Orthogonal Polynomials and Continued Fractions

Continued fractions, studied since Ancient Greece, only became a powerful tool in the eighteenth century, in the hands of the great mathematician Euler. This book tells how Euler introduced the idea of orthogonal polynomials and combined the two subjects, and how Brouncker's formula of 1655 can be derived from Euler's efforts in Special Functions and Orthogonal Polynomials. The most interesting applications of this work are discussed, including the great Markoff's Theorem on the Lagrange spectrum, Abel's Theorem on integration in finite terms, Chebyshev's Theory of Orthogonal Polynomials, and very recent advances in Orthogonal Polynomials on the unit circle. As continued fractions become more important again, in part due to their use in finding algorithms in approximation theory, this timely book revives the approach of Wallis, Brouncker and Euler and illustrates the continuing significance of their influence. A translation of Euler's famous paper 'Continued Fractions, Observation' is included as an Addendum.

• Considers the modern state of continued fractions and orthogonal polynomials from Euler's point of view, giving a full account of his work on the subject
• Outlines Brouncker's formula; Euler's discoveries of the Gamma and Beta functions; Markoff's Theorem on the Lagrange spectrum and its relation with Jean Bernoulli sequences; Brouncker's method as a solution to Fermat's question on Pell's equation
• Contains the first English translation of Euler's 'Continued Fractions, Observation', 1739, with comments relating it to Brouncker's proof