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HeunT

The Heun Triconfluent function

HeunTPrime

The derivative of the Heun Triconfluent function

 Calling Sequence HeunT($\mathrm{\alpha }$, $\mathrm{\beta }$, $\mathrm{\gamma }$, z) HeunTPrime($\mathrm{\alpha }$, $\mathrm{\beta }$, $\mathrm{\gamma }$, z)

Parameters

 $\mathrm{\alpha }$ - algebraic expression $\mathrm{\beta }$ - algebraic expression $\mathrm{\gamma }$ - algebraic expression z - algebraic expression

Description

 • The HeunT function is the solution of the Heun Triconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunT are
 ${\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({z}\right){-}\left({3}{}{{z}}^{{2}}{+}{\mathrm{\gamma }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({z}\right)\right){-}\left(\left({-}{\mathrm{\beta }}{+}{3}\right){}{z}{-}{\mathrm{\alpha }}\right){}{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}{0}\right\}\right)$ (1)
 • The HeunT($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z) function is a local solution to Heun's Triconfluent equation, computed as a standard power series expansion around the origin, a regular point. Because the single singularity is located at $\mathrm{\infty }$, this series converges in the whole complex plane.
 • The Triconfluent Heun Equation (THE) above is obtained from the Doubleconfluent Heun Equation (DHE) through a confluence process, that is, a process where two singularities coalesce, performed by redefining parameters and taking limits. In this case the two irregular singularities of the DHE are coalesced into one irregular singularity at $\mathrm{\infty }$. The resulting Heun Triconfluent equation, thus, has the structure of singularities f of the 0F1 hypergeometric equation and so can be related to the Airy functions.
 • A special case happens when in HeunT($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z), the second parameter satisfies $\mathrm{\beta }=3\left(n+1\right)$, where $n$ is a positive integer. In this case the $n$th+1, $n$th+2 and $n$th+3 coefficients form a polynomial system for the remaining parameters $\mathrm{\alpha }$ and $\mathrm{\gamma }$; when this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree $n$ for HeunT. Remark: for $n=0$ this situation leads to a constant, for $n=1$ HeunT will also be a constant since its series expansion satisfies $\mathrm{HeunT}\text{'}$ at 0 = 0 and for $n=2$ the polynomial system for $\mathrm{\alpha }$ and $\mathrm{\gamma }$ is inconsistent. So the non-trivial polynomial forms of HeunT are of degree $3\le n$.

Examples

Heun's Triconfluent equation,

 > $\mathrm{THE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)=\left(3{z}^{2}+\mathrm{\gamma }\right)\mathrm{diff}\left(y\left(z\right),z\right)+\left(\left(3-\mathrm{\beta }\right)z-\mathrm{\alpha }\right)y\left(z\right)$
 ${\mathrm{THE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){=}\left({3}{}{{z}}^{{2}}{+}{\mathrm{\gamma }}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right)\right){+}\left(\left({3}{-}{\mathrm{\beta }}\right){}{z}{-}{\mathrm{\alpha }}\right){}{y}{}\left({z}\right)$ (2)

can be transformed into another version of itself, that is, an equation with one regular and one irregular singularities respectively located at 0 and $\mathrm{\infty }$ through transformations of the form

 > $z=\mathrm{\kappa }t,y\left(z\right)=\mathrm{exp}\left(\frac{\frac{1}{2}t\left(\mathrm{\kappa }-1\right)\left({\mathrm{\kappa }}^{2}+\mathrm{\kappa }+1\right)\left(\mathrm{\gamma }+{t}^{2}\right)}{{\mathrm{\kappa }}^{3}}\right)u\left(t\right)$
 ${z}{=}{\mathrm{\kappa }}{}{t}{,}{y}{}\left({z}\right){=}{{ⅇ}}^{\frac{{t}{}\left({\mathrm{\kappa }}{-}{1}\right){}\left({{\mathrm{\kappa }}}^{{2}}{+}{\mathrm{\kappa }}{+}{1}\right){}\left({{t}}^{{2}}{+}{\mathrm{\gamma }}\right)}{{2}{}{{\mathrm{\kappa }}}^{{3}}}}{}{u}{}\left({t}\right)$ (3)

