## First order analytic difference equations and integrable quantum systems.(English)Zbl 0877.39002

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.

### MSC:

 39A10 Additive difference equations 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals 81U40 Inverse scattering problems in quantum theory
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### References:

 [1] DOI: 10.1515/crll.1869.70.258 · JFM 02.0122.04 [2] DOI: 10.1007/BF01167832 · Zbl 0379.39003 [3] DOI: 10.1016/0003-4916(86)90097-7 · Zbl 0608.35071 [4] DOI: 10.1007/BF01207363 · Zbl 0673.58024 [5] DOI: 10.1016/0370-1573(83)90018-2 [6] DOI: 10.1016/0003-4916(79)90391-9 [7] DOI: 10.2977/prims/1195164440 · Zbl 0842.58050
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