Classification of hypergeometric identities for π and other logarithms of algebraic numbers

This paper provides transcendental and algebraic framework for the classification of identities expressing π and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of (p+1)F(p) hypergeometric functions and their values are analyzed. The existing identities are explained, and new exhaustive classes of new ones are presented.