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Key research themes
1. How can the Gamma function be generalized and extended to unify special functions and address applications in fractional calculus and applied sciences?
Research on generalizing the Gamma function focuses on developing extended integral and series formulations with additional parameters or structures. These advances enable connections to fractional calculus operators, new special functions (e.g., k-gamma, generalized Mittag-Leffler), and wider classes of applications including stochastic models, diffraction, and physical processes. This theme is vital for enhancing the analytical toolkit provided by the Gamma function and for capturing more complex phenomena mathematically.
Key finding: The study proposes a novel generalized gamma function defined via infinite products involving powers of logarithms and connects this construction to generalized Euler-Mascheroni constants. It establishes essential properties... Read more
Key finding: This paper introduces the k-version of the Bessel function built on the k-gamma function, a generalization of the classical gamma. It demonstrates relationships between the k-Bessel function and other extended functions like... Read more
2. What are the properties and inequalities involving derivatives and expansions of generalized Gamma-related functions and their parameter dependencies?
Many investigations target the behavior of the Gamma function and its generalizations through asymptotic expansions, bounds, monotonicity, and derivatives with respect to parameters—especially in the context of special functions like digamma, trigamma, and the Whittaker functions. These studies develop inequalities, completely monotonic properties, and analytic expansions that inform approximation, numerical evaluation, and theoretical insights critical to mathematical analysis and applied problems.
Key finding: The authors derive a detailed asymptotic expansion for a class of generalized gamma functions Γ_μ(v) involving parameter μ. They rigorously demonstrate the function's completely monotonic properties with respect to its... Read more
Key finding: Extending classical inequalities for the Gamma and digamma functions, this paper proves that the trigamma function ψ'(z) exhibits geometric convexity (GG-convexity) and GA-convexity on (0, ∞). Among its main results is that... Read more
Key finding: This paper systematically derives parameter derivatives of the Whittaker M function via confluent hypergeometric function expansions. The differentiation leads to infinite series involving quotients of digamma and gamma... Read more
Key finding: Extending the previous work on Whittaker parameter derivatives, this study analyzes the W variant of Whittaker functions with respect to κ and μ parameters. It develops infinite series and integral analytic formulae,... Read more
3. How do the Gamma function and its variants inform probability distributions, statistical modeling, and applications in data analysis?
The Gamma function underpins numerous continuous probability distributions, including gamma, beta, chi-square, F, Student’s t, and various logistic distributions. Research in this theme develops new modified or generalized Gamma-based models and investigates their probabilistic properties, parameter estimation methods, and applications to real-world data such as medical statistics, reliability, and environmental processes. This theme highlights the Gamma function’s foundational role in statistical theory and applied data analysis.
Key finding: The authors propose a modified Gamma (MG) distribution introducing a risk intensity control to modulate the right tail behavior for data exhibiting light tails relative to classical gamma. They derive analytical properties of... Read more
Key finding: This paper unifies and extends type I and II generalized logistic distributions by introducing the gamma generalized logistic distribution (GGLD) with three parameters, capturing skewness and kurtosis flexibility. The authors... Read more
Key finding: This paper reviews the derivation of several fundamental continuous probability distributions—including gamma, beta, chi-square, Student’s t, and F distributions—directly via the Euler Gamma function. The expository work... Read more