Using the concept of exponential monomial on Sturm–Liouville hypergroups we show that an important subclass of exponential monomials, the class of special exponential monomials has a linear independence property. The result can be reformulated as the linear independence of the derivatives with respect to the parameter of the solutions of eigenvalue problems for second order linear differential equations.

We present an algorithm for computing A-annihilated elements of the form QI[1] in H*QS0 where I runs through admissible sequences of positive excess. This is algorithm with polynomial time complexity to address a sub-problem of an unsolved problem in algebraic topology known as the hit problem of Peterson which is likely to be NP-hard.

Exponential monomials are the basic building bricks of spectral analysis and spectral synthesis on Abelian groups. Recently there have been some attempts to extend the most important spectral analysis and spectral synthesis results from groups to hypergroups. For this purpose it is necessary to introduce a reasonable concept of exponential monomials. In this work we reconsider this problem, and using a ring-theoretical approach we prove characterization theorems for particular function classes, which can be considered as “exponential monomials” on commutative hypergroups.

Let $I\subseteq\mathbb{R}$ be an interval and $\beta\colon I\to I$ a strictly increasing continuous function with a unique fixed point $s_0\in I$ satisfying $(t-s_0)(\beta(t)-t)\le 0$ for all $t\in I$. Hamza et al. introduced the general quantum difference operator $D_{\beta}$ by $D_{\beta}f(t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t}$ if $t\ne s_0$ and $D_{\beta}f(t):=f'(s_0)$ if $t=s_0$.   In this paper, we establish results concerning Taylor's formula associated with $D_{\beta}$. For this, we define two types of monomials and then present our main results. The obtained results are new in the literature and are useful for further research in the field.

Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103–120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122–130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

In the current work, we implement the meshless local radial point interpolation (MLRPI) method to find numerical solution of one-dimensional linear telegraph equations with variable coefficients. The MLRPI method, as a meshless technique, does not require any background integration cells and all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. Weak form formulation of the discretized equations has been constructed on local subdomains, hence the domain and boundary integrals in the weak form methods can easily be evaluated over the regularly shaped subdomains by some numerical quadratures. Radial basis functions augmented with monomials are used in to create shape functions. These shape functions have delta function property. Also the time derivatives is eliminated by using two-step finite differences approximation. Two illustrative numerical examples are given to show the stability and accuracy of the present method.

Let f be a transcendental entire function de, ned in the open complex plane C. A di, erence-monomial generated by f is an expression of the form F = fn(fm 􀀀,1) Yd j=1 (f(z + cj)), j, where n, m and , j are all non-negative integers. Now for the sake of de, niteness let us take, Mi[f] = fn(fm 􀀀,1) Yi j=1 (f(z + cj)), j, where 1 ,i ,d: If M1[f], M2[f], : : :,Mn[f] are such monomials in f as de, ned above, then [f] = a1M1[f] + a2M2[f] +: : : + anMn[f] where ai 6= 0 (i = 1,2, : : :, n) is called a di, erence-polynomial generated by f. In this paper, we compare the Valiron defect with the relative Nevan-linna defect of a particular type of di, erential-di, erence polynomial generated by a transcendental entire function with respect to integrated moduli of logarithmic derivative. Some examples are provided in order to justify the results obtained.