A New Class of Continued Fraction Expansions for the Ratios of Hypergeometric Functions

1956 ◽  
Vol 81 (2) ◽  
pp. 453 ◽  
Author(s):  
Evelyn Frank
Keyword(s):  
Hypergeometric Functions ◽  
Continued Fraction ◽  
New Class ◽  
Continued Fraction Expansions

scholarly journals Branched Continued Fraction Expansions of Horn’s Hypergeometric Function H3 Ratios

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 148
Author(s):  
Tamara Antonova ◽  
Roman Dmytryshyn ◽  
Victoriia Kravtsiv
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Special Functions ◽  
Convergence Criteria ◽  
Several Variables ◽  
Connected Set ◽  
Continued Fraction Expansions ◽  
Branched Continued Fraction

The paper deals with the problem of construction and investigation of branched continued fraction expansions of special functions of several variables. We give some recurrence relations of Horn hypergeometric functions H3. By these relations the branched continued fraction expansions of Horn’s hypergeometric function H3 ratios have been constructed. We have established some convergence criteria for the above-mentioned branched continued fractions with elements in R2 and C2. In addition, it is proved that the branched continued fraction expansions converges to the functions which are an analytic continuation of the above-mentioned ratios in some domain (here domain is an open connected set). Application for some system of partial differential equations is considered.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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