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.

We examined an alkaline ring complex in the Eastern Desert of Egypt, Wadi Dib ring complex (WDRC), to understand formation mechanisms of the intimately related ring structure and chemical diversity. The WDRC consists of multiple circular rings of the oldest volcanic units, the middle-stage plutonic unit, and the youngest dike unit, and these units show overlapping whole-rock major element compositions. The compositional variation of the volcanic and plutonic units can be accounted for by a stepwise fractional crystallization starting with a trachytic magma without significant magma replenishment or crustal contamination. From the margin to the center and oldest to youngest, the plutonic unit consists of an outer ring (syenite), inner rings 1 (quartz-bearing syenite) and 2 (quartz syenite), and a granitic core (syenogranite). The whole-rock chemical composition of the plutonic unit is progressively more fractionated inwards from the outer ring to the granitic core through the inner rings. Syenites from the innermost outer ring show high degree of deformation, which gradually decreases outwards in the outer ring and abruptly decreases inwards in the outermost inner ring 1. The deformed syenites show microstructures suggesting reactive melt transportation. The country rocks neighboring the ring complex and equivalent blocks present in the periphery of the outer ring, the overlying volcanic unit and equivalent blocks present in the inner rings all show microstructures indicative of pyrometamorphism. Spatial variations in the microstructures in the plutonic unit indicate an increase in cooling rate from the outer ring to the granitic core and thus with time …