{"title":"关于具有非单调Taylor系数二阶商的Laguerre Polya I类的整体函数","authors":"Thu Hien Nguyen, A. Vishnyakova","doi":"10.30970/ms.56.2.149-161","DOIUrl":null,"url":null,"abstract":"For an entire function $f(z) = \\sum_{k=0}^\\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \\frac{a_{k-1}^2}{a_{k-2}a_k}, k \\geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \\in \\mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients\",\"authors\":\"Thu Hien Nguyen, A. Vishnyakova\",\"doi\":\"10.30970/ms.56.2.149-161\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For an entire function $f(z) = \\\\sum_{k=0}^\\\\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \\\\frac{a_{k-1}^2}{a_{k-2}a_k}, k \\\\geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \\\\in \\\\mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\\\\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-07-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.56.2.149-161\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.56.2.149-161","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients
For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.