [<\/mml:mo>\n k<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n Q_n(k) \\in {\\mathbb Z}[k]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is explicitly defined and the polynomials\n \n \n \n \n \n P<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n \n \n \n P<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n \n \u2208\n \n <\/mml:mo>\n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n [<\/mml:mo>\n k<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n P_n(k), \\widehat {P}_n(k)\\in {\\mathbb Z}[k]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n can be computed recursively. We provide explicit formulas for the leading coefficients of\n \n \n \n \n \n P<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n P_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \n P<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widehat {P}_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and compute the coefficients of these polynomials for\n \n \n \n \n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 300<\/mml:mn>\n <\/mml:mrow>\n n \\leq 300<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Numerical verification shows that\n \n \n \n \n \n P<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n P_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \n P<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widehat {P}_n(k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n take non-negative values for all\n \n \n \n \n k<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n k \\in \\mathbb {N}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n n<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 300<\/mml:mn>\n <\/mml:mrow>\n n\\le 300<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , offering further evidence in support of conjectures by Alan Sokal.\n <\/p>","DOI":"10.1090\/mcom\/4131","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T14:47:19Z","timestamp":1753282039000},"page":"2917-2939","source":"Crossref","is-referenced-by-count":0,"title":["On series expansions of zeros of the deformed exponential function"],"prefix":"10.1090","volume":"95","author":[{"given":"Alexey","family":"Kuznetsov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"1","unstructured":"NIST Digital Library of Mathematical Functions, F. 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