Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 4000–4014 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Quasi associated continued fractions and Hankel determinants of Dixon elliptic functions via Sumudu transform Adem Kilicman a,∗ , Rathinavel Silambarasan b , Omer Altun c a Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia. b M. Tech IT-Networking, Department of Information Technology, School of Information Technology and Engineering, VIT University, Vellore, Tamilnadu, India. c Department of Mathematics and Institute for Mathematical research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia. Communicated by X.-J. Yang Abstract In this work, Sumudu transform of Dixon elliptic functions for higher arbitrary powers sm N (x, α); N  1, sm N (x, α)cm(x, α); N  0 and sm N (x, α)cm 2 (x, α); N  0 by considering modulus α = 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. Keywords: Dixon elliptic functions, quasi associated continued fractions, Hankel determinants, Sumudu transform, three term recurrence. 2010 MSC: 11A55, 11C20, 33E05, 44A10. 1. Introduction Jacobi elliptic functions sn(x, k), cn(x, k) and dn(x, k), there ratios were applied with Laplace trans- form to calculate Hankel determinants from their continued fractions in [26] and Fourier series of those functions were used to derive the different ways of writing square and triangular numbers [26]. Contin- ued fractions related to Hahn and orthogonal polynomials from the results of Laplace transform of Jacobi elliptic functions such as difference equations derived in [16]. Bimodular Jacobi elliptic functions treated with Laplace transform to solve as continued fractions and by using modular transformation results were traced back to Jacobi elliptic functions in [9]. Determinants of Bernoulli numbers, coefficients of Maclau- rin series of Jacobi elliptic functions were calculated through continued fractions in [2] and orthogonal polynomials from Fourier series of Jacobi elliptic functions in [8]. On studying the cubic curve x 3 + y 3 - 3αxy = 1 in [11] gave raise to elliptic functions sm(x, α) and cm(x, α) named after Dixon whose functions are doubly periodic, related properties and examples given ∗ Corresponding author Email addresses: akilic@upm.edu.my (Adem Kilicman), silambu_vel@yahoo.co.in (Rathinavel Silambarasan), omeraltun11@yahoo.com (Omer Altun) doi:10.22436/jnsa.010.07.49 Received 2017-02-11