The five Second Order Linear Heun equations and the corresponding Heun function solutions - Maple Programming Help

The five Second Order Linear Heun equations and the corresponding Heun function solutions

Description

 The five multiparameter Heun equations have been popping up with surprising frequency in applications during the last 15 years. Heun equations include as particular cases the Lame, Mathieu, spheroidal wave, hypergeometric, and with them most of the known equations of mathematical physics.
 Five Heun functions are defined as the solutions to each of these five Heun equations, computed as power series solutions around the origin satisfying prescribed initial conditions.

Examples

The five Heun equations and their singularities

 The General Heun equation, with four regular singular points, is
 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{singularities}\right)$
 $\left[{\mathrm{singularities}}\right]$ (1)
 > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(y\left(z\right),\mathrm{prime}=z,\mathrm{quiet}\right)$
 > $\mathrm{GHE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)+\left(\frac{\mathrm{\gamma }}{z}+\frac{\mathrm{\delta }}{z-1}+\frac{\mathrm{\epsilon }}{z-a}\right)\mathrm{diff}\left(y\left(z\right),z\right)+\frac{\mathrm{\alpha }\mathrm{\beta }z-q}{z\left(z-1\right)\left(z-a\right)}y\left(z\right)=0$
 ${\mathrm{GHE}}{≔}{\mathrm{y\text{'}\text{'}}}{+}\left(\frac{{\mathrm{\gamma }}}{{z}}{+}\frac{{\mathrm{\delta }}}{{z}{-}{1}}{+}\frac{{\mathrm{\epsilon }}}{{z}{-}{a}}\right){}{\mathrm{y\text{'}}}{+}\frac{\left({\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{-}{q}\right){}{y}}{{z}{}\left({z}{-}{1}\right){}\left({z}{-}{a}\right)}{=}{0}$ (2)
 > $\mathrm{singularities}\left(\mathrm{GHE}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{a}{,}{\mathrm{\infty }}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (3)
 The solution to this equation is implemented in Maple as the HeunG function. The sum of the exponents of the singularities of Heun's equation is equal to two and the parameter $\mathrm{\epsilon }$ is expressed in terms of the other ones by
 > $\mathrm{\epsilon }=\mathrm{\alpha }+\mathrm{\beta }+1-\mathrm{\gamma }-\mathrm{\delta }$
 ${\mathrm{\epsilon }}{=}{\mathrm{\alpha }}{+}{\mathrm{\beta }}{+}{1}{-}{\mathrm{\gamma }}{-}{\mathrm{\delta }}$ (4)
 The choice of standard form for the GHE is uniform in the literature except in one particular: the numerator of the coefficient of $y$ is sometimes presented as $\mathrm{\alpha }\mathrm{\beta }\left(z-q\right)$. The Maple choice of standard form, and so the definition of HeunG implemented, follows Chapter 1 in [2] (see references at the end) and has the small but significant advantage that you can take $\mathrm{\alpha }=0$ (or $\mathrm{\beta }=0$) without eliminating the term in $y$ completely.
 The other four Heun equations are confluent cases, obtained from the general Heun equation above through confluence processes. These are: the Heun Confluent equation
 > $\mathrm{CHE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)+\left(\frac{\mathrm{\gamma }}{z}+\frac{\mathrm{\delta }}{z-1}-\mathrm{\epsilon }\right)\mathrm{diff}\left(y\left(z\right),z\right)+\left(\frac{q-\mathrm{\alpha }\mathrm{\beta }}{z-1}-\frac{q}{z}\right)y\left(z\right)=0$
 ${\mathrm{CHE}}{≔}{\mathrm{y\text{'}\text{'}}}{+}\left(\frac{{\mathrm{\gamma }}}{{z}}{+}\frac{{\mathrm{\delta }}}{{z}{-}{1}}{-}{\mathrm{\epsilon }}\right){}{\mathrm{y\text{'}}}{+}\left(\frac{{-}{\mathrm{\alpha }}{}{\mathrm{\beta }}{+}{q}}{{z}{-}{1}}{-}\frac{{q}}{{z}}\right){}{y}{=}{0}$ (5)
 > $\mathrm{singularities}\left(\mathrm{CHE}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (6)
 having for solution the HeunC function; the Biconfluent equation
 > $\mathrm{BHE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)+\left(-2z-\mathrm{\beta }+\frac{1+\mathrm{\alpha }}{z}\right)\mathrm{diff}\left(y\left(z\right),z\right)+\left(\mathrm{\gamma }-\mathrm{\alpha }-2-\frac{\frac{1}{2}\left(\left(1+\mathrm{\alpha }\right)\mathrm{\beta }+\mathrm{\delta }\right)}{z}\right)y\left(z\right)=0$
 ${\mathrm{BHE}}{≔}{\mathrm{y\text{'}\text{'}}}{+}\left({-}{2}{}{z}{-}{\mathrm{\beta }}{+}\frac{{1}{+}{\mathrm{\alpha }}}{{z}}\right){}{\mathrm{y\text{'}}}{+}\left({\mathrm{\gamma }}{-}{\mathrm{\alpha }}{-}{2}{-}\frac{\left({1}{+}{\mathrm{\alpha }}\right){}{\mathrm{\beta }}{+}{\mathrm{\delta }}}{{2}{}{z}}\right){}{y}{=}{0}$ (7)
 > $\mathrm{singularities}\left(\mathrm{BHE}\right)$
 ${\mathrm{regular}}{=}\left\{{0}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (8)
 having for solution the HeunB function; the Doubleconfluent equation
 > $\mathrm{DHE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)-\frac{\mathrm{\alpha }+2z+{z}^{2}\mathrm{\alpha }-2{z}^{3}}{{\left(z+1\right)}^{2}{\left(z-1\right)}^{2}}\mathrm{diff}\left(y\left(z\right),z\right)+\frac{\mathrm{\delta }+\left(2\mathrm{\alpha }+\mathrm{\gamma }\right)z+\mathrm{\beta }{z}^{2}}{{\left(z-1\right)}^{3}{\left(z+1\right)}^{3}}y\left(z\right)$
 ${\mathrm{DHE}}{≔}{\mathrm{y\text{'}\text{'}}}{-}\frac{\left({\mathrm{\alpha }}{}{{z}}^{{2}}{-}{2}{}{{z}}^{{3}}{+}{\mathrm{\alpha }}{+}{2}{}{z}\right){}{\mathrm{y\text{'}}}}{{\left({z}{+}{1}\right)}^{{2}}{}{\left({z}{-}{1}\right)}^{{2}}}{+}\frac{\left({\mathrm{\delta }}{+}\left({2}{}{\mathrm{\alpha }}{+}{\mathrm{\gamma }}\right){}{z}{+}{\mathrm{\beta }}{}{{z}}^{{2}}\right){}{y}}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}$ (9)
 > $\mathrm{singularities}\left(\mathrm{DHE}\right)$
 ${\mathrm{regular}}{=}{\varnothing }{,}{\mathrm{irregular}}{=}\left\{{-1}{,}{1}\right\}$ (10)
 having for solution the HeunD function, and the Triconfluent equation
 > $\mathrm{THE}≔\mathrm{diff}\left(y\left(z\right),z,z\right)+\left(-\mathrm{\gamma }-3{z}^{2}\right)\mathrm{diff}\left(y\left(z\right),z\right)+\left(\mathrm{\alpha }+z\mathrm{\beta }-3z\right)y\left(z\right)=0$
 ${\mathrm{THE}}{≔}{\mathrm{y\text{'}\text{'}}}{+}\left({-}{3}{}{{z}}^{{2}}{-}{\mathrm{\gamma }}\right){}{\mathrm{y\text{'}}}{+}\left({\mathrm{\beta }}{}{z}{+}{\mathrm{\alpha }}{-}{3}{}{z}\right){}{y}{=}{0}$ (11)
 > $\mathrm{singularities}\left(\mathrm{THE}\right)$
 ${\mathrm{regular}}{=}{\varnothing }{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (12)
 having for solution the HeunT function.
 The standard form of the four confluent equations is not uniform in the literature. The Maple choice of standard form follows [1], the classic reference for these equations, except for one particular: for the Doubleconfluent equation (DHE above) (and so the definition of HeunD) Maple uses the so-called Jaffe form, see [3], so that the two irregular singular points are symmetrically located at $z=-1$ and $z=1$.

