1303.0830 (Yoon Seok Choun)
Yoon Seok Choun
We consider the power series expansion of Heun function (infinite series and polynomial), very precisely in an arrogant way, including all higher terms of $A_n$'s; applying three term recurrence formula by Choun. The Heun function has such a rich structure and include as particular so many functions; the Mathieu, Lame, Spheroidal Wave and hypergeometric $_2F_1$, $_1F_1$ and $_0F_1$ functions, the interrelationships between them and the Heun ones are a source of many nontrivial identities between the former. Even if the Heun equation was found by K. Heun in 1889, it was mostly disregarded in theoretical physics until lately. However, the Heun functions start to appear in modern theoretical physics currently; For example, in the Schrodinger equation with anharmonic potential, in the Stark effect, the Eguchi-Hanson case, Teukolsky equation in Kerr-Newman-de Sitter geometries, Kerr-Shen black hole problem, in crystalline materials, in three-dimensional waves in atmosphere, e.t.c. Besides, Heun function can be described as $_2F_1$ function in its integral forms analytically; We will publish the integral form of Heun function soon.
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http://arxiv.org/abs/1303.0830
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