No Arabic abstract
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We present a new implementation of dAlemberts reduction of order method, applying it to linear second-order recursion equations. Further, we introduce an iterative method to obtain a general solution, giving two linearly independent polynomial solutions to the recurrence relation. In the case of a particular confluent hypergeometric function for which the standard second solution is not independent of the first, i.e. the solutions are degenerate, we use the corresponding differential equation and apply the extended Cauchy-integral method to find a polynomial second solution for the difference equation. We show that the standard dAlembert method also generates this polynomial solution.
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.
This contribution deals with the sequence ${mathbb{U}_{n}^{(a)}(x;q,j)}_{ngeq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide sever
We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-th order differential equation and give a simple method for obtaining the differential equation. Also we determine the exponents of the differential equation and derive a bound for the degree of the polynomials, which are coefficients in the differential equation. The bound corresponds to the order of differential equation satisfied by the n-fold convolution of the Fourier transform of the function. These results are applied to some probability density functions used in statistics.