A simple algorithm for expanding a power series as a continued fraction

Abstract

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).

arXiv commentLaTeX2e, 48 pages. Version 2 contains a few additional historical remarks, and adds a new Remark 4 at the end of Section 10. To be published in Expositiones Mathematicae

References in corpus (9)

• De seriebus divergentibus
• The leading root of the partial theta function
• Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions
• Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions
• The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion
• The Euler and Springer numbers as moment sequences
• De transformatione seriei divergentis 1 - mx + m(m+n)x^2 - m(m+n)(m+2n)x^3 + etc. in fractionem continuam
• Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials
• Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

Cited by in corpus (4)

• When does a hypergeometric function ${}_p{F}_q$ belong to the Laguerre--Pólya class $L{P}^{+}$?
• Lattice paths and branched continued fractions. III. Generalizations of the Laguerre, rook and Lah polynomials
• Continued fractions for cycle-alternating permutations
• Continued fractions for $q$-deformed real numbers, $\{-1,0,1\}$-Hankel determinants, and Somos-Gale-Robinson sequences