Applications of the ( q , τ ) -Gamma function in fractional calculus and special functions

The ( q , τ ) -Gamma function provides a two-parameter extension of the classical and q -deformed Gamma functions, enabling enhanced modeling of memory and scaling effects in fractional calculus and special functions. This paper presents the definition, key properties, and computational aspects of the ( q , τ ) -Gamma function Applications in generalized fractional derivatives, integral transforms, and ( q , τ ) -special polynomials are described, with selected examples and discussion of future directions.