Related papers: Binomial transform and the backward difference
The Fractional Fourier Transform is a ubiquitous signal processing tool in basic and applied sciences. The Fractional Fourier Transform generalizes every property and application of the Fourier Transform. Despite the practical importance of…
By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting…
An identity for binomial symbols modulo an odd positive integer $n$ relating to the least prime factor of $n$ is proved. The identity is discussed within the context of Pell conics.
We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra,…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
In this short article, we study different problems described as initial value problems of discrete differential equations and develop a a transform method called the sigma transform, a discrete version of the continuous Laplace transform to…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's…
In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…
We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the…
The subject of time-band-limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
In this paper, we study some combinations of the degenerate and incomplete Stirling numbers of the second kind. We use a combinatorial approach and provide some asymptotic results.
In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like…
Stirling number of the first and the second kinds have seen many generalizations and applications in various areas of mathematics. We introduce some combinatorial parameters which realize $q$-analogues and Broder's $r$-variants of Stirling…
An index transform, involving the square of Whittaker's function is introduced and investigated. The corresponding inversion formula is established. Particular cases cover index transforms of the Lebedev type with products of the modified…
For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. However, we observe that if $ f\left( x\right) $ is a sum of two functions $f\left( x\right) =f_{1}\left( x\right)…
In the framework leading to the multiplicative anomaly formula ---which is here proven to be valid even in cases of known spectrum but non-compact manifold (very important in Physics)--- zeta-function regularisation techniques are shown to…
In this study, we apply "r" times the binomial transform to k-Lucas sequence. Also, the Binet formula, summation, generating function of this transform are found using recurrence relation. Finally, we give the properties of iterated…
Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic…