Some notes on Sonine-Gegenbauer integrals

Area of Science:

• Mathematical analysis
• Special functions
• Integral calculus

Background:

• Sonine-Gegenbauer integrals are important in mathematical physics.
• Existing literature lacks an explicit formula for a specific type of these integrals.
• Understanding dependence relations of integrals aids in simplifying complex mathematical expressions.

Purpose of the Study:

• To derive a novel explicit formula for a Sonine-Gegenbauer integral.
• To establish a dependence relation for another class of Sonine-Gegenbauer integrals.
• To explicitly calculate coefficient functions within this relation.

Main Methods:

• Symbolic integration techniques.
• Derivation of explicit mathematical formulas.
• Analysis of functional dependencies.

Main Results:

• An explicit formula for a Sonine-Gegenbauer integral, new to the literature, was successfully derived.
• A dependence relation over rational functions was established for a related integral type.
• The coefficient functions involved in the dependence relation were explicitly calculated.

Conclusions:

• The study introduces a significant new formula in the field of special functions.
• The established dependence relation offers a new perspective on the structure of these integrals.
• The explicit calculation of coefficients facilitates further theoretical and applied research.