where $\left\{t,u\left(t\right)\right\}$ are new variables and ${\mathrm{\kappa }}^{6}=1$. Under this transformation, the HeunT parameters transform according to $\mathrm{\alpha }$ -> $\frac{\mathrm{\alpha }}{{\mathrm{\kappa }}^{2}}$, $\mathrm{\beta }$ -> $\frac{\mathrm{\beta }}{{\mathrm{\kappa }}^{3}}$, $\mathrm{\gamma }$ -> $\frac{\mathrm{\gamma }}{{\mathrm{\kappa }}^{4}}$. These transformations form a group of six elements and imply on identities, among which you have

 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{HeunT}\right)$
 $\left[\left[{\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{z}\right){=}{\mathrm{HeunT}}{}\left({j}{}{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{{j}}^{{2}}{}{\mathrm{\gamma }}{,}{j}{}{z}\right){,}{{j}}^{{3}}{=}{1}\right]{,}\left[{\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{z}\right){=}{\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{-}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{-}{z}\right){}{{ⅇ}}^{{{z}}^{{3}}}{,}{\mathrm{\gamma }}{=}{0}\right]\right]$ (4)

When, in HeunT($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z), $\mathrm{\beta }=3\left(n+1\right)$, where $n$ is a positive integer, the $n$th+1, $n$th+2 and $n$th+3 coefficients form a polynomial system for the remaining parameters $\mathrm{\alpha }$ and $\mathrm{\gamma }$. When this system is identically satisfied all the subsequent coefficients cancel too and the series truncates, resulting in a polynomial form of degree $n$ for HeunT. For example, this is the necessary condition for a polynomial form

 > $\mathrm{HeunT}\left(\mathrm{\alpha },3n+3,\mathrm{\gamma },z\right)$
 ${\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{3}{}{n}{+}{3}{,}{\mathrm{\gamma }}{,}{z}\right)$ (5)

Considering the first non-trivial case, for $n=3$, the function is

 > $\mathrm{HT}≔\mathrm{subs}\left(n=3,\right)$
 ${\mathrm{HT}}{≔}{\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{12}{,}{\mathrm{\gamma }}{,}{z}\right)$ (6)

So the coefficients of ${z}^{m}$ for $m$ equal to 4, 5, and 6 in the series expansion are

