q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson's transformation formula. A formal extension of Bailey's summation and an extension of the Sears terminating balanced transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears transformation formula and Heine's transformation formula.     

Some integrals of the Dedekind η-function

This note presents selected values of the class of integrals

     

On p-adic q-l-functions and sums of powers

In this paper, we give an explicit p-adic expansion of

     

Dual generalized Bernstein basis

The generalized Bernstein basis in the space Π_n of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78],

We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials P_k(x;a,b,ω/q;q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial (0jn) as a linear combination of min(j,n-j)+1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346].     

On p-adic interpolating function for q-Euler numbers and its derivatives

In this paper we study a two-variable p-adic q-l-function lp,q(s,t|χ) for Dirchlet's character χ, with the property that

     

Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials

For each the nth Laguerre polynomial has an m-fold zero at the origin when α=−m. As the real variable α→−m, it has m simple complex zeros which approach 0 in a symmetric way. This symmetry leads to a finite value for the limit of the sum of the reciprocals of these zeros. There is a similar property for the zeros of the q-Laguerre polynomials and of the Jacobi polynomials and similar results hold for sums of other negative integer powers.     

The L-L estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

We first obtain the L^p-L^q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation

     

Asymptotical behavior of one class of p-adic singular Fourier integrals

We study the asymptotical behavior of the p-adic singular Fourier integrals

     

Eigenvalues of p(x)-Laplacian Dirichlet problem

This paper studies the eigenvalues of the p(x)-Laplacian Dirichlet problem

     

Well-posedness of linear partial differential equations with unbounded delay operators

In this paper we show the well-posedness of the following constant delay equation:

     

Functional evolution equations

Of concern is the functional evolution equation

     

Positive solutions for a class of p-Laplacian systems with multiple parameters

Consider the system

     

Asymptotic stability in perturbed delay difference systems

Consider the delay difference system