In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of Levy processes. In more detail, for a subordinator (a non-decreasing Levy process) $(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t e^{-X_s}ds $, evaluated at a finite, deterministic time $t>0$. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time $t$ which under very minor restrictions is shown to be equivalent to an infinite series. We believe this work can be regarded as a stepping stone towards a more in-depth study of general exponential functionals of Levy processes on a finite time horizon.

中文翻译：

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二元 Bernstein-Gamma 函数和从属函数的指数函数的矩

在本文中，我们将最近关于我们称为 Bernstein-gamma 的函数的工作扩展到二元 Bernstein-gamma 函数类。在更一般的双变量设置中，我们确定了斯特林型渐近边界，它概括、改进和简化了为单变量 Bernstein-gamma 函数找到的边界。然后，我们通过应用到 Levy 过程的指数函数来证明这些结果的重要性和力量。更详细地，对于从属（非递减征费过程）$(X_s)_{s\geq 0}$，我们研究其\textit{指数函数}，$\int_0^te^{-X_s}ds $ , 在有限的、确定性的时间 $t>0$ 计算。我们在这里的主要结果是一个明确的无限卷积公式，用于指数函数的梅林变换（复矩）直到时间 $t$，在非常小的限制下显示等效于无限级数。我们相信这项工作可以被视为在有限时间范围内更深入研究 Levy 过程的一般指数函数的垫脚石。