Sumudu transform of Dixon elliptic functions with non-zero modulus as Quasi C fractions and its Hankel determinants

Sumudu transform of the Dixon elliptic function with non zero modulus α , 0 for arbitrary powers smN(x,α) ; N ≥ 1 , smN(x,α)cm(x,α) ; N ≥ 0 and smN(x,α)cm2(x,α) ; 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 α = 0 gives the Hankel determinants of regular C fraction. The derived results were back tracked to the Laplace transform of sm(x,α) , cm(x,α) and sm(x,α)cm(x,α).


Introduction
To determine the coeffecients in the Maclaurin series of Jacobi elliptic functions, continued fractions and the Heilermann correspondence the relation employing Formal Power Series (FPS) and its continued fraction to calculate Hankel determinants are used in [2], also determinants of Bernoulli numbers were calculated from the correspondence in [2]. By using continued fraction and Fourier series expansions of Jacobi elliptic functions in [13] obtained orthogonal polynomials which are related to each other through multiplication formulas of Jacobi elliptic functions in [13]. Laplace transform of Jacobi elliptic functions expanded as continued fractions and shown their coeffecients are orthogonal polynomials and derived dual Hahn polynomials in [19]. Fourier series and continued fractions expansions of ratis of Jacobi elliptic functions and their Hankel determinants are given in [25] from which different ways of representing sum of square numbers derived in determinant forms in [25]. Laplace transform of bimodular Jacobi elliptic functions expanded as continued fractions in [14] and by modular transformation results were back tracked to unimodular Jacobi elliptic functions in [14].
A. C. Dixon studied the cubic curve x 3 + y 3 − 3αxy = 1 ; α −1 for the orthogonal polynomials, where the curve has double period in [16] which then give raise to two set of elliptic functions sm(x, α) and cm(x, α) now known as Dixon Elliptic Functions (DEF). The examples, its relation to hypergeometric series, modular transformation and formulae for their ratio given in [17]. When α = 0 in the above cubic curve, their series expansions and transformations studied in [18]. DEF were used in the study of conformal mapping and geographical structure of world maps in [1] addition and multiplication formulae for DEF are derived in [1]. Laplace transform applied for DEF for both the cases of α = 0 and α 0 to expand as set of continued fractions in [14]. The above cubis curve and its relation to Fermat curve is studied for the Urn representation and combinatorics in [15]. Number theory related results followed by [25] for factorial of numbers using DEF given in [4]. DEF relation to trefoil curves and relation to Weierstrass and its derivative functions shown in [23].
Fractional heat equations are solved using Sumudu transform in [3]. Sumudu transform embedded in decomposition method in [27] and in homotopy perturbation method to solve Klein-Gordon equations in [26]. Fractional Maxwell's equations solved with Sumudu transform in [28] and some differential equations with Sumudu transform in [29]. Fractional gas dynamics differential equations using Sumudu transform is solved in [5]. Sumudu transform definition for trigonometric functions and its infinite series expansions proved with examples comprising tables and properties in [6]. Maxwell's coupled equations solved with Sumudu transform for magnetic field solutions in TEMP waves given in [7]. Without using any of decomposition, perturbation (or) analysis techniques Sumudu transform of functions calculated by differentiating the function in [8]. Symbolic C++ program for Sumudu transform given in [8]. Sumudu transform applied for bimodular Jacobi elliptic functions [14] for arbitrary powers in [9] as associated continued fraction and their Hankel determinants. Applying modular transformation, Sumudu transform of tan(x) and sec(x) derived in [9].
In this work Sumudu transform applied for DEF of 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 and expanded as Quasi C Fractions (QCF). Using the numerator coeffecients of QCF, Hankel determinants are calculated by the correspondence connecting FPS and Regular C fractions through Sumudu transform.
When the sequence β (n) is constant then C-fraction is called Regular C fraction. And QCF has the following form. .
Sometimes the coeffecient a n = a n (u) thus the coeffecients are functions of u which can be seen in the main results of this work.
Then the following m × m matrices are defined [14,20,25], whose determinants are denoted by respective H The matrix for χ m is obtained from the matrix for H The relation between FPS and Regular C fraction is given as lemma [14], (Theorem 7.2, pp 223-226, [20]), [25]. Lemma 1. When the Regular C fraction converges to FPS.
Assume denominators of Theorem 1 to 1. Now applying Eq (8) of Lemma 1 to the coeffecients of Theorem 1 where H (.) are the Hankel determinants of quasi associated continued fraction given in [11]. Now iterating and simplifying Hankel determinants of Eqs (9) -(13) given by respective enumerates of Theorem 4.
(.) are the Hankel determinants of Quasi associated continued fraction given in [11]. Iterating and simplifying leads to the Hankel determinants of Eqs (22) - (27) by respective enumerates of Theorem 5. Theorem 6. Hankel determinants corresponding to Theorem 3 given by the following equations:

Conclusion
In this research work we applied the Sumudu integral transform to non-zero modulus Dixon elliptic functions to derive general three term recurrences from which Quasi C fractions expanded. Next by assuming the functions in the denominator of QCF we calculated the Hankel determinants H 2 m of non-zero DEF without expanding their Maclaurin's series by using Lemma 1 in which Hankel determinants H 1 m are used from authors previous work [11]. Results and discussions section ensures our asumptions are correct and gives the previous results. It remains the open query of Sumudu transform of cm 3 (x, α) and other higher powers as they lead to four term recurrences. Secondly if the assumption of denominators to 1 is restricted for the Hankel determinants of QCF is another open query which will be the further study from this work.