奇点附近的Fox-Wright函数与分支割
The Fox-Wright function near the singularity and branch cut
摘要：
Fox-Wright函数是广义超几何函数的进一步推广，它是通过在被加数中的Γ函数的自变量中引入任意正的比例因子而得到的。它的重要性主要来自于它在分数微积分中的作用，尽管也存在其他有趣的应用。如果顶部和底部参数中的比例因子之和相等，则定义Fox-Wright函数的级数具有有限的非零收敛半径。Braaksma在1964年证明了Fox-Wright函数可以在复平面上延拓为一个全纯函数，该复平面沿着从收敛圆盘边界上的正点到无穷远无穷远点的一条射线切割。本文研究了Fox-Wright函数在这个正奇点邻域内的行为。在一定的限制下，我们给出了一个收敛的展开与递归计算的系数完全表征这种行为。我们进一步计算了Fox-Wright函数在分支割边上的跳跃和平均值。
英文摘要：
The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in fractional calculus although other interesting applications also exist. If the sums of the scaling factors in the top and bottom parameters are equal, the series defining the Fox-Wright function has a finite non-zero radius of convergence. It was demonstrated by Braaksma in 1964 that the Fox-Wright function can then be extended to a holomorphic function in the complex plane cut along a ray from the positive point on the boundary of the disk of convergence to the point at infinity. In this paper we study the behavior of the Fox-Wright function in the neighborhood of this positive singular point. Under certain restrictions we give a convergent expansion with recursively computed coefficients completely characterizing this behavior. We further compute the jump and the average value of the Fox-Wright function on the banks of the branch cut.
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