On some explicit integrals related to “fractal foothills”

Key Points

• The aim is to derive closed-form expressions for partial loop counting functions that measure returns to the origin in random walks.
• Focused on partial loop counting functions V that track returns to the origin.
• Analyzed the mathematical relationships of V with closed-form integrals.
• Explored connections to complete loop counting functions and Bernoulli polynomials.
• Derived explicit formulas for various integrals related to partial loop counting functions V.
• Demonstrated links between complete loop counting functions and Bernoulli polynomials.