The main focus of this paper is to develop certain types of fundamental theorems using q, $q(\alpha )$, and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, $q(\alpha )$, and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and $q(\alpha )$ are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and $q(\alpha )$ symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and $q(\alpha )$ difference equations.

This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The $\nu$th-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to nth-sum for a class of delta integrable functions, that is, functions with both discrete integration and nth-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞-order delta integrable functions, the discrete integration related to the $\nu$th-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

In this paper, we develop theorems on finite and infinite summation formulas by utilizing the $\mathfrak{q}$ and $(\mathfrak{q},\mathfrak{h})$ anti-difference operators, and also we extend these core theorems to ${\mathfrak{q}}_{\left(\alpha \right)}$ and ${(\mathfrak{q},\mathfrak{h})}_{\alpha }$ difference operators. Several integer order theorems based on $\mathfrak{q}$ and ${\mathfrak{q}}_{\left(\alpha \right)}$ difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for $\mathfrak{q}$ and ${\mathfrak{q}}_{\left(\alpha \right)}$ difference operators. In order to develop the fractional order anti-difference equations for $\mathfrak{q}$ and ${\mathfrak{q}}_{\left(\alpha \right)}$ difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for $\mathfrak{q}$ and ${\mathfrak{q}}_{\left(\alpha \right)}$ difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the $(\mathfrak{q},\mathfrak{h})$ and ${(\mathfrak{q},\mathfrak{h})}_{\alpha }$ difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the $(\mathfrak{q},\mathfrak{h})$ and ${(\mathfrak{q},\mathfrak{h})}_{\alpha }$ difference operators for verification.

In this research, we proposed a new concept called as the $\mathcal{H}$-Nacci sequence. The $\mathcal{H}$-Nacci sequence (Fibonacci sequences of length h) is a collection of numbers developed from the coefficients of the generalized m-th Fibonacci equation. After that, we determined the golden ratio for each type of $\mathcal{H}$-Nacci sequence, which also coincided with an existing Fibonacci sequence. As each Fibonacci sequence has a unique advantage, first of all, we have applied the $\mathcal{H}$-Nacci sequence to the virus mutation process to show the key benefits of the $\mathcal{H}$-Nacci sequence, and then we found the Fibonacci risk model to analyze the risk factor of each mutant virus using the $\mathcal{H}$-Nacci sequence.

In this article, we introduce a three-dimensional q-difference operator with its equation. Also we derive certain theorems using three-dimensional q-difference operator. Suitable examples veri ed by MATLAB are inserted to illustrate our findings.