Ruijsenaars, S. N. M. First order analytic difference equations and integrable quantum systems. (English) Zbl 0877.39002 J. Math. Phys. 38, No. 2, 1069-1146 (1997). Author’s abstract: We present a new solution method for a class of first order analytic difference equations. The method yields explicit “minimal” solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type — Euler’s gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (\(u\) - and \(w\) -functions) associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these \(u\) - and \(w\) -functions, exploiting the difference equations they satisfy. Reviewer: A.D.Mednykh (Novosibirsk) Cited in 5 ReviewsCited in 133 Documents MSC: 39A10 Additive difference equations 33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals 81U40 Inverse scattering problems in quantum theory Keywords:generalized gamma functions; scattering functions; weight functions; first order analytic difference equations; minimal solutions; integrable quantum systems × Cite Format Result Cite Review PDF Full Text: DOI Link References: [1] DOI: 10.1515/crll.1869.70.258 · JFM 02.0122.04 · doi:10.1515/crll.1869.70.258 [2] DOI: 10.1007/BF01167832 · Zbl 0379.39003 · doi:10.1007/BF01167832 [3] DOI: 10.1016/0003-4916(86)90097-7 · Zbl 0608.35071 · doi:10.1016/0003-4916(86)90097-7 [4] DOI: 10.1007/BF01207363 · Zbl 0673.58024 · doi:10.1007/BF01207363 [5] DOI: 10.1016/0370-1573(83)90018-2 · doi:10.1016/0370-1573(83)90018-2 [6] DOI: 10.1016/0003-4916(79)90391-9 · doi:10.1016/0003-4916(79)90391-9 [7] DOI: 10.2977/prims/1195164440 · Zbl 0842.58050 · doi:10.2977/prims/1195164440 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.