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