{"title":"具有一个催化变量的正泛函方程的普遍渐近性质","authors":"Michael Drmota, Eva-Maria Hainzl","doi":"10.1007/s44007-023-00063-0","DOIUrl":null,"url":null,"abstract":"Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.","PeriodicalId":74051,"journal":{"name":"La matematica","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Universal Asymptotic Properties of Positive Functional Equations with One Catalytic Variable\",\"authors\":\"Michael Drmota, Eva-Maria Hainzl\",\"doi\":\"10.1007/s44007-023-00063-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.\",\"PeriodicalId\":74051,\"journal\":{\"name\":\"La matematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"La matematica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s44007-023-00063-0\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"La matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s44007-023-00063-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Universal Asymptotic Properties of Positive Functional Equations with One Catalytic Variable
Functional equations with one catalytic variable appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions, the dominant singularity of the solution has a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we—usually—have a singularity of type 3/2.