The integral formalism and the generating function of grand confluent hypergeometric function

Abstract

Biconfluent Heun (BCH) function, a confluent form of Heun function, is the special case of Grand Confluent Hypergeometric (GCH) function: this has a regular singularity at x=0, and an irregular singularity at infinity of rank 2. In this paper I apply three term recurrence formula (3TRF) [arXiv:1303.0806] to the integral formalism of GCH function including all higher terms of A_n's and the generating function for the GCH polynomial which makes B_n term terminated. I show how to transform power series expansion in closed forms of GCH equation to its integral representation analytically. This paper is 10th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 6 for all the papers in the series. The previous paper in the series describes the power series expansion in closed forms of GCH equation and its asymtotic behaviours. [arXiv:1303.0813]

31 pages, final version. arXiv admin note: substantial text overlap with arXiv:1303.0813, arXiv:1303.0820, arXiv:1303.0873, arXiv:1303.0879, arXiv:1303.0876, arXiv:1310.7811, arXiv:1302.7309, arXiv:1303.0878

References in corpus (8)

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

Cited by in corpus (8)

• Generalization of the three-term recurrence formula and its applications
• Asymptotic behavior of Heun function and its integral formalism
• Analytic solution for 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)