The confluence process and the confluent Heun equation

 To illustrate the confluence process leading to the four confluent Heun equations, the CHE is obtained from the general equation as follows.
 Redefine the parameters entering the GHE using
 > $\mathrm{\beta }=\mathrm{\beta }a,\mathrm{\epsilon }=\mathrm{\epsilon }a,q=qa$
 ${\mathrm{\beta }}{=}{\mathrm{\beta }}{}{a}{,}{\mathrm{\epsilon }}{=}{\mathrm{\epsilon }}{}{a}{,}{q}{=}{q}{}{a}$ (13)
 > $\mathrm{eval}\left(\mathrm{GHE},\left[\right]\right)$
 ${\mathrm{y\text{'}\text{'}}}{+}\left(\frac{{\mathrm{\gamma }}}{{z}}{+}\frac{{\mathrm{\delta }}}{{z}{-}{1}}{+}\frac{{\mathrm{\epsilon }}{}{a}}{{z}{-}{a}}\right){}{\mathrm{y\text{'}}}{+}\frac{\left({a}{}{\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{-}{q}{}{a}\right){}{y}}{{z}{}\left({z}{-}{1}\right){}\left({z}{-}{a}\right)}{=}{0}$ (14)
 Take the limit $a$ -> $\mathrm{\infty }$
 > $\mathrm{Limit}\left(,a=\mathrm{\infty }\right)$
 $\underset{{a}{\to }{\mathrm{\infty }}}{{lim}}{}\left({\mathrm{y\text{'}\text{'}}}{+}\left(\frac{{\mathrm{\gamma }}}{{z}}{+}\frac{{\mathrm{\delta }}}{{z}{-}{1}}{+}\frac{{\mathrm{\epsilon }}{}{a}}{{z}{-}{a}}\right){}{\mathrm{y\text{'}}}{+}\frac{\left({a}{}{\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{-}{q}{}{a}\right){}{y}}{{z}{}\left({z}{-}{1}\right){}\left({z}{-}{a}\right)}{=}{0}\right)$ (15)
 Evaluating this limit, the singularity at $a$ "coalesces" with the one at $\mathrm{\infty }$ resulting in
 > $\mathrm{value}\left(\right)$
 $\frac{{-}{y}{}{\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{-}{\mathrm{y\text{'}}}{}{\mathrm{\epsilon }}{}{{z}}^{{2}}{+}{\mathrm{y\text{'}}}{}{\mathrm{\delta }}{}{z}{+}{\mathrm{y\text{'}}}{}{\mathrm{\epsilon }}{}{z}{+}{\mathrm{y\text{'}}}{}{\mathrm{\gamma }}{}{z}{+}{\mathrm{y\text{'}\text{'}}}{}{{z}}^{{2}}{-}{\mathrm{y\text{'}}}{}{\mathrm{\gamma }}{+}{y}{}{q}{-}{\mathrm{y\text{'}\text{'}}}{}{z}}{{{z}}^{{2}}{-}{z}}{=}{0}$ (16)
 Rearranging the terms you get the CHE
 > $\left(\mathrm{lhs}-\mathrm{rhs}\right)\left(\mathrm{collect}\left(\mathrm{isolate}\left(,\mathrm{diff}\left(y\left(z\right),z,z\right)\right),\left[y\left(z\right),\mathrm{diff}\left(y\left(z\right),z\right)\right],u↦\mathrm{convert}\left(u,\mathrm{parfrac},z\right)\right)\right)=0$
 ${\mathrm{y\text{'}\text{'}}}{-}\left(\frac{{q}}{{z}}{+}\frac{{\mathrm{\alpha }}{}{\mathrm{\beta }}{-}{q}}{{z}{-}{1}}\right){}{y}{-}\left({\mathrm{\epsilon }}{-}\frac{{\mathrm{\gamma }}}{{z}}{-}\frac{{\mathrm{\delta }}}{{z}{-}{1}}\right){}{\mathrm{y\text{'}}}{=}{0}$ (17)
 Since the two singularities being coalesced are regular, the resulting single singularity at $\mathrm{\infty }$ will be irregular (this is typical; there are exceptions though)
 > $\mathrm{singularities}\left(\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (18)

