New closed form pFq hypergeometric solutions for families of the General, Confluent and Bi-Confluent Heun differential equations.
Edgardo S. Cheb-Terrab, MITACS and MaplesoftMonday March 22nd, 2004 in K9509 at 3:30pm.
The General Heun equation, /gamma delta epsilon\ (alpha beta x - q) y y'' + |----- + ----- + -------| y' + -------------------- = 0 \ x x - 1 x - a / x (x - 1) (x - a) was first studied by K. Heun in 1885 as a generalization of the hypergeometric pFq second order equation. Heun's equation has four regular singularities, while the pFq equation has three. Through "confluence" processes, where singularities coalesce, four different confluent Heun equations can be obtained, namely: the Confluent, Biconfluent, Doubleconfluent and Triconfluent equations. These five multiparameter Heun equations include as particular cases the Lame, Mathieu, spheroidal wave and other well known equations of mathematical physics. The Heun family of equations has been popping up with surprising frequency in applications during the last 10 years, for example in general relativity, quantum, plasma ,atomic, molecular, and nano physics, to mention but a few. This has been pressing for related mathematical developments, and from some point of view, it would not be wrong to think that Heun equations will represent - in the XXI century - what the hypergeometric equations represented in the XX century. That is: a vast source of ideas for linear differential equations and developments for special functions. The solutions to these five Heun equations, however, are a matter of current research in various places, with results being presented every year. In this framework, this talk presents a completely new connection between Heun and hypergeometric pFq equations, which solves in terms of pFq functions the largest subfamilies of Heun's equations known at present. If time permits, the impact of these new solutions in a sample of works presented after 2000 in top-level Physics journals will be discussed.