Owais Khan

We introduce two new families of mathematical functions, called parametric poly-Genocchi polynomials, blending advanced algebraic tools (polylogarithms) and trigonometric shapes. These functions enable modeling of natural processes exhibiting both rhythmic (oscillatory) changes and memory effects, such as the heat dynamics in living tissues. We rigorously establish their fundamental properties—including summation identities, differential relations, and connections to Voigt functions—and apply them in developing a fractional bio-heat transfer model for human tissues. This model, wherein these polynomials capture oscillatory metabolic heating with memory effects, demonstrates a 90\% reduction in RMSE relative to the classical Pennes model. Analytical and numerical validations bridge special function theory with biomedical engineering, offering powerful tools for optimizing thermal therapies such as hyperthermia. These results present an interdisciplinary advance, providing mathematical frameworks adaptable to diverse real-world systems characterized by oscillatory and memory-driven behaviors. 2010 Mathematics Subject Classification: 33E12, 11B83, 11B68, 33E20, 05A15.

The main objective is to introduce a novel kind of beta function known as the beta logarithmic function using extended beta functions and logarithmic mean. Further, we study its essential properties and investigate various formulas of beta logarithmic functions such as integral representation, summation formula, transform formula and their statistical properties. Based on this concept, we introduce new hypergeometric and confluent hypergeometric functions and study their properties.

In this article our main object to compute image formulas of generalized fractional hypergeometric operators, involving the product of multivariable Srivastava polynomial and multiindex Bessel function. The results obtained provide unification and an extension of known results given earlier by Agarwal and Nieto ^[1], Agarwal et al. ^[2] Mishra et al. ^[18], Saxena and Saigo ^[26], Suthar et al. ^[32]. We also consider certain special cases of derived results by specializing suitable value of the parameters.

Fractional calculus is allowing integrals and derivatives of any positive order (the term `fractional' kept only for historical reasons), which can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power-law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this line, our main object to investigate image formulas of generalized fractional hypergeometric operators involving the product of Mathieu-type series and generalized Mittag-Leffler function. We also consider some interesting special cases of derived results by specializing suitable value of the parameters.