In this paper, we derive some classical and fractional properties of the rRs matrix function
by using the Hilfer fractional operator. The theory of special matrix functions is the theory of
those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss
hypergeometric matrix functions. We will also show the relationship with other generalized special
matrix functions in the context of the Konhauser and Laguerre matrix polynomials.

The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for Θ,α,θ,Ψ contraction, rational Θ,α,θ,Ψ contraction and almost Θ,α,θ,Ψ contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of Θ,θ,Ψυ0, several fixed circle solutions are provided in the setting of a G-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated.

The main object of this paper is to deduce the bibasic Humbert functions Ξ1 and Ξ2
Some interesting results and elementary summations technique that was successfully employed,
q?recursion, q?derivatives relations, the q?differential recursion relations, the q?integral representations
for Ξ1 and Ξ2 are given. The summation formula derives a list of p?analogues
of transformation formulas for bibasic Humbert functions that have been studied, also some
hypergeometric functions properties of some new interesting special cases have been given.