Hyperbolic functions obtained from k-Jacobsthal sequences

In this paper, an expansion of the classical hyperbolic functions is presented and studied. Also, many features of the [Formula: see text]-Jacobsthal hyperbolic functions are given. Finally, we introduced some graph and curved surfaces related to the [Formula: see text]-Jacobsthal hyperbolic functions.

Exponentials of general multivector in 3D Clifford algebras

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

Approximations of Tangent Polynomials, Tangent –Bernoulli and Tangent – Genocchi Polynomials in terms of Hyperbolic Functions

Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.

Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

Low-Latency Hardware Implementation of High-Precision Hyperbolic Functions sinhx and coshx Based on Improved CORDIC Algorithm

CORDIC algorithm is used for low-cost hardware implementation to calculate transcendental functions. This paper proposes a low-latency high-precision architecture for the computation of hyperbolic functions sinhx and coshx based on an improved CORDIC algorithm, that is, the QH-CORDIC. The principle, structure, and range of convergence of the QH-CORDIC are discussed, and the hardware circuit architecture of functions sinhx and coshx using the QH-CORDIC is plotted in this paper. The proposed architecture is implemented using an FPGA device, showing that it has 75% and 50% latency overhead over the two latest prior works. In the synthesis using TSMC 65 nm standard cell library, ASIC implementation results show that the proposed architecture is also superior to the two latest prior works in terms of total time (latency × period), ATP (area × total time), total energy (power × total time), energy efficiency (total energy/efficient bits), and area efficiency (efficient bits/area/total time). Comparison of related works indicates that it is much more favorable for the proposed architecture to perform high-precision floating-point computations on functions sinhx and coshx than the LUT method, stochastic computing, and other CORDIC algorithms.

Robust adaptive filtering algorithms based on (inverse)hyperbolic sine function

Recently, adaptive filtering algorithms were designed using hyperbolic functions, such as hyperbolic cosine and tangent function. However, most of those algorithms have few parameters that need to be set, and the adaptive estimation accuracy and convergence performance can be improved further. More importantly, the hyperbolic sine function has not been discussed. In this paper, a family of adaptive filtering algorithms is proposed using hyperbolic sine function (HSF) and inverse hyperbolic sine function (IHSF) function. Specifically, development of a robust adaptive filtering algorithm based on HSF, and extend the HSF algorithm to another novel adaptive filtering algorithm based on IHSF; then continue to analyze the computational complexity for HSF and IHSF; finally, validation of the analyses and superiority of the proposed algorithm via simulations. The HSF and IHSF algorithms can attain superior steady-state performance and stronger robustness in impulsive interference than several existing algorithms for different system identification scenarios, under Gaussian noise and impulsive interference, demonstrate the superior performance achieved by HSF and IHSF over existing adaptive filtering algorithms with different hyperbolic functions.

On generalized Bessel–Maitland function

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by $\mathcal{J}_{\nu , \lambda }^{\mu }$ J ν , λ μ , where $\mu >0$ μ > 0 and $\lambda ,\nu \in \mathbb{C\ }$ λ , ν ∈ C get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions $\mathcal{I}_{\nu ,\lambda }^{\mu }$ I ν , λ μ defined by $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$ I ν , λ μ ( z ) = i − 2 λ − ν J ν , λ μ ( i z ) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$ I ν , λ μ ( z ) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

On a half-discrete Hilbert-type inequality related to hyperbolic functions

By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.