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Exact solution to a class of functional difference equations with application to a moving contact line flow. (English) Zbl 0934.39005

For the Barnes double gamma function \(G\), the authors derive a new integral representation: \(G(s,\delta)=\) \[ \begin{split} \exp \Bigl\{ \int^1_0\biggl[\frac {r^{s-1}}{ (r-1)(r^\delta -1)}-\frac {s^2}{2 \delta} r^{\delta-1}- sr^{\delta -1 }\bigl (\frac {2-r^\delta} {r^\delta -1 }-\frac {1} {2\delta}\bigr)\\ -r^{\delta-1}+\frac {1} {r-1}-\frac {r^{\delta -1}} {(r-1)(r^\delta-1)}\biggr] \frac{dr}{\ln r}\Bigr \}, \end{split} \] which is valid for \(\text{Re}(s)>0\) and \(\delta >0\) and is easily computable for real \(s>0\). Furthermore, they show how the solutions of the functional difference equations \(A(\alpha,s) f(s)-B(\alpha, s)f(s+ 1)=0\), where \(A(\alpha, s)\) and \(B(\alpha, s)\) are finite products of trigonometric functions of the form \(\sin \{\alpha (s+\beta)\}\) and \(\cos\{(\alpha (s+ \gamma) \}\) with real \(\beta\) and \(\gamma\), which play a central role in many boundary problems arising from fluid mechanics, may be expressed in terms of quotients of \(G\). Besides this, the article contains interesting applications of the theory and an appendix about the main properties of the Barnes double gamma function.

MSC:

39A10 Additive difference equations
33B15 Gamma, beta and polygamma functions
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
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