On a continued fraction of Ramanujan, two and four-square theorems and bibasic q-Appell functions

2021 ◽  
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Continued Fraction ◽  
Four Square ◽  
Appell Functions

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals Factors associated with dynamic balance in people with Persistent Postural Perceptual Dizziness (PPPD): a cross-sectional study using a virtual-reality Four Square Step Test

2021 ◽  
Vol 18 (1) ◽  
Author(s):  
Moshe M. H. Aharoni ◽  
Anat V. Lubetzky ◽  
Liraz Arie ◽  
Tal Krasovsky
Keyword(s):  
Virtual Reality ◽  
Trait Anxiety ◽  
Dynamic Balance ◽  
Step Test ◽  
Simulator Sickness ◽  
Balance Confidence ◽  
Task Duration ◽  
Head Kinematics ◽  
Perceived Disability ◽  
Four Square

Abstract Background Persistent postural-perceptual dizziness (PPPD) is a condition characterized by chronic subjective dizziness and exacerbated by visual stimuli or upright movement. Typical balance tests do not replicate the environments known to increase symptoms in people with PPPD—crowded places with moving objects. Using a virtual reality system, we quantified dynamic balance in people with PPPD and healthy controls in diverse visual conditions. Methods Twenty-two individuals with PPPD and 29 controls performed a square-shaped fast walking task (Four-Square Step Test Virtual Reality—FSST-VR) using a head-mounted-display (HTC Vive) under 3 visual conditions (empty train platform; people moving; people and trains moving). Head kinematics was used to measure task duration, movement smoothness and anterior–posterior (AP) and medio-lateral (ML) ranges of movement (ROM). Heart rate (HR) was monitored using a chest-band. Participants also completed a functional mobility test (Timed-Up-and-Go; TUG) and questionnaires measuring anxiety (State-Trait Anxiety Inventory; STAI), balance confidence (Activities-Specific Balance Confidence; ABC), perceived disability (Dizziness Handicap Inventory) and simulator sickness (Simulator Sickness Questionnaire). Main effects of visual load and group and associations between performance, functional and self-reported outcomes were examined. Results State anxiety and simulator sickness did not increase following testing. AP-ROM and HR increased with high visual load in both groups (p < 0.05). There were no significant between-group differences in head kinematics. In the high visual load conditions, high trait anxiety and longer TUG duration were moderately associated with reduced AP and ML-ROM in the PPPD group and low ABC and  high perceived disability were associated with reduced AP-ROM (|r| =  0.47 to 0.53; p < 0.05). In contrast, in controls high STAI-trait, low ABC and longer TUG duration were associated with increased AP-ROM (|r| = 0.38 to 0.46; p < 0.05) and longer TUG duration was associated with increased ML-ROM (r = 0.53, p < 0.01). Conclusions FSST-VR may shed light on movement strategies in PPPD beyond task duration. While no main effect of group was observed, the distinct associations with self-reported and functional outcomes, identified using spatial head kinematics, suggest that some people with PPPD reduce head degrees of freedom when performing a dynamic balance task. This supports a potential link between spatial perception and PPPD symptomatology.

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.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Calculating astronomical refraction by means of continued fractions

1979 ◽  
Vol 89 ◽  
pp. 95-101
Author(s):  
S. Mikkola
Keyword(s):  
Asymptotic Expansion ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Astronomical Refraction

A continued fraction was derived for the summation of the asymptotic expansion of astronomical refraction. Using simple approximations for the last denominator of the fraction, accurate formulae, useful down to the horizon, were obtained. The method is not restricted to any model of the atmosphere and can thus be used in calculations based on actual aerological measurements.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

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