This presentation is a reconsideration of solutions to second-order linear differential equations with polynomial coefficients. The general form of the equation is

$A_2\left(z\right)\frac{d^{2}F}{d{z}^2}+A_1\left(z\right)\frac{dF}{dz}+A_0\left(z\right)F=0A_i=\sum _k{a}_{ik}{z}^k$

Solutions to these equations fall into groups based upon singular points in the equation, and relations between solutions to different cases of the equation can be deduced from them as well. The existence of singular points is also why a single general solution as a function of the constant coefficients is not possible. The appearance of a new singular point alters the resulting solution, separating it from solutions to equations with fewer singularities.

The equation is said to have a singular point if there are singularities in the ratios of coefficients

$\frac{A_1\left(z\right)}{A_2\left(z\right)}\frac{A_0\left(z\right)}{A_2\left(z\right)}$

For polynomial coefficients, the singularities at finite values can only be poles from the coefficient of the second-order derivative. A singularity at z_s is said to be regular if the two combinations

$(z-z_{\mathrm{s}})\frac{A_1\left(z\right)}{A_2\left(z\right)}(z-z_{\mathrm{s}}{)}^2\frac{A_0\left(z\right)}{A_2\left(z\right)}$

do not have singularities, otherwise the singularity is said to be irregular.

There is generally another singularity at the point at infinity in the complex plane, whose nature is found by mapping the point at infinity to the origin via $z=\frac{1}{w}$ . The change of variable of differentiation has the effect

$\begin{array}{c}\frac{d}{dz}=\frac{d}{d\left(\frac{1}{w}\right)}=-w^2\frac{d}{dw}\\ \frac{d^2}{d{z}^2}=w^2\frac{d}{dw}{w}^2\frac{d}{dw}=w^4\frac{d^2}{d{w}^2}+2w^3\frac{d}{dw}\end{array}$

so that the differential equation becomes

$w^4{A}_2\left(\frac{1}{w}\right)\frac{d^{2}F}{d{w}^2}+[2w^3{A}_2\left(\frac{1}{w}\right)-w^2{A}_1\left(\frac{1}{w}\right)]\frac{dF}{dw}+A_0\left(\frac{1}{w}\right)F=0$

where the solution function depends on the new independent variable. A singularity at the origin in this equation now corresponds to a singularity at the point at infinity.

The simplest second-order equation is one with constant coefficients:

$a_{20}\frac{d^{2}F}{d{z}^2}+a_{10}\frac{dF}{dz}+a_{00}F=0$

This equation does not have finite singularities. The well-known solutions to this equation are exponentials $e^{\lambda z}$ in the independent variable. Substituting this form in the differential equation gives the auxiliary equation and solutions

$a_{20}{\lambda }^2+a_{10}\lambda +a_{00}=0{\lambda }_{\pm }=\frac{-a_{10}\pm \sqrt{a_{10}^2-4a_{20}{a}_{00}}}{2a_{20}}$

If the discriminant under the square root is nonzero, the general solution is

$F\left(z\right)=c_{1}exp\left[{\lambda }_{+}z\right]+c_{2}exp\left[{\lambda }_{-}z\right]$

and if the discriminant is zero the general solution is

$F\left(z\right)=c_{1}exp\left[\lambda z\right]+c_{2}zexp\left[\lambda z\right]\lambda =-\frac{a_{10}}{2a_{20}}$

Mapping the point at infinity to the origin, the equation with constant coefficients becomes

$a_{20}{w}^4\frac{d^{2}F}{d{w}^2}+(2a_{20}{w}^3-a_{10}{w}^2)\frac{dF}{dw}+a_{00}F=0$

Forming the coefficient ratios

$\frac{2}{w}-\frac{a_{10}}{a_{20}{w}^2}\frac{a_{00}}{a_{20}{w}^4}$

indicates that there is a nasty singular point at infinity, which is irregular because the combinations

$w[\frac{2}{w}-\frac{a_{10}}{a_{20}{w}^2}]w^2\frac{a_{00}}{a_{20}{w}^4}$

are still singular. This seems a bit puzzling at first, since the exponential is one of the best behaved functions in mathematics, until one writes the general solution in terms of the mapped variable,

$F\left(w\right)=c_{1}exp\left[\frac{{\lambda }_{+}}{w}\right]+c_{2}exp\left[\frac{{\lambda }_{-}}{w}\right]$

and it becomes immediately clear that there is indeed a nasty irregular singular point at infinity. The presence of this singularity at infinity does not prevent a complete solution to the differential equation. The form of this solution is typical for describing the behavior of functions with an irregular singularity, which will contain an exponential of a polynomial.

The singularity at infinity becomes regular if $a_{00}=a_{10}=0$ , but then the differential equation and its general solution appear to change radically,

$a_{20}\frac{d^{2}F}{d{z}^2}=0F=c_1+c_{2}z=c_1+\frac{c_2}{w}$

although this form is completely consistent with the solution for discriminant zero.

The next simplest second-order equation is the Euler equation

$a_{22}{z}^2\frac{d^{2}F}{d{z}^2}+a_{11}z\frac{dF}{dz}+a_{00}F=0$

whose solutions are powers $z^{\lambda }$ of the independent variable. Substituting this form in the differential equation gives the auxiliary equation and solutions

$a_{22}\lambda (\lambda -1)+a_{11}\lambda +a_{00}=0{\lambda }_{\pm }=\frac{a_{22}-a_{11}\pm \sqrt{(a_{22}-a_{11}{)}^2-4a_{22}{a}_{00}}}{2a_{22}}$

If the discriminant under the square root is nonzero, the general solution is

$F\left(z\right)=c_1{z}^{{\lambda }_{+}}+c_2{z}^{{\lambda }_{-}}$

and if the discriminant is zero the general solution is

$F\left(z\right)=c_1{z}^{\lambda }+c_2{z}^{\lambda }lnz\lambda =\frac{a_{22}-a_{11}}{2a_{22}}$

Mapping the point at infinity to the origin, the Euler equation becomes

$a_{22}{w}^2\frac{d^{2}F}{d{w}^2}+(2a_{22}-a_{11})w\frac{dF}{dw}+a_{00}F=0$

and the singularity at infinity is seen to be regular. The form of this solution is typical for functions with a regular singularity, which contain powers of the independent variable.

The case $a_{00}=a_{11}=0$ has the two distinct values one and zero for the exponent,

$a_{22}\frac{d^{2}F}{d{z}^2}=0F=c_{1}z+c_2=\frac{c_1}{w}+c_2$

consistent with the solution to the altered differential equation.

For equations whose solutions cannot be written in terms of known functions, it is useful to have some idea about the behavior of the function at singular points. For a regular singular point at a finite location, the leading behavior is a power of the independent variable, and the value of its exponent is given by an indicial equation. Taking derivatives of the approximation $F\sim (z-z_{\mathrm{s}}{)}^{\lambda }$ and incorporating the definition above for a regular singular point, the indicial equation is

$\lambda (\lambda -1)+p_0\lambda +q_0=0$

where the coefficients at the singularity are given by

$\begin{array}{l}{p}_0=\mathrm{Constant}\left[(z-z_{\mathrm{s}})