The present work consists of two parts. First, methods for proving non-holonomicity of sequences and functions are discussed. Holonomic sequences, which are defined by linear difference equations with polynomial coefficients, are a popular object of study in combinatorics. The presentation of methods for finding out whether a given sequence is in this class or not takes the reader through number theory, complex analysis, and analytic combinatorics. The second part of the book is devoted to inequalities involving recursively defined sequences. It contains a new computer algebra assisted proving procedure. This approach, which is based on cylindrical algebraic decomposition, allows to prove many classical inequalities mechanically. Moreover, the oscillation behavior of constant coefficient recurrences is investigated with number-theoretic tools, Fourier series and results from the theory of uniform distribution. The intended audience consists of professional mathematicians with an interest in combinatorics, number theory, or computer algebra.

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