Alpha fractional frequency Laplace transform through multiseries

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.


Introduction
Digital signal processing (DSP) has revolutionized many areas in science and engineering such as space, medicine, commerce, military, technology, and communication. The Laplace transform (LT) and discrete Laplace transform (DLT) effectively change a signal (function) from time domain to frequency domain with the factor e -st . Several applications of LT and DLT were discussed by many authors [6,[8][9][10]. The applications of the n-dimensional Laplace transform appear in heat equations, wave equations, and modeling in fluid dynamics [13,14,21]. In [22,23], the authors considered some mathematical logistic models of fractional operators. Recently, the authors found the solutions of fractional difference equations [24][25][26]. Some more findings on fractional order models with the numerical simulation are discussed in [27][28][29][30][31][32][33]. For recent development in the theory of fractional difference operators, we refer to [15][16][17][18][19][20].
The LT and DLT are respectively defined as L[u(t)] = ∞ 0 u(t)e -st dt and L[u(n)] = ∞ n=0 u(n)e -sn , s > 0. From the basic difference identity -1 x n | ∞ 0 = ∞ n=0 x n [4] the DLT can be expressed as L[u(n)] = -1 u(n)e -sn . In the literature the Laplace transform in discrete calculus comes from the time scale definition of the Laplace transform and has a strong re-frequency Laplace transform (LFT) with tuning factor α and frequency s 1/ν as When α = 1 and h = 1, transform (1) becomes the discrete Laplace transform. When α = 1 and h → 0, (1) becomes the Laplace transform [5,6]. To develop LFT, we can study the operators h and α(h) and their inverses [7,11].
In 2011, the authors in [1] defined the alpha difference operator as In this research work, we extend the results on multiseries by using α-difference operators and then analyze LFT for signals of algebraic and geometric type functions.

Preliminaries
In [3] the authors introduced t (m) h = m-1 r=0 (trh) and obtained the following expressions: where s m r and S m r denote the Stirling numbers.
has a solution in the following infinite series form: and

Multiseries inverse of product of two and three functions
In this section, we derive a finite summation formula and m-series formula. Also, we present the m-series inverse of the product of two and three functions. Replacing Successively substituting all these expressions into (10), we arrive at

Theorem 4.4 For the functions u(t) and v(t), we have
Proof From the definition of α(h) we have Now (14), we obtain (13).
Proof After changing the powers of sin and cos into linear, we obtain After simplification, we get Applying -1 α(h) to both sides of equation (16), we get  Continuing this process up to m-inverse for n 1 , we get equation (17).

Laplace transforms and its applications
Here,] we derive the fractional frequency Laplace transform for the input functions (signals) in multiseries (m = 1) and analyze the results by MATLAB. Proof Taking the limit from 0 to ∞ in (15) gives the Laplace transform of sin n 2 pt cos n 3 qt.
Similarly, we can find results for other cases (odd-even, even-odd, even-even).
In the following example, we analyze LTT using MATLAB. The results are analyzed with input and output signals. Figure 1 shows the input signal (function) for the product of sine and cosine functions. Figure 2 shows the output signal  Figure 3 shows the output signal for = 0.5 with varying α. Figure 4 is the output signal for = 0.8 with varying α. (ii) α -(r+1) (t + rh) n 1 e -s 1/ν (t+rh) sin n 2 p(t + rh) cos n 3 q(t + rh) (e -s 1/ν h + α 2 e s 1/ν h -2α cos(UV )h) 1+s 1 .
Proof Taking the limit 0 to ∞ in (18) gives the Laplace transform of t n 1 sin n 2 pt cos n 3 qt.
Similarly, we can find results for other cases (odd-even, even-odd, even-even). The results are analyzed with input and output signals. Figure 5 shows the input signal (function) for the product of polynomial, sine, and cosine functions.  shows the output signal for = 0.2 with varying α. Figure 7 shows the output signal for = 0.3 and varying α. Figure 8 shows the output signal for = 0.8 with varying α.

Conclusions
We proposed formulas for the frequency Laplace transforms of the products of two and three functions and a multiseries formula for circular functions. Further, LFT is employed on circular functions to get appropriate results numerically and also analyzed the findings for different values of tuning factor α and fractional frequency factor s 1/ν . We also observed with the help of the diagrams generated by MATLAB that LFT gives innumerable outcomes for the given input signal, and this enables us to make a choice for an optimal