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HeunD

The Heun Doubleconfluent function

HeunDPrime

The derivative of the Heun Doubleconfluent function

 Calling Sequence HeunD($\mathrm{\alpha }$, $\mathrm{\beta }$, $\mathrm{\gamma }$, $\mathrm{\delta }$, z) HeunDPrime($\mathrm{\alpha }$, $\mathrm{\beta }$, $\mathrm{\gamma }$, $\mathrm{\delta }$, z)

Parameters

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

Description

 • The HeunD function is the solution of the Heun Doubleconfluent equation. Following the first reference (at the end), the equation and the conditions at the origin satisfied by HeunD are
 ${\mathrm{HeunD}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({z}\right){-}\frac{\left({\mathrm{\alpha }}{}{{z}}^{{4}}{-}{2}{}{{z}}^{{5}}{+}{4}{}{{z}}^{{3}}{-}{\mathrm{\alpha }}{-}{2}{}{z}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_Y}}{}\left({z}\right)\right)}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}{-}\frac{\left({-}{{z}}^{{2}}{}{\mathrm{\beta }}{+}\left({-}{\mathrm{\gamma }}{-}{2}{}{\mathrm{\alpha }}\right){}{z}{-}{\mathrm{\delta }}\right){}{\mathrm{_Y}}{}\left({z}\right)}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}{0}\right\}\right)$ (1)
 • The HeunD($\mathrm{\alpha }$,$\mathrm{\beta }$,$\mathrm{\gamma }$,z) function is a local solution to Heun's Doubleconfluent equation, computed as a standard power series expansion around the origin, a regular point. Because of the presence of two irregular singularities located at -1 and 1, the radius of convergence of this series is $\left|z\right|<1$. An analytic continuation of HeunD outside the unit circle is obtained through the identity
 $\left[{\mathrm{HeunD}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{HeunD}}{}\left({-}{\mathrm{\alpha }}{,}{-}{\mathrm{\delta }}{,}{-}{\mathrm{\gamma }}{,}{-}{\mathrm{\beta }}{,}\frac{{1}}{{z}}\right)\right]$ (2)
 • The Doubleconfluent Heun Equation (DHE) above is obtained from the Confluent Heun Equation (CHE) through an additional confluence process, with the two regular singularities of the CHE coalescing into one irregular singularity at the origin. The resulting Heun equation, with two irregular singularities at 0 and $\mathrm{\infty }$, is further transformed using $x$ -> $\frac{x+1}{x-1}$, relocating these singularities symmetrically at -1 and 1, leaving the origin as a regular point. The Doubleconfluent equation, thus, has a structure of singularities that can be transformed into that of the 0F1 hypergeometric equation and particular cases of HeunD are related to the Bessel functions.

Examples

Heun's Doubleconfluent equation,

 > $\mathrm{DHE}≔\mathrm{diff}\left(y\left(z\right),\mathrm{}\left(z,2\right)\right)=\frac{-2{z}^{5}+4{z}^{3}+4\mathrm{\alpha }{z}^{4}-2z-4\mathrm{\alpha }}{{\left(z-1\right)}^{3}{\left(z+1\right)}^{3}}\mathrm{diff}\left(y\left(z\right),z\right)+\frac{\left(-4\mathrm{\beta }-2{\mathrm{\alpha }}^{2}-4\mathrm{\delta }-4\mathrm{\gamma }+1\right){z}^{2}+8\left(\mathrm{\beta }-\mathrm{\alpha }-\mathrm{\gamma }\right)z-4\mathrm{\gamma }+4\mathrm{\delta }-4\mathrm{\beta }+2{\mathrm{\alpha }}^{2}-1}{{\left(z-1\right)}^{3}{\left(z+1\right)}^{3}}y\left(z\right)$
 ${\mathrm{DHE}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right){=}\frac{\left({4}{}{\mathrm{\alpha }}{}{{z}}^{{4}}{-}{2}{}{{z}}^{{5}}{+}{4}{}{{z}}^{{3}}{-}{4}{}{\mathrm{\alpha }}{-}{2}{}{z}\right){}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({z}\right)\right)}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}{+}\frac{\left(\left({-}{2}{}{{\mathrm{\alpha }}}^{{2}}{-}{4}{}{\mathrm{\beta }}{-}{4}{}{\mathrm{\delta }}{-}{4}{}{\mathrm{\gamma }}{+}{1}\right){}{{z}}^{{2}}{+}{8}{}\left({\mathrm{\beta }}{-}{\mathrm{\alpha }}{-}{\mathrm{\gamma }}\right){}{z}{-}{4}{}{\mathrm{\gamma }}{+}{4}{}{\mathrm{\delta }}{-}{4}{}{\mathrm{\beta }}{+}{2}{}{{\mathrm{\alpha }}}^{{2}}{-}{1}\right){}{y}{}\left({z}\right)}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}$ (3)

