ANTIQUANTIZATION AND INTEGRAL RELATIONS FOR HEUN EQUATIONS AND SYMMETRIES FOR THE CORRESPONDING PAINLEVE EQUATIONS
Sergey Yuryevich Slavyanov, Alexandr Ya. Kazakov, Filip R. Vukajlovic
Last modified: 2010-03-30
Abstract
Special functions play significant role in Computer
Algebra packages. Here we can mention all-purpose
packages like Mathematika or Maple as well as
specialized packages as SFTools.
Further development would without doubt be focused
on more sophisticated Heun functions and closely
related Painleve transcendents. Partly the
relationship between Heun equations and Painleve
equations is presented in the
package SFTools. However new studies induce revision
of presentation of these relations. The
items of these revisions are the following.
1. Relations between equations belonging to Heun class,
the corresponding deformed equations with added
apparent singularity and the corresponding 2x2 systems
and 3x3 systems of first order equations.
2. Integral transforms linking
different equations belonging to Heun class.
3. Antiquantization as a tool of derivation
Painleve-type equations from linear equations.
4. Derivation of Okamoto-type symmetries
for Painleve equations.
Algebra packages. Here we can mention all-purpose
packages like Mathematika or Maple as well as
specialized packages as SFTools.
Further development would without doubt be focused
on more sophisticated Heun functions and closely
related Painleve transcendents. Partly the
relationship between Heun equations and Painleve
equations is presented in the
package SFTools. However new studies induce revision
of presentation of these relations. The
items of these revisions are the following.
1. Relations between equations belonging to Heun class,
the corresponding deformed equations with added
apparent singularity and the corresponding 2x2 systems
and 3x3 systems of first order equations.
2. Integral transforms linking
different equations belonging to Heun class.
3. Antiquantization as a tool of derivation
Painleve-type equations from linear equations.
4. Derivation of Okamoto-type symmetries
for Painleve equations.