## Analytic solution for grand confluent hypergeometric function    [PDF]

Yoon Seok Choun
We consider the exact analytic solution of grand confluent hypergeometric function including all higher terms of $A_n$'s; applying three term recurrence formula by Choun. We prove an approximative solution of this function only up to one term of $A_n$'s by Choun \textit{et} \textit{al}.The current paper extends this function more than one term of $A_n$'s mathematically. Also, in general, most of well-known special function only has one eigenvalue for the polynomial case. However, this new function has two species eigenvalues, further one makes $A_n$'s term terminated at specific value of index n and latter one makes $B_n$'s term terminated at specific value of index n. Also, the number of each species eigenvalue are infinity surprisingly: because it involves three term recurrence formula proved by Choun. Biconfluent Heun function is the special case of this function replacing $\mu$ and $\varepsilon \omega$ by 1 and $-q$: this has a regular singularity at $x=0$, and an irregular singularity at $\infty$ of rank 2. For example, Biconfluent Heun function is included in the radial Schrodinger equations associated to some quantum-mechanical systems (rotating harmonic oscilator, confinement potentials): recently it's appeared in many physics and mathematics areas.
View original: http://arxiv.org/abs/1303.0813