Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Polynomial solutions of differential equations | Azad et al. Advances in Difference Equations 2011 2011 58 http content 2011 1 58 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Polynomial solutions of differential equations H Azad A Laradji and M T Mustafa Correspondence hassanaz@ Department of Mathematics Statistics King Fahd University of Petroleum Minerals Dhahran Saudi Arabia Springer Abstract A new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the k N form L y ak x y k where ak is a polynomial of degree k over an infinite field k 0 F has all eigenvalues in F in the space of polynomials of degree at most n for all n. If these eigenvalues are distinct then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L for every non-negative integer n. Specializing to the real field the potential of the method is illustrated by recovering Bochner s classification of second order ODEs with polynomial coefficients and polynomial solutions as well as cases missed by him - namely that of Romanovski polynomials which are of recent interest in theoretical physics and some Jacobi type polynomials. An important feature of this approach is the simplicity with which the eigenfunctions and their orthogonality and norms can be determined resulting in significant reduction in computational complexity of such problems. 2000 MSC 33C45 34A05 34A30 34B24. 1 Introduction Polynomial solutions of differential equations is a classical subject going back to Routh 1 Bochner 2 and Brenke 3 and it continues to be of interest in applications as in . 4 5 . The idea we wish to present in this article is to conduct the discussion of differential equations with polynomial coefficients in a linear algebraic context. It is surprising that by such a change of view point one can add more than what is available in the .
