Quasi associated continued fractions and Hankel determinants of Dixon elliptic functions via Sumudu transform
TL;DR: In this paper, the Sumudu transform of Dixon elliptic functions for higher arbitrary powers was used for Hankel determinants calculations by connecting the formal power series (Maclaurin series) and associated continued fractions.
Abstract: In this work, Sumudu transform of Dixon elliptic functions for higher arbitrary powers smN(x,α);N > 1, smN(x,α)cm(x,α); N > 0 and smN(x,α)cm2(x,α);N > 0 by considering modulus α 6= 0 is obtained as three term recurrences and hence expanded as product of quasi associated continued fractions where the coefficients are functions of α. Secondly the coefficients of quasi associated continued fractions are used for Hankel determinants calculations by connecting the formal power series (Maclaurin series) and associated continued fractions. c ©2017 All rights reserved.
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TL;DR: In this paper, a discrete inverse Sumudu transform was used to solve an algebro-geometric equation and two new sets of exact analytical and complex solutions were obtained through a discrete inverted sumudu transformation, and Maple complex graphs were drawn to show the new solution simulations in the complex plane which were compared to the existing solutions.
Abstract: The discrete inverse Sumudu transform method is designed to solve ordinary differential equations and tested for an algebro-geometric equation. The two new sets of exact analytical and complex solutions are gotten through a discrete inverse Sumudu transform, and Maple complex graphs are drawn to show the new solution simulations in the complex plane which are compared to the existing solutions. The list of inverse Sumudu transforms is added in the sequel to strengthen the study.
5 citations
06 Jun 2018
TL;DR: In this paper, Hankel determinants of the Dixon elliptic function with non zero modulus a ≠ 0 for arbitrary powers are derived by product of Quasi C fractions.
Abstract: Sumudu transform of the Dixon elliptic function with non zero modulus a ≠ 0 for arbitrary powers smN(x,a) ; N ≥ 1 ; smN(x,a)cm(x,a) ; N ≥ 0 and smN(x,a)cm2(x,a) ; N ≥ 0 is given by product of Quasi C fractions. Next by assuming denominators of Quasi C fraction to 1 and hence applying Heliermann correspondance relating formal power series (Maclaurin series of Dixon elliptic functions) and regular C fraction, Hankel determinants are calculated and showed by taking a = 0 gives the Hankel determinants of regular C fraction. The derived results were back tracked to the Laplace transform of sm(x,a) ; cm(x,a) and sm(x,a)cm(x,a).
Cites background or methods from "Quasi associated continued fraction..."
...) are the polynomials in u and α given in [11]....
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...As QCF is dealing in this article, Quasi Associated continued fractions are considered which are discussed in detail in [11]....
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...Next by assuming the functions in the denominator of QCF we calculated the Hankel determinants H2 m of non-zero DEF without expanding their Maclaurin’s series by using Lemma 1 in which Hankel determinants H1 m are used from authors previous work [11]....
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...) are the Hankel determinants of Quasi associated continued fraction given in [11]....
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...) are polynomials in u and α given in [11]....
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17 Apr 2019
TL;DR: In this paper, the Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions.
Abstract: The Sumudu transform of the Dixon elliptic function with non-zero modulus α ≠ 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating formal power series (Maclaurin series of the Dixon elliptic function) and the regular C fraction, the Hankel determinants are calculated for the non-zero Dixon elliptic functions and shown by taking α = 0 to give the Hankel determinants of the Dixon elliptic function with zero modulus. The derived results were back-tracked to the Laplace transform of Dixon elliptic functions.
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Book•
01 Jun 1967
TL;DR: In this article, a convergence theory of positive definite continued fractions is presented. But the convergence theory is not a generalization of the Stieltjes convergence theorem, and the convergence of continued fractions whose partial denominators are equal to unity is not discussed.
Abstract: Part I: Convergence Theory: The continued fraction as a product of linear fractional transformations Convergence theorems Convergence of continued fractions whose partial denominators are equal to unity Introduction to the theory of positive definite continued fractions Some general convergence theorems Stieltjes type continued fractions Extensions of the parabola theorem The value region problem Part II: Function Theory: J-fraction expansions for rational functions Theory of equations J-fraction expansions for power series Matrix theory of continued fractions Continued fractions and definite integrals The moment problem for a finite interval Bounded analytic functions Hausdorff summability The moment problem for an infinite interval The continued fraction of Gauss Stieltjes summability The Pade table Bibliography Index.
1,640 citations
Book•
01 Jan 1989
TL;DR: In this paper, Jacobi's Zeta and Epsilon functions are presented as functions of the squared modulus of the Equation (1) of the Squared Modulus.
Abstract: 1 Theta Functions.- 2 Jacobi's Elliptic Functions.- 3 Elliptic Integrals.- 4 Geometrical Applications.- 5 Physical Applications.- 6 Weierstrass's Elliptic Function.- 7 Applications of the Weierstrass Functions.- 8 Complex Variable Analysis.- 9 Modular Transformations..- Appendix A Fourier Series for a Periodic Analytic Function.- Appendix B Calculation of a Definite Integral.- Appendix C BASIC Program for Reduction of Elliptic Integral to Standard Form.- Appendix D Computation of Tables.- Table A. Theta Functions.- Table B. Nome and Complete Integrals of the First and Second Kinds as Functions of the Squared Modulus.- Table D. Legendre's Incomplete Integrals of First and Second Kinds.- Table E. Jacobi's Zeta and Epsilon Functions.- Table F. Sigma Functions.
853 citations
"Quasi associated continued fraction..." refers methods in this paper
...In [22] Dixon functions connecting trefoil curve and Weierstrass elliptic functions [22, 23] have been described....
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Book•
09 Mar 2009
TL;DR: In this paper, the authors present a method for representing analytic functions by continued fractions and apply it to the birth-death process (BDP) in the context of continuous fractions.
Abstract: 1. Introduction 2. Elementary Properties of Continued Fractions 3. Continued Fractions 4. Convergence of Continued Fractions 5. Methods for Representing Analytic Functions by Continued Fractions 6. Representations of Analytic Functions by Continued Fractions 7. Types of Corresponding Continued Fractions and Related Algorithms 8. Truncation-Error Analysis 9. Asymptotic Expansions and Moment Problems 10. Numerical Stability in Evaluating Continued Fractions 11. Application of Continued Fractions to Birth-Death Processes 12. Miscellaneous Results.
705 citations
"Quasi associated continued fraction..." refers methods in this paper
...Then the following m ×m matrices are defined [9, 17, 26], whose determinants are denoted by respective H m and χm:...
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...The matrix for χm is obtained from the matrix for H (1) m+1 by deleting the last row and next to last column [9, 17, 26]....
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Book•
01 Jan 1960
640 citations
"Quasi associated continued fraction..." refers background in this paper
...Determinants H (n) m and χm [27] are named as persymmetric determinants (or) Turanian determinants (or) Hankel determinants....
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TL;DR: In this paper, the authors introduce the concept of continuoustime fractions and three-term recurrence relations, and propose convergence criteria and convergence expansions of continued fraction expansions, and apply them in number theory.
Abstract: I. Introductory Examples. II. More Basics. III. Convergence Criteria. IV. Continued Fractions and Three-Term Recurrence Relations. V. Correspondence of Continued Fractions. VI. Hypergeometric Functions. VII. Moments and Orthogonality. VIII. Pade Approximants. IX. Some Applications in Number Theory. X. Zero-free Regions. XI. Digital Filters and Continued Fractions. XII. Applications to Some Differential Equations. Appendix: Some Continued Fraction Expansions. References. Index.
383 citations