In this study, the Volterra-FredholmIntegro-Differential Equations (VFIDEs) are solved computationally using first-kind Chebyshev polynomials as basis functions. The technique converts VFIDEs into a system of unknown Chebyshev coefficients linear algebraic equations, which are then solved via matrix inversion. Five numerical examples were solved in order to assess the scheme's correctness and effectiveness. The findings reveal that the approach outperforms the comparison results in the literature

The aim of this paper is to define the generalized discrete proportional derivative (GDPD) and illustrate the application of the Leibniz theorem, the binomial expansion, and Montmort’s formulas in the context of the generalized discrete proportional case. Furthermore, we introduce the generalized discrete proportional Laplace transform and determine the GDPLT of various functions using the inverse operator. The results obtained are showcased through relevant examples and validated using MATLAB.

In this study, we present a generalized higher-order delta operator with the co-efficient of falling factorial and its inverse, both of which allow us to get more generalized k(ε)-Fibonacci sequences along with their sums, a few theorems, and some intriguing conclusions regarding the sum of more generalized terms of k(ε)-Fibonacci sequences with falling factorial. In addition, the n-fold Fibonacci ratio and a few applications in life sciences and crystal growth were presented in the work. In addition, suitable examples are presented using MATLAB to show our results.

In this paper, we introduce second order difference operator with polynomial factorial and its inverse by which we obtain advanced Fibonacci sequence and its sum. Some theorems and interesting results on the sum of the terms of second order Fibonacci sequence are derived. Suitable examples are provided to illustrate our results and verified by MATLAB.

‎This study presents a general third-order nabla difference operator that allows us to get \(\overline{\varphi({\tt{x}})}\)-Tribonacci sequences‎, ‎Tribonacci numbers‎, ‎and their sum using the coefficients of different trigonometric functions and their‎ ‎inverse‎. ‎In this section‎, ‎we examined the numerical solutions and \(C^*\)-solutions of the‎ ‎\(\overline{\varphi({\tt{x}})}\)-Tribonacci sequences for different functions‎. ‎In addition‎, ‎some interesting conclusions and theorems are obtained for the sum of the terms of the Tribonacci‎ ‎sequence‎. ‎Also‎, ‎we offer appropriate examples to show how to use MATLAB to demonstrate our results‎. ‎

This paper introduces a general nabla operator of order two that includes coefficients of various trigonometric functions. We also introduce its inverse, which leads us to derive the second-order $ \overline{\theta({\tt{t}})} $-Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the $ \overline{\theta({\tt{t}})} $-Fibonacci polynomial using a proportional derivative. Furthermore, this study presents derived theorems and intriguing findings on the summation of terms in the second-order Fibonacci sequence, and we have investigated the bifurcation analysis of the $ \overline{\theta({\tt{t}})} $-Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.