Unrestricted Lichtenberg Hybrid Sequences

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

In this paper, bicomplex $k$-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex $k$-Fibonacci quaternions are investigated. For example, the summation formula, generating functions, Binet's formula, the Honsberger identity, the d'Ocagne's identity, Cassini's identity, Catalan's identity for these quaternions are given. In the last part, a different way to find $n-th$ term of the bicomplex $k$-Fibonacci quaternion sequence was given using the determinant of a tridiagonal matrix.

The aim of the study is to obtain new binomial transforms for the $k-$ Narayana sequence. The first of these is the binomial transform, which is its normal form, and in the first step, after finding the recurrence relation of this new binomial transform, the generating function and Binet formula were obtained. Finally, Pascal's triangle was calculated. In the rest of the article, $k-$binomial transform was performed for the $k-$ Narayana sequence and the recurrence relation, generating function, Binet formula and Pascal's triangle were examined for the new sequence obtained. Then, by performing the falling binomial transform and the rising binomial transform, the features listed above were found again for these sequences.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform.

In this paper, we introduced the extended generalized Fibonacci polynomial sequence \{$Y_{2,n}$\} and extended generalized Tribonacci polynomial \{$Y_{3,n}$\} with arbitrary initial values and established a recursive matrix and then presented some properties of these. Further, we investigated some well-known identities like Binet's formula, Catalan's identity, Cassini's identity, d'Ocagne's identity, generating function, explicit sum formula, sum of first $n$ terms for the extended generalized Fibonacci polynomial sequence and extended generalized Tribonacci polynomial sequence.

Recently Lavoie, Grondin and Rathie obtained ten results closely related to the classical Kummer's theorem as special cases from generalized Whipple's theorem on the sum of a ${}_3F_2$ with unit argument. The aim of this paper is to provide general summation formulas contiguous to the Kummer's theorem by simply using a known integral representation of ${}_2F_1$. As by-product, two classes of summation formulas closely related to the Kummer's theorem were obtained. Some simplified special cases were also given for later easy use.

In this paper, we investigate the hyperbolic Fibonacci sequence and the hyperbolic Fibonacci numbers. Furthermore, we give recurrence relations, the golden ratio and Binet's formula for the hyperbolic Fibonacci sequence and Lorentzian inner product, cross product and mixed product for the hyperbolic Fibonacci vectors.

The processes involved in the biogeochemical cycle of Fe in the oceans have been intensely discussed in recent decades because this element is limiting to primary productivity in most oceanic regions. From biogeochemical and ecological perspectives, inputs from anthropogenic sources, especially mining activities, may be more representative than natural inputs in coastal areas affected by metal loads from tailings. Here we provide a review of all the stages related to Fe behaviour in marine ecosystems, including Fe input sources, which may be of natural and (or) anthropogenic origin; input rates; chemical speciation; bioavailability; and changes in the phytoplankton community structure. To allow conceptualization of the anthropogenic processes, the collapse of the Fundão tailings dam (southeast Brazil) was used as a case study of one of the worst environmental disasters of the mining industry. From this perspective, the interrelations among the chemical, biological, and ecological components were discussed. Regarding the chemical component, Fe speciation must be determined by the input of several other materials, mainly organic compounds that can be complexed to this element and increase its solubility. From a biological perspective, the biochemical and physiological processes used for the assimilation of this element, such as the reduction in cell membranes and the production of chelating substances (such as siderophores), will also determine the forms of this element present in the water column. On the other hand, the groups that obtain a competitive advantage due to these assimilation strategies must be dominant in the system. Synergistic effects are also expected with other materials such as the inorganic nutrients, organic compounds, and metals that are carried to the coastal region together with Fe. In the specific case of mine tailings, the accumulation of this material in the river banks and bed should cause an increase in Fe input from other sources, such as atmospheric transport and submarine groundwater discharge, as well as river discharge by erosion and transport under increased river flow conditions. The iron fluxes from mining areas to coastal oceans and the effects of these loads to phytoplankton ecological aspects should be investigated.

Polynomial representation formulae for power sums of the extended Fibonacci-Lucas numbers are established, which include, as special cases, four formulae for odd power sums of Melham type on Fibonacci and Lucas numbers, obtained recently by Ozeki and Prodinger (2009).

In this paper, the Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are obtained by using the recurrence relation satisfied by them. Then, some linear and quadratic relations are given between Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials.

The Euler–Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler–Maclaurin summation formula and the Poisson summation formula as special cases. By making use of the modified formula, a possible numerical summation method is obtained and the remainder can be controlled. The modified formula is also generalized from one dimension to two dimensions. Approximate expressions of partition functions of a classical particle in square well in 1D and 2D and that of a quantum rotator are obtained with error estimation.

A q-summation formula, the continuous q-Hahn polynomials and the big q-Jacobi polynomials

Local equilibrium of winds and wind-waves is discussed as a basis for research of the drag coefficient of the water surface as well as for the spectral growth of wind-waves. This hypothesis assumes, in a narrow sense, that statistical properties are determined from four physical quantities, which represent winds and wind-waves: the friction velocityu *, the gravitational accelerationg, the powerE of the surface displacement, and the peak frequencyω p of a wind-wave spectrum. Then one has only one nondimensionalcontrol parameter, which may be either the wave age or wave nonlinearity (slope) of dominant waves. In a wide-sense, one can take into account viscosity and surface tension in terms of one more additional parameter by virtue of the virtual invariance of those material constants; that parameter describes the scale ratio between dominant waves and the short waves for which viscosity or surface tension is important. A unified expression for the roughness height according to this hypothesis turns out to include Charnock's and Toba's formulas as special cases. On the basis of a preliminary analysis of the experimental data, a new empirical formula is proposed.

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

Summation and recurrence formula involving the central factorial numbers and zeta function