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On a class of functional difference equations: explicit solutions, asymptotic behavior and applications. (English) Zbl 07811713

The article is devoted to the development of the theory of functional-difference equations. The author continues the research in this area initiated by E. W. Barnes [Proc. Lond. Math. Soc. (2) 2, 438–469 (1905; JFM 36.0406.04)], B. A. Vasil’ev [Differ. Equations 21, 554–557 (1985; Zbl 0631.35009); translation from Differ. Uravn. 21, No. 5, 815–819 (1985)], V. A. Solonnikov and E. V. Frolova [Algebra Anal. 2, No. 4, 213–241 (1990; Zbl 0711.35036)], B. V. Bazaliy and A. Friedman [J. Differ. Equations 216, No. 2, 387–438 (2005; Zbl 1075.76022)], as well as her own research.
In this paper, the author studied an equation of the form \[ (a_1\sigma+a_2\sigma^{\nu})\mathcal{Y}(z+\beta,\sigma)-\Omega(z)\mathcal{Y}(z,\sigma)={F}(z,\sigma),\qquad z\in \mathbb{C}, \] where \(\mathcal{Y}\) is an unknown complex-valued function, and the remaining parameters and functions satisfy certain conditions. Under weaker assumptions (compared to previous works) the author construct explicit solutions of the above equation and describe their asymptotic behavior as \(|z|\to +\infty\). Other equations of this type are also studied and applications to the theory of boundary value problems are given.

MSC:

39A06 Linear difference equations
39B32 Functional equations for complex functions
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