Analytic solution for grand confluent hypergeometric function

Abstract

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 relation of GCH function. In this paper I will apply three term recurrence formula [3] to the power series expansion in closed forms of GCH function (infinite series and polynomial) including all higher terms of A_n's. In general most of well-known special function with two recursive coefficients only has one eigenvalue for the polynomial case. However this new function with three recursive coefficients has infinite eigenvalues that make B_n's term terminated at specific value of index n because of three term recurrence formula [3]. This paper is 9th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. Previous paper in series deals with generating functions of Lame polynomial in the Weierstrass's form [28]. The next paper in the series describes the integral formalism and the generating function of GCH function [30].

16 pages, final version. arXiv admin note: substantial text overlap with arXiv:1303.0819, arXiv:1303.0879, arXiv:1302.7309, arXiv:1303.0878, arXiv:1303.0876, arXiv:1303.0873

References in corpus (9)

• Heun Functions and Some of Their Applications in Physics
• Asymptotic behavior of Heun function and its integral formalism
• Generalization of the three-term recurrence formula and its applications
• The generating functions of Lame equation in Weierstrass's form
• The integral formalism and the generating function of grand confluent hypergeometric function
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Lame equation in the algebraic form
• Power series and integral forms of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)

Cited by in corpus (8)

• Generalization of the three-term recurrence formula and its applications
• Asymptotic behavior of Heun function and its integral formalism
• The integral formalism and the generating function of grand confluent hypergeometric function
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Lame equation in the algebraic form
• The generating functions of Lame equation in Weierstrass's form
• Power series and integral forms of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)