BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anna Vishnyakova (V. N. Karazin Kharkiv National University\, Ukra ine and Holon Institute of Technology\, Israel and Holon Institute of Technology\, Israel) DTSTART:20230124T140000Z DTEND:20230124T150000Z DTSTAMP:20260422T201115Z UID:CAvid/93 DESCRIPTION:Title: N ecessary and sufficient conditions for entire functions to belong to the L aguerre-Polya class\nby Anna Vishnyakova (V. N. Karazin Kharkiv Nation al University\, Ukraine and Holon Institute of Technology\, Israel an d Holon Institute of Technology\, Israel) as part of CAvid: Complex Analys is video seminar\n\nLecture held in N/A.\n\nAbstract\nThe famous Laguerre- Polya class consists of\nentire functions which are uniform on the\ncompac ts limits of real polynomials having all\nreal zeros. The Laguerre-Polya c lass is of interest\nto many areas of mathematics such as complex analysis \,\nstatistical physics\, combinatorics\, asymptotic analysis\,\nthe theor y of mock modular forms and others. We present\nnew necessary and new suff icient conditions for an entire\nfunction to belong to the Laguerre-Polya class in terms\nof Taylor coefficients of the function. The partial theta- function\n$g_a(z) =\\sum_{k=0}^{\\infty} \\frac {z^k}{a^{k^2}}\, a>1\,$\np lays an important role in our investigations. It is known\nthat there exis ts a constant $ q_\\infty\\approx 3{.}23363666\,$\nsuch that the partial t heta-function belongs to the Laguerre-Polya\nclass if and only if $a^2 \\g eq q_\\infty.$ The following statement\nis an example of our results. Let $f(z)=\\sum_{k=0}^\\infty a_k z^k $\nbe an entire function with positive coefficients. Suppose that the\nsequence $\\frac{a_n^2}{a_{n-1} a_{n+1}}$ is decreasing in $n$\,\nand the limit of this sequence is greater than or equal to\n$\\ q_\\infty.$ Then the function $f$ belongs to the\nLaguerre-P olya class.\n LOCATION:https://researchseminars.org/talk/CAvid/93/ END:VEVENT END:VCALENDAR