## Special Functions: A Unified Theory Based on SingularitiesSergej Ju. Slavjanov, Sergej Jur'evič Slavjanov, Sergeĭ I{u0361}Urʹevich Slavi{u0361}anov, Sergeĭ I͡Urʹevich Slavi͡anov, Sergej U. Slavianov, Sergei Yuryevitsh Slavyanov, S. Iu Slavianov, Wolfgang Lay The subject of this book is the theory of special functions, not considered as a list of functions exhibiting a certain range of properties, but based on the unified study of singularities of second-order ordinary differential equations in the complex domain. The number and characteristics of the singularities serve as a basis for classification of each individual special function. Links between linear special functions (as solutions of linear second-order equations), and non-linear special functions (as solutions of Painleve equations) are presented as a basic and new result. Many applications to different areas of physics are shown and discussed. The book is written from a practical point of view and will address all those scientists whose work involves applications of mathematical methods. Lecturers, graduate students and researchers will find this a useful text and reference work. " |

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### Contents

Linear secondorder ODEs with polynomial coefficients | 1 |

The hypergeometric class of equations | 55 |

The Heun class of equations | 97 |

Application to physical sciences | 163 |

The Painleve class of equations | 232 |

### Common terms and phrases

arbitrary asymptotic behaviour asymptotic factors asymptotic series avoided crossings Birkhoff set Birkhoff solutions calculation canonical form central two-point connection characteristic exponents coefficients confluence process confluent equations confluent Heun equation confluent hypergeometric function convergence corresponding CTCP denoted difference equation eigenfunctions eigensolution eigenvalue eigenvalue condition eigenvalue curves elementary singularities equation of Poincare-Perron equations belonging expansion finite formulae Frobenius solutions Fuchsian equation Fuchsian singularity gamma function Gegenbauer polynomials gravitational singularity Heun class Heun equation hypergeometric class hypergeometric equation independent variable integral equation Jacobi polynomials Laguerre polynomials Lemma linear matrix elements Mobius transformation movable singularities normal obtained Painleve equations parameters particular solutions plane Poincare-Perron type potential recurrence relation reduced confluent regular singularity relevant singularities result s-homotopic transformation s-rank second-order self-adjoint form SFTools singly confluent singularity at infinity solution of eqn special functions Stokes studied Suppose Theorem Thome solutions triconfluent two-point connection problem values zero