The five Heun functions

 The Heun functions, HeunG, HeunC, HeunB, HeunD and HeunT, are defined as the solutions to the corresponding General, Confluent, Biconfluent, Doubleconfluent and Triconfluent Heun equations. These solutions are constructed as power series solutions around the origin, for certain initial conditions. The five equations and the initial conditions used to define the corresponding five Heun functions can be seen through the FunctionAdvisor
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{HeunG}\right)$
 ${\mathrm{HeunG}}{}\left({a}{,}{q}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{{{\mathrm{_Y}}}_{{z}{,}{z}}{-}\frac{\left(\left({\mathrm{\alpha }}{+}{\mathrm{\beta }}{+}{1}\right){}{{z}}^{{2}}{+}\left(\left({-}{\mathrm{\delta }}{-}{\mathrm{\gamma }}\right){}{a}{-}{\mathrm{\alpha }}{+}{\mathrm{\delta }}{-}{\mathrm{\beta }}{-}{1}\right){}{z}{+}{\mathrm{\gamma }}{}{a}\right){}{{\mathrm{_Y}}}_{{z}}}{{z}{}\left({z}{-}{1}\right){}\left({-}{z}{+}{a}\right)}{-}\frac{\left({\mathrm{\alpha }}{}{\mathrm{\beta }}{}{z}{-}{q}\right){}{\mathrm{_Y}}{}\left({z}\right)}{{z}{}\left({z}{-}{1}\right){}\left({-}{z}{+}{a}\right)}\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}\frac{{q}}{{\mathrm{\gamma }}{}{a}}\right\}\right)$ (19)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{HeunC}\right)$
 ${\mathrm{HeunC}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{\mathrm{\eta }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{{{\mathrm{_Y}}}_{{z}{,}{z}}{-}\frac{\left({-}{{z}}^{{2}}{}{\mathrm{\alpha }}{+}\left({-}{\mathrm{\beta }}{+}{\mathrm{\alpha }}{-}{\mathrm{\gamma }}{-}{2}\right){}{z}{+}{\mathrm{\beta }}{+}{1}\right){}{{\mathrm{_Y}}}_{{z}}}{{z}{}\left({z}{-}{1}\right)}{-}\frac{\left(\left(\left({-}{\mathrm{\beta }}{-}{\mathrm{\gamma }}{-}{2}\right){}{\mathrm{\alpha }}{-}{2}{}{\mathrm{\delta }}\right){}{z}{+}\left({\mathrm{\beta }}{+}{1}\right){}{\mathrm{\alpha }}{+}\left({-}{\mathrm{\gamma }}{-}{1}\right){}{\mathrm{\beta }}{-}{2}{}{\mathrm{\eta }}{-}{\mathrm{\gamma }}\right){}{\mathrm{_Y}}{}\left({z}\right)}{{2}{}{z}{}\left({z}{-}{1}\right)}\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}\frac{\left({-}{\mathrm{\alpha }}{+}{1}{+}{\mathrm{\gamma }}\right){}{\mathrm{\beta }}{+}{\mathrm{\gamma }}{-}{\mathrm{\alpha }}{+}{2}{}{\mathrm{\eta }}}{{2}{}\left({\mathrm{\beta }}{+}{1}\right)}\right\}\right)$ (20)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{HeunB}\right)$
 ${\mathrm{HeunB}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{{{\mathrm{_Y}}}_{{z}{,}{z}}{-}\frac{\left({\mathrm{\beta }}{}{z}{+}{2}{}{{z}}^{{2}}{-}{\mathrm{\alpha }}{-}{1}\right){}{{\mathrm{_Y}}}_{{z}}}{{z}}{-}\frac{\left(\left({2}{}{\mathrm{\alpha }}{-}{2}{}{\mathrm{\gamma }}{+}{4}\right){}{z}{+}{\mathrm{\beta }}{}{\mathrm{\alpha }}{+}{\mathrm{\beta }}{+}{\mathrm{\delta }}\right){}{\mathrm{_Y}}{}\left({z}\right)}{{2}{}{z}}\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}\frac{{\mathrm{\beta }}{}{\mathrm{\alpha }}{+}{\mathrm{\beta }}{+}{\mathrm{\delta }}}{{2}{}{\mathrm{\alpha }}{+}{2}}\right\}\right)$ (21)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{HeunD}\right)$
 ${\mathrm{HeunD}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{{{\mathrm{_Y}}}_{{z}{,}{z}}{-}\frac{\left({\mathrm{\alpha }}{}{{z}}^{{4}}{-}{2}{}{{z}}^{{5}}{+}{4}{}{{z}}^{{3}}{-}{\mathrm{\alpha }}{-}{2}{}{z}\right){}{{\mathrm{_Y}}}_{{z}}}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}{-}\frac{\left({-}{{z}}^{{2}}{}{\mathrm{\beta }}{+}\left({-}{\mathrm{\gamma }}{-}{2}{}{\mathrm{\alpha }}\right){}{z}{-}{\mathrm{\delta }}\right){}{\mathrm{_Y}}{}\left({z}\right)}{{\left({z}{-}{1}\right)}^{{3}}{}{\left({z}{+}{1}\right)}^{{3}}}\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}{0}\right\}\right)$ (22)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{HeunT}\right)$
