Stieltjes Transforms and R-Transforms Associated with Two-Parameter Lambert-Tsallis Functions.

Area of Science:

• Mathematics
• Probability and Statistics
• Mathematical Physics

Background:

• The study investigates Stieltjes transformations, which are fundamental in complex analysis and have applications in random matrix theory.
• Holomorphic Lambert-Tsallis functions, a generalization of the Lambert function, are introduced as a relevant framework.
• These transformations are connected to eigenvalue distributions in statistically sparse random matrix models.

Purpose of the Study:

• To analyze a two-parameter family of Stieltjes transformations associated with holomorphic Lambert-Tsallis functions.
• To determine the conditions under which these functions represent Stieltjes transformations of probabilistic measures.
• To derive an explicit formula for the corresponding R-transformations.

Main Methods:

• Utilizing concepts from complex analysis and function theory.
• Applying methods from random matrix theory to analyze eigenvalue distributions.
• Deriving necessary and sufficient conditions for probabilistic measure representation.

Main Results:

• A two-parameter family of Stieltjes transformations related to generalized Lambert functions is studied.
• A precise condition is identified for these functions to be Stieltjes transformations of probabilistic measures.
• An explicit formula for the associated R-transformations is provided.

Conclusions:

• The research provides a comprehensive analysis of a specific class of Stieltjes transformations.
• The findings contribute to understanding probabilistic measures in the context of random matrix theory and sparse models.
• The derived R-transformation formula offers a valuable tool for further theoretical development.