Mittag-Leffler function is a natural generalization of the exponential function. Recent applications of Mittag-Leffler function have reshaped the scientific literature due to its fractional effects that cannot be obtained by using exponential function. Present motivation is to define a new special function by modification in the original gamma function with Mittag-Leffler function. Properties of this modified function are discussed by investigating a new series representation involving delta function. Hence, the results are also validated with the earlier obtained results for gamma function as special cases. Furthermore, the new function is used to generate a probability density function, and its statistical properties are explored. Similar properties of existing distributions can be deduced.

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使用 Mittag-Leffler 函数修正 Gamma 函数

Mittag-Leffler 函数是指数函数的自然推广。Mittag-Leffler 函数的最新应用已经重塑了科学文献，因为它的分数效应无法通过使用指数函数获得。目前的动机是通过使用 Mittag-Leffler 函数修改原始伽马函数来定义一个新的特殊函数。通过研究涉及 delta 函数的新级数表示来讨论此修改函数的属性。因此，结果也用先前获得的伽马函数结果作为特殊情况进行了验证。此外，新函数用于生成概率密度函数，并探索其统计特性。可以推导出现有分布的类似属性。