On a class of differential-difference equations arising in number theory
Functions defined by a differential-difference equation of the type (1) \(uf ' (u) + af(u) + bf(u-1) = 0\), where \(a\) and \(b\) are constants, arise frequently in number theory. Probably the best known examples are the Dickman function \(\rho(u)\) and the Buchstab function \(\omega (u)\). Functions satisfying (1) when \(a+b\) is an integer arise in sieve theory and in this context have been investigated by various authors. The equation (1) with general coefficients \(a\) and \(b\) have been studied by \textit{H. Iwaniec} [Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)] and \textit{F. Wheeler} [Trans. Am. Math. Soc. 318, 491-523 (1990; Zbl 0697.10035)]. The object of the present paper is to describe, for any given pair of complex coefficients \((a,b)\) with \(b \neq 0\) the structure and asymptotic behavior of the general solution to \((1)\). Equation (1) with the initial condition (2) \(f(u) = \varphi(u)\) \((u_ 0 - 1 \leq u \leq u_ 0)\), where \(\varphi (u)\) is any given continuous function on \([u_ 0 - 1,u_ 0]\), has a unique continuous solution \(f(u) = f(u;\varphi)\) for \(u \geq u_ 0\). In the paper, the authors construct a set of ``fundamental'' solutions \(F(u)\) and \(F_ n(u)\) \((n \in \mathbb{Z})\) and the solution \(f(u)\) can be expressed as a convergent series \(f(u) = \alpha F(u) + \sum_{n \in \mathbb{Z}} \alpha_ n F_ n (u)\) with suitable coefficients \(\alpha\) and \(\alpha_ n\). The functions \(F\) and \(F_ n\) are defined by means of a contour integral, which can be estimated rather precisely. To state the result we set \[ \Phi (u,s) = {\exp \{-us + bI(s)\}s^{a+b-1} \over \sqrt {2 \pi u(1-1/s)}}, \] where \(I(s) = \int^ s_ 0{e^ z-1 \over z} dz\), and then we have the following result: For any fixed non-zero integer \(n\) and \(u \geq u_ 0 (\varepsilon,n)\) we have \[ F_ n(u) = (1+O({1 \over u})) \Phi (u, \zeta_ n) \] where \(\zeta_ n = \zeta_ n (u/b)\) is a certain complex solution of the equation \(e^ \zeta = 1 + {u \over b} \zeta\), and the implied constant depends at most on \(\varepsilon\) and \(n\). The principal result of the paper is as follows. Let \(\varphi (u)\) be a continuous function on \([u_ 0 - 1,u_ 0]\) and let \(f(u) = f(u;\varphi)\) be the unique continuous solution to (1) and (2). Then we have \[ f(u) = \alpha F(u) + \sum_{\alpha \in \mathbb{Z}} \alpha_ n F_ n(u), \quad u>u_ 0+1 \tag{3} \] where \(\alpha = \langle \varphi, G \rangle\), \(\alpha_ n = \langle \varphi, G_ n \rangle\) and the series in (3) is uniformly convergent for \(u \geq u_ 0 + 1 + \delta\), for any fixed \(\delta>0\). Moreover, the authors derive several corollaries from the principal result.
scientific article; zbMATH DE number 528673
scientific article