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In this research, we proposed a new concept called as the H-Nacci sequence. The 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 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 H-Nacci sequence to the virus mutation process to show the key benefits of the H-Nacci sequence, and then we found the Fibonacci risk model to analyze the risk factor of each mutant virus using the H-Nacci sequence.

The summation and exact form of the solutions related to the special type of difference equations are established in this paper by using the inverse of the delta and alpha-delta operators. As an application, the solutions of the population growth model, particularly, fractional order kinetic equation, are obtained by this method. Also, by using the summation form and Mittag–Leffler factorial functions, the alpha-delta integrations have been applied for solving the fractional order difference equations involving the factorial polynomials. Numerical examples are provided to validate the theoretical results.

Abstract Our main goal in this work is to derive the frequency Laplace transforms of the products of two and three functions with tuning factors. We propose the Laplace transform for certain types of multiseries of circular functions as well. For use in numerical results, we derive a finite summation formula and m-series formulas. Moreover, we discuss various explanatory examples.

Abstract This article aims to develop fractional order convolution theory to bring forth innovative methods for generating fractional Fourier transforms by having recourse to solutions for fractional difference equations. It is evident that fractional difference operators are used to formulate for finding the solutions of problems of distinct physical phenomena. While executing the fractional Fourier transforms, a new technique describing the mechanism of interaction between fractional difference equations and fractional differential equations will be introduced as h tends to zero. Moreover, by employing the theory of discrete fractional Fourier transform of fractional calculus, the modeling techniques will be improved, which would help to construct advanced equipments based on fractional transforms technology using fractional Fourier decomposition method. Numerical examples with graphs are verified and generated by MATLAB.