New closed form pFq hypergeometric solutions for families of the General, Confluent and Bi-Confluent Heun differential equations.
Edgardo S. Cheb-Terrab, MITACS and Maplesoft
Monday 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.