Heun's Biconfluent equation,
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can be transformed into another version of itself, that is, an equation with one regular and one irregular singularity respectively located at 0 and through transformations of the form
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where are new variables, and . Under this transformation, the HeunB parameters transform according to -> , -> , -> and -> . These transformations form a group and imply on a number of identities, among which you have
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A relation between HeunB and the confluent 1F1 hypergeometric function is
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When, in HeunB(,,,,z), , with a positive integer, the th coefficient in the series expansion is a polynomial in of order . If is a root of that polynomial, that th coefficient and the subsequent ones are zero. The series then truncates and HeunB reduces to a polynomial. For example, this is the necessary condition for a polynomial form
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Considering the first non-trivial case, for , the function is
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So the coefficient of in the series expansion is
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solving for , requesting from solve to return using RootOf, you have
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substituting in we have
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When the function admits a polynomial form, as is the case of by construction, to obtain the actual polynomial of degree (in this case ) use
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| (15) |