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In earlier papers [5,6,7,8] Gürsey 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 [7]. The theory for vanishing mass was solved using Confluent Hypergeometric functions [8].In this paper I construct the normalized wave function for the spin-free Hamiltonian with light quark masses (only up to the first order of the mass of quark). I develop the new kind of special function theory in mathematics that generalize all existing theories of Confluent Hypergeometric types. I call it the Grand Confluent Hypergeometric (GCH) Function. My solution produces previously unknown extra hidden radial quantum numbers relevant for description of supersymmetry and for generating new mass formulas.This paper is 1st out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The next paper in the series describes generalization of three term recurrence relation in linear ordinary differential equations and its applications [12].

Heun functions generalize well-known special functions such as Spheroidal Wave, Lamé, Mathieu, and hypergeometric-type functions. They are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solutions of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.We consider the radius of convergence of the Heun function, and we show why the Poincaré-Perron (P-P) theorem is not available for the absolute convergence since it is applied to the Heun's equation. Moreover, we construct the absolute convergence test in which is suitable for the three term recurrence relation in a power series.

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The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2 F 1 , 1 F 1 and 0 F 1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.In this paper I will apply three term recurrence formula (Choun, Y.S., arXiv:1303.0806., 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of A n 's. This paper is 3rd out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 5 for all the papers in the series. The previous paper in series deals with three term recurrence formula (3TRF). The next paper in the series describes the integral forms of Heun function and its asymptotic behaviors analytically.