 ${\mathrm{HeunT}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{z}\right){=}{\mathrm{DESol}}{}\left(\left\{{{\mathrm{_Y}}}_{{z}{,}{z}}{-}\left({3}{}{{z}}^{{2}}{+}{\mathrm{\gamma }}\right){}{{\mathrm{_Y}}}_{{z}}{-}\left(\left({-}{\mathrm{\beta }}{+}{3}\right){}{z}{-}{\mathrm{\alpha }}\right){}{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({z}\right)\right\}{,}\left\{{\mathrm{_Y}}{}\left({0}\right){=}{1}{,}{\mathrm{D}}{}\left({\mathrm{_Y}}\right){}\left({0}\right){=}{0}\right\}\right)$ (23)
 The power series solutions at the base of the functions' definitions have restricted radius of convergence in the HeunG, HeunC and HeunD cases, where the numerical evaluation is done using analytic extensions, exploring closed form identities satisfied by these functions, as well as series expansions around different singularities. For arbitrary values of the parameters, however, closed form formulas for the connection constants relating series expansions around different singularities, are not known.
 As an example of the use of identities, for HeunD,
 > $\mathrm{FunctionAdvisor}\left(\mathrm{identities},\mathrm{HeunD}\right)$
 $\left[{\mathrm{HeunD}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\gamma }}{,}{\mathrm{\delta }}{,}{z}\right){=}{\mathrm{HeunD}}{}\left({-}{\mathrm{\alpha }}{,}{-}{\mathrm{\delta }}{,}{-}{\mathrm{\gamma }}{,}{-}{\mathrm{\beta }}{,}\frac{{1}}{{z}}\right)\right]$ (24)
 Hence, the evaluation of the function in the complex plane outside the unit circle is performed by mapping the problem into that of evaluating the function inside that circle. A similar situation is described in the help page for HeunG.
 Remark: The coefficients entering the series expansions represented by the Heun functions satisfy three term recurrence relations. A solution to these recursion equations is not known in the general case, so a closed form for the series's coefficients is not available and the computation of - say - the $n$th coefficient requires the explicit computation of all the previous ones.

Some important facts about Heun functions

 • They are more general than the rest of the functions of the mathematical language in that they contain most of them as particular cases. Consequently, the Heun equations cannot have their solution expressed (but as infinite sum power series) without using the corresponding Heun functions.
 • The Heun functions have a rich structure and so satisfy a rather large number of identities.
 • Because they have such a rich structure and include as particular so many functions, including the Mathieu, Lame, Spheroidal Wave and hypergeometric 2F1, 1F1 and 0F1 functions, the interrelations between them and the Heun ones are a source of many nontrivial identities between the former.
 • Due to the enlarged structure of singularities (if compared for instance with hypergeometric functions) the Heun functions are increasingly appearing in the modeling of different types of problems in applied mathematics.

References

 [1] 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. I-II. (1978): 53-78.
 [2] Ronveaux, A., ed. Heun's Differential Equations. Oxford, England: Oxford University Press, 1995.
 [3] Slavyanov, S.Y., and Lay W. Special Functions, A Unified Theory Based on Singularities. Oxford, England: Oxford Mathematical Monographs, 2000.