Inversion of a Class of Singular Integral Operators on Entire Functions

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Contribution

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Abstract

Given constants $x, \nu \in \mathbb{C}$ and the space $\mathscr{H}_0$ of entire functions in $\mathbb{C}$ vanishing at $0$, we consider the integro-differential operator $$ \mathfrak{L} = \left ( \frac{x \, \nu(1-\nu)}{1-x} \right ) \; \delta \circ \mathfrak{M}\, , $$ with $\delta = z \, \mathrm{d}/\mathrm{d}z$ and $\mathfrak{M}:\mathscr{H}_0 \rightarrow \mathscr{H}_0$ defined by $$ \mathfrak{M}f(z) = \int_0^1 e^{-z t^{-\nu}(1-(1-x)t)} \, f \left (z \, t^{-\nu}(1-t) \right ) \, \frac{\mathrm{d}t}{t}, \qquad z \in \mathbb{C}, $$ for any $f \in \mathscr{H}_0$. Operator $\mathfrak{L}$ originates from an inversion problem in Queuing Theory. Bringing the inversion of $\mathfrak{L}$ back to that of $\mathfrak{M}$ translates into a singular Volterra integral equation, but with no explicit kernel. In this paper, the inverse of operator $\mathfrak{L}$ is derived through a new inversion formula recently obtained for infinite matrices with entries involving Hypergeometric polynomials. For $x \notin \mathbb{R}^- \cup \{1\}$ and $\mathrm{Re}(\nu) < 0$, we then show that the inverse $\mathfrak{L}^{-1}$ of $\mathfrak{L}$ on $\mathscr{H}_0$ has the integral representation $$ \mathfrak{L}^{-1}g(z) = \frac{1-x}{2i\pi x} \, e^{z} \int_1^{(0+)} \frac{e^{-xtz}}{t(t-1)} \, g \left (z \, (-t)^{\nu}(1-t)^{1-\nu} \right ) \, \mathrm{d}t, \qquad z \in \mathbb{C}, $$ for any $g \in \mathscr{H}_0$, where the bounded integration contour in the complex plane starts at point 1 and encircles the point 0 in the positive sense. Other related integral representations of $\mathfrak{L}^{-1}$ are also provided.