The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function.

本文致力于研究广义 Kratzel 函数，即 1 型和 2 型路径变换的核函数。研究了这些函数的各种分析特性，例如 Lipschitz 连续性、不动点特性和可积性。此外，本文还介绍了与广义 Kratzel 函数相关的两个新的不等式。还得到了各种分数算子的复合公式，如Grünwald-Letnikov分数阶微分算子、Riemann-Liouville分数积分以及具有这些函数的微分算子。此外，已经证明这些核函数与 Heaviside 阶跃函数、Dirac delta 函数和 Riemann zeta 函数相关。还获得了核函数的可计算级数表示。通过使用 Mittag-Leffler 函数修改核函数来指出进一步的工作范围。