{"title":"Applications of Spectral Theory to Special Functions","authors":"E. Koelink","doi":"10.1017/9781108908993.004","DOIUrl":"https://doi.org/10.1017/9781108908993.004","url":null,"abstract":"Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. \u0000Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.","PeriodicalId":413584,"journal":{"name":"Lectures on Orthogonal Polynomials and Special Functions","volume":"41 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132822953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Elliptic Hypergeometric Functions","authors":"V. Spiridonov","doi":"10.1017/9781108908993.005","DOIUrl":"https://doi.org/10.1017/9781108908993.005","url":null,"abstract":"This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. \u0000The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. \u0000It contains an outline of a general heuristic scheme for building univariate special functions through self-similar reductions of spectral transformation chains, which allowed construction of the differential-difference q-Painleve equations, as well as of the most general known set of elliptic biorthogonal functions comprising all classical orthogonal polynomials and biorthogonal rational functions. One of the key results of the thesis consists in the discovery of genuinely transcendental elliptic hypergeometric functions determined by the elliptic hypergeometric integrals. The whole theory of such integrals can be built from the univariate elliptic beta integral -- the most complicated known definite integral with exact evaluation, which generalizes the ordinary binomial theorem and its q-extension, Euler's beta integral, the measure for Askey-Wilson polynomials, and many other previously established results on ordinary and q-hypergeometric functions.","PeriodicalId":413584,"journal":{"name":"Lectures on Orthogonal Polynomials and Special Functions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128211414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}