On Binomial Transform of the Generalized Fifth Order Pell Sequence

In this paper, we define the binomial transform of the generalized fifth order Pell sequence and as special cases, the binomial transform of the fifth order Pell and fifth order Pell-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’s formulas, generating functions, Simson formulas, recurrence properties, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.

Binomial Transform of the Generalized Fourth Order Pell Sequence

In this study, we define the binomial transform of the generalized fourth order Pell sequenceand as special cases, the binomial transform of the fourth order Pell and fourth order Pell-Lucassequences will be introduced. We investigate their properties in details.

Analysis of Decoherence in Linear and Cyclic Quantum Walks

We theoretically analyze the case of noisy Quantum walks (QWs) by introducing four qubit decoherence models into the coin degree of freedom of linear and cyclic QWs. These models include flipping channels (bit flip, phase flip and bit-phase flip), depolarizing channel, phase damping channel and generalized amplitude damping channel. Explicit expressions for the probability distribution of QWs on a line and on a cyclic path are derived under localized and delocalized initial states. We show that QWs which begin from a delocalized state generate mixture probability distributions, which could give rise to useful algorithmic applications related to data encoding schemes. Specifically, we show how the combination of delocalzed initial states and decoherence can be used for computing the binomial transform of a given set of numbers. However, the sensitivity of QWs to noisy environments may negatively affect various other applications based on QWs.

Notes on Binomial Transform of the Generalized Narayana Sequence

In this paper, we define the binomial transform of the generalized Narayana sequence and as special cases, the binomial transform of the Narayana, Narayana-Lucas, Narayana-Perrin sequences will be introduced. We investigate their properties in details.

ON THE BINOMIAL TRANSFORMS OF THE HORADAM QUATERNION 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 dierent type of binomial transform.

Binomial Transform of the Generalized Tribonacci Sequence

In this paper, we define the binomial transform of the generalized Tribonacci sequence and as special cases, the binomial transform of the Tribonacci, Tribonacci-Lucas, Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.

Special equation from Binomial transform and nth order finite difference

Using numerical analysis and tables, nth order backward difference of exponential function is obtained. Further analyzing the obtained equation yields a special identity given as \[\sum\limits_{k\, = \,0}^n {{{( - 1)}^k}\,\frac{{{{(x - \,k)}^{n\,\, - m}}}}{{(n\, - \,k - m)!\,k!}}\,} \, = \,\,\sum\limits_{k\, = \,0}^{n\, - m} {{{( - 1)}^k}\,\frac{{{{(x - \,k)}^{n\, - m}}}}{{(n\, - \,k - m)!\,k!}}\,} = \,\,1\,\,\,\,\,\,:\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\,\,\,\,\,\,\,\,\,\,\,\,\,x \in R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{n \in \,W{\rm{ }}}\\{m\, \in \,W\,\,{\rm{:}}\,\,m \le \,n}\end{array}} \right.\]This equation yields the value of negative integer factorial, zero factorial and zero to the power zero which is currently indeterminate forms along with other unknown values. With this equation, we also conclude the product of zero and infinity is unity within certain parameters.

On a generalized function-to-sequence transform

By attaching a sequence {?n}n?N0 to the binomial transform, a new operator D? is obtained. We use the same sequence to define a new transform T? mapping derivatives to the powers of D?, and integrals to D-1?. The inverse transform B? of T? is introduced and its properties are studied. For ?n = (-1)n, B? reduces to the Borel transform. Applying T? to Bessel's differential operator d/dx x d/dx, we obtain Bessel's discrete operator D?nN?. Its eigenvectors correspond to eigenfunctions of Bessel's differential operator.