can be transformed into another version of itself, that is, an equation with two irregular singularities located at -1 and 1 through transformations of the form

 > $z=\frac{\left(\mathrm{\sigma }+1\right)t+\mathrm{\sigma }-1}{\left(\mathrm{\sigma }-1\right)t+\mathrm{\sigma }+1},y\left(z\right)=\mathrm{exp}\left(\frac{\mathrm{\kappa }\left(t+1\right)}{t-1}+\frac{\mathrm{\rho }}{t+1}\left(t-1\right)\right)u\left(t\right)$
 ${z}{=}\frac{\left({\mathrm{\sigma }}{+}{1}\right){}{t}{+}{\mathrm{\sigma }}{-}{1}}{\left({\mathrm{\sigma }}{-}{1}\right){}{t}{+}{\mathrm{\sigma }}{+}{1}}{,}{y}{}\left({z}\right){=}{{ⅇ}}^{\frac{{\mathrm{\kappa }}{}\left({t}{+}{1}\right)}{{t}{-}{1}}{+}\frac{{\mathrm{\rho }}{}\left({t}{-}{1}\right)}{{t}{+}{1}}}{}{u}{}\left({t}\right)$ (4)

where $\left\{t,u\left(t\right)\right\}$ are new variables, and ${\mathrm{\sigma }}^{4}=1$, ${\mathrm{\kappa }}^{2}=\frac{\mathrm{\sigma }\mathrm{\epsilon }\mathrm{\alpha }\mathrm{\kappa }}{4}$. Under this transformation, the HeunD parameters transform according to $\mathrm{\alpha }$ = $2\mathrm{\kappa }-\frac{\mathrm{\sigma }\mathrm{\epsilon }\mathrm{\alpha }}{4}$, $\mathrm{\beta }$ = $\frac{\mathrm{\epsilon }\mathrm{\gamma }+\mathrm{\beta }+\mathrm{\delta }}{16\mathrm{\sigma }}$, $\mathrm{\gamma }$ = $\frac{\left(-\mathrm{\epsilon }\mathrm{\gamma }+\mathrm{\beta }+\mathrm{\delta }\right)\mathrm{\sigma }}{16}$ and $\mathrm{\delta }$ = $\frac{\left(\frac{\left(\mathrm{\beta }-\mathrm{\delta }-\frac{{\mathrm{\epsilon }}^{2}{\mathrm{\alpha }}^{2}}{4}\right)}{2}+1\right)}{4}$, where ${\mathrm{\epsilon }}^{2}=1$.

These transformations form a group of 32 elements and imply on identities, among which you have

 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{HeunD}\right)$
 $\left[{\mathrm{HeunD}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{HeunD}}{}\left({-}{\mathrm{\alpha }}{,}{-}{\mathrm{\delta }}{,}{-}{\mathrm{\gamma }}{,}{-}{\mathrm{\beta }}{,}\frac{{1}}{{z}}\right)\right]$ (5)

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.