Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions

Abstract

We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different expansions obey four-, five-, or six-term recurrence relations that are reduced to ones involving less number of terms only in a few exceptional cases. The conditions for deriving finite-sum solutions via termination of the series are discussed.

References in corpus (11)

• Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions
• On reducing the Heun equation to the hypergeometric equation
• Fifteen classes of solutions of the quantum two-state problem in terms of the confluent Heun function
• New solutions of Heun general equation
• Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial
• Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
• Parametric transformations between the Heun and Gauss hypergeometric functions
• Incomplete beta-function expansions of the solutions to the confluent Heun equation
• Confluent and Double-Confluent Heun Equations: Convergence of Solutions in Series of Coulomb Wavefunctions
• Integral representation of solutions to Fuchsian system and Heun's equation
• Transformations of Heun's equation and its integral relations

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• Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions
• The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential
• The third exactly solvable hypergeometric quantum-mechanical potential
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• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients
• Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations
• A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations
• Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
• On the derivatives of the Heun functions
• A conditionally integrable Schrödinger potential of a bi-confluent Heun class
• Series solutions of confluent Heun equations in terms of incomplete Gamma-functions
• Generalized-hypergeometric solutions of the biconfluent Heun equation