International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 94204, 10 pages
Identity-Type Functions and Polynomials
1Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of the College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan
Received 6 June 2006; Revised 30 November 2006; Accepted 4 March 2007
Academic Editor: Heinrich Begehr
Copyright © 2007 M. Aslam Chaudhry and Asghar Qadir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a noncommuting product of functions, similar to convolutions, an “identity-type function” leaving a specific function invariant is defined. It is evaluated for any choice of function on which it acts by solving a functional equation. A closed-form representation for the identity-type function of is obtained, which is a solution of a second-order linear differential equation with given boundary conditions. It yields orthogonal polynomials whose graphs are also given. The relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed.