Heun's Triconfluent equation,
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can be transformed into another version of itself, that is, an equation with one regular and one irregular singularities respectively located at 0 and through transformations of the form
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where are new variables and . Under this transformation, the HeunT parameters transform according to -> , -> , -> . These transformations form a group of six elements and imply on identities, among which you have
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When, in HeunT(,,,z), , where is a positive integer, the th+1, th+2 and th+3 coefficients form a polynomial system for the remaining parameters and . When this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree for HeunT. 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 coefficients of for equal to 4, 5, and 6 in the series expansion are
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solving for and , requesting from solve to return using RootOf, you have
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substituting for instance the first of these two solutions in HT we have
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When the function admits a polynomial form, as is the case of HT_polynomial by construction, to obtain the actual polynomial of degree n (in this case n=3) use
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