A new conformable nabla derivative and its application on arbitrary time scales

In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form $(t-a)^{(\eta )}$ ( t − a ) ( η ) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ y ( t ) = y ( t , f ( t ) ) , and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

On the finiteness of solutions for polynomial-factorial Diophantine equations

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that {l!} is represented by {N_{A}(x)}, where {N_{A}} is a norm form constructed from the field norm of a field extension {K/\mathbf{Q}}. We also deal with the equation {N_{A}(x)=l!_{S}}, where {l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of {\mathbf{Z}} and for any polynomial {P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation {P(x)=l!_{S}} has only finitely many solutions {(x,l)}. For some special infinite subsets S of {\mathbf{Z}}, we can show the finiteness of solutions for the equation {P(x)=l!_{S}} unconditionally.

Performances Analysis of Novel Proposed Code for SAC-OCDMA System

The development of SAC-OCDMA system is based on the creation of new code structure. A novel proposed code structure is called ZFD that is based on the combination of various matrix creation methods and especially on the factorial function. This approach has not been exploited before in the OCDMA system. This novel code has its own advantage that represented in its possibility to generate code-words in flexible way by setting variable basic codes at fixed code weight. The performance of this proposed code is demonstrated via mathematical methods. It is represented by its optimal cross-correlation to avoid the multiple access interference (MAI). Consequently, the effects of the contributed noises using direct detection method have been considered. These noises are represented by shot noise and thermal noise. In addition, a system of three users was simulated with ZFD code under the C band (1530–1565 nm) for the upstream signal with different channel spacing (0.8, 1 and 1.2 nm). In this paper, several parameter effects such as fiber length, effective source power and data rate have been studied and compared to prove the effectiveness of this code. Also, ZFD code has been compared to some other codes such as RD, EDW and MDW according to the previous parameters. The suggested code has reached good results in terms of the bit error rate (BER) and the eye diagram.

A Study on Novel Extensions for the $p$-adic Gamma and $p$-adic Beta Functions

In this paper, we introduce the (ρ,q)-analogue of the p-adic factorial function. By utilizing some properties of (ρ,q)-numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic (ρ,q)-gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2. We also derive more representations of the p-adic (ρ,q)-gamma function in general case. Moreover, we consider the p-adic (ρ,q)-Euler constant derived from the derivation of p-adic (ρ,q)-gamma function at x = 1. Furthermore, we provide a limit representation of aforementioned Euler constant based on (ρ,q)-numbers. Finally, we consider (ρ,q)-extension of the p-adic beta function via the p-adic (ρ,q)-gamma function and we then investigate various formulas and identities.

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities.

Hurwitz-Lerch Type Multi-Poly-Cauchy Numbers

In this paper, we define Hurwitz–Lerch multi-poly-Cauchy numbers using the multiple polylogarithm factorial function. Furthermore, we establish properties of these types of numbers and obtain two different forms of the explicit formula using Stirling numbers of the first kind.

Improvements of asymptotic approximation formulas for the factorial function

Asymptotic expansions of the gamma function are studied and new accurate approximations for the factorial function are given.