graham higman has defined coset diagrams for psl(2,ℤ). these diagrams are composed of fragments, and the fragments are further composed of two or more circuits. q. mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

Hermite polynomials are one of the Apell and various results were found by researchers. Using Hermit combined with q-numbers, we derive different types differential equations study these equations. From equations, investigate some identities properties q-Hermite polynomials. We also find position roots under certain conditions their stacked structures. Furthermore, locate forms according to loo...

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using factorization approach. Families of differential equations, including differential, integro–differential, partial obtained these operators. The Volterra integral is also discovered.

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zgraded polynomials in the nth q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solvin...

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

Those homogeneous polynomials P are characterized for which for arbitrary lower order polynomials Q the partial differential operator (P+Q)(D) admits a continuous linear right inverse if regarded as an operator from the space of all C∞functions on R into itself. It is shown that P has this property if and only if P is of principal type and real up to a complex constant and has no elliptic factor.

In this paper, we deal with the distribution of zeros of q-shift difference polynomials of transcendental entire functions of zero order. At the same time we also investigate the uniqueness problems when two difference products of entire functions share one value with finite weight. The results of the paper improve and generalize some recent results due to Xu, Liu and Cao [Math. Commun. 20 (201...

Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p; q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p; q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p; q) are constructed, and three-dimensional first-...

in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...