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Ince's limits for confluent and double-confluent Heun equations

B. D. Bonorino Figueiredo
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Abstract

We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable $z$, while the other is given by a series of modified Bessel functions and converges for $|z|>|z_{0}|$, where $z_{0}$ denotes a regular singularity. For short, the preceding limit is called Ince's limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when $z_{0}$ tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schrödinger equation for inverse fourth and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince's limit, respectively.

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B. D. Bonorino Figueiredo. 2005-09-08. Ince's limits for confluent and double-confluent Heun equations. https://doi.org/10.1063/1.2104267

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