 > $Q≔\mathrm{simplify}\left(\mathrm{series}\left(\mathrm{HT},z,7\right),\mathrm{size}\right)$
 ${Q}{≔}{1}{-}\frac{{1}}{{2}}{}{\mathrm{\alpha }}{}{{z}}^{{2}}{+}\left({-}\frac{{\mathrm{\gamma }}{}{\mathrm{\alpha }}}{{6}}{-}\frac{{3}}{{2}}\right){}{{z}}^{{3}}{+}\left(\frac{{1}}{{24}}{}{{\mathrm{\alpha }}}^{{2}}{-}\frac{{1}}{{24}}{}{{\mathrm{\gamma }}}^{{2}}{}{\mathrm{\alpha }}{-}\frac{{3}}{{8}}{}{\mathrm{\gamma }}\right){}{{z}}^{{4}}{-}\frac{{1}}{{120}}{}\left({\mathrm{\gamma }}{}{\mathrm{\alpha }}{+}{9}\right){}\left({{\mathrm{\gamma }}}^{{2}}{-}{2}{}{\mathrm{\alpha }}\right){}{{z}}^{{5}}{+}\left({-}\frac{{{\mathrm{\alpha }}}^{{3}}}{{720}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}{}{{\mathrm{\alpha }}}^{{2}}}{{240}}{+}\frac{\left({-}{{\mathrm{\gamma }}}^{{4}}{+}{27}{}{\mathrm{\gamma }}\right){}{\mathrm{\alpha }}}{{720}}{-}\frac{{{\mathrm{\gamma }}}^{{3}}}{{80}}\right){}{{z}}^{{6}}{+}{O}{}\left({{z}}^{{7}}\right)$ (7)
 > $\mathrm{c4},\mathrm{c5},\mathrm{c6}≔\mathrm{coeff}\left(Q,z,4\right),\mathrm{coeff}\left(Q,z,5\right),\mathrm{coeff}\left(Q,z,6\right)$
 ${\mathrm{c4}}{,}{\mathrm{c5}}{,}{\mathrm{c6}}{≔}\frac{{1}}{{24}}{}{{\mathrm{\alpha }}}^{{2}}{-}\frac{{1}}{{24}}{}{{\mathrm{\gamma }}}^{{2}}{}{\mathrm{\alpha }}{-}\frac{{3}}{{8}}{}{\mathrm{\gamma }}{,}{-}\frac{\left({\mathrm{\gamma }}{}{\mathrm{\alpha }}{+}{9}\right){}\left({{\mathrm{\gamma }}}^{{2}}{-}{2}{}{\mathrm{\alpha }}\right)}{{120}}{,}{-}\frac{{{\mathrm{\alpha }}}^{{3}}}{{720}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}{}{{\mathrm{\alpha }}}^{{2}}}{{240}}{+}\frac{\left({-}{{\mathrm{\gamma }}}^{{4}}{+}{27}{}{\mathrm{\gamma }}\right){}{\mathrm{\alpha }}}{{720}}{-}\frac{{{\mathrm{\gamma }}}^{{3}}}{{80}}$ (8)

solving for $\mathrm{\alpha }$ and $\mathrm{\gamma }$, requesting from solve to return using RootOf, you have

 > $\mathrm{_EnvExplicit}≔\mathrm{false}$
 ${\mathrm{_EnvExplicit}}{≔}{\mathrm{false}}$ (9)
 > $\mathrm{subs}\left(\mathrm{ga}=\mathrm{\gamma },\left[\mathrm{solve}\left(\mathrm{subs}\left(\mathrm{\gamma }=\mathrm{ga},\left\{\mathrm{c4},\mathrm{c5},\mathrm{c6}\right\}\right),\left\{\mathrm{\alpha },\mathrm{ga}\right\}\right)\right]\right)$
 $\left[\left\{{\mathrm{\alpha }}{=}{0}{,}{\mathrm{\gamma }}{=}{0}\right\}{,}\left\{{\mathrm{\alpha }}{=}\frac{{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{36}\right)}^{{2}}}{{2}}{,}{\mathrm{\gamma }}{=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{3}}{+}{36}\right)\right\}\right]$ (10)

substituting for instance the first of these two solutions in HT we have

 > $\mathrm{HT_polynomial}≔\mathrm{subs}\left(\left[1\right],\mathrm{HT}\right)$
 ${\mathrm{HT_polynomial}}{≔}{\mathrm{HeunT}}{}\left({0}{,}{12}{,}{0}{,}{z}\right)$ (11)

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

 > $\mathrm{eval}\left(,\mathrm{HeunT}=\mathrm{HeunT}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\right)$
 ${1}{-}\frac{{3}{}{{z}}^{{3}}}{{2}}$ (12)

References

 Decarreau, A.; Dumont-Lepage, M.C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes Canoniques de Equations confluentes de l'equation de Heun". Annales de la Societe Scientifique de Bruxelles. Vol. 92 I-II, (1978): 53-78.
 Ronveaux, A. ed. Heun's Differential Equations. Oxford University Press, 1995.
 Slavyanov, S.Y., and Lay, W. Special Functions, A Unified Theory Based on Singularities. Oxford Mathematical Monographs, 2000.