Hypergeometric type functions and their symmetries

Abstract

We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, we discuss recurrence relations of their solutions, their integral representations and discrete symmetries.

94 pages, 12 figures, revised version, accepted by Annales Henri Poincare

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• From Heun Class Equations to Painlevé Equations
• From Conformal Group to Symmetries of Hypergeometric Type Equations
• Point potentials on Euclidean space, hyperbolic space and sphere in any dimension
• Symbolic analysis of second-order ordinary differential equations with polynomial coefficients
• Effective binding potential from Casimir interactions: the case of the Bose gas
• Series solutions of linear ODEs by Newton-Raphson method on quotient $D$-modules
• Exactly solvable Schrödinger operators related to the confluent equation
• Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals
• Generalized integrals and point interactions
• On the Weyl symbol of the resolvent of the harmonic oscillator
• Equations of hypergeometric type in the degenerate case