Related papers: The integral formalism and the generating function…
In previous paper I construct an approximative solution of the power series expansion in closed forms of Grand Confluent Hypergeometric (GCH) function only up to one term of A_n's [4]. And I obtain normalized constant and orthogonal…
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in…
In the previous series "Special functions and three term recurrence formula (3TRF)", I generalize the three term recurrence relation in the linear differential equation for the infinite series and polynomial which makes B_n term terminated…
Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent…
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric_2F_1,_1F_1 and_0F_1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice…
In the first series "Special functions and three term recurrence formula (3TRF)", I show how to obtain power series solutions of Heun, Grand Confluent Hypergeoemtric (GCH), Mathieu and Lame equations for an infinite series and a polynomial…
I consider the power series expansion of Lame function in the Weierstrass's form and its integral forms applying three term recurrence formula[1]. I investigate asymptotic expansions of Lame function for the cases of infinite series and…
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a…
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The…
We review the series solutions of the general and single-confluent Heun equations in terms of powers, ordinary-hypergeometric and confluent-hypergeometric functions. The conditions under which the expansions reduce to finite sums as well as…
We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric…
In earlier papers [3,4,5,6] Gursey et al. showed development of a bilocal baryon-meson field from two quark-antiquark fields. The Hamiltonian in the case of vanishing quark masses was shown to have a very good agreement with experiments…
We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the…
We show that in the particular case when a characteristic exponent of the singularity at infinity is zero the solution of the general Heun equation can be expanded in terms of the incomplete Beta functions. By means of termination of the…
The confluent Heun equation is one of 4 confluent forms of Heun's differential equation in which is the Fuchsian equation of second order with four regular singularities. A confluent Heun function is applicable to diverse areas such as…
We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. Infinitely many cases for which a solution of the biconfluent equation is presented as an irreducible linear…
We show that there exist infinitely many particular choices of parameters for which the three-term recurrence relations governing the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions…
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different…
We derive a concise closed-form solution for a linear three-term recurrence relation. Such recurrence relations are very common in the quantitative sciences, and describe finite difference schemes, solutions to problems in Markov processes…