scholarly journals Continued Fractions and Hankel Determinants from Hyperelliptic Curves

2020 ◽  
Author(s):  
Andrew N. W. Hone
Keyword(s):  
Continued Fractions ◽  
Hyperelliptic Curves ◽  
Hankel Determinants

scholarly journals Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

2019 ◽  
pp. 56-101
Author(s):  
Laura Capuano ◽  
Peter Jossen ◽  
Christina Karolus ◽  
Francesco Veneziano
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Hyperelliptic Curves ◽  
Pell Equation ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Algebraic Functions ◽  
Generalized Jacobians

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.

scholarly journals On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions

2020 ◽  
Vol 16 (10) ◽  
pp. 2187-2212
Author(s):  
Yining Hu ◽  
Guoniu Wei-Han
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Period Doubling ◽  
Converse Problem ◽  
Seminal Paper ◽  
Hankel Determinants ◽  
Automatic Sequence ◽  
Power Series Expansions ◽  
Partial Quotients ◽  
Continued Fraction Expansions

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo [Formula: see text]. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo [Formula: see text] are [Formula: see text]-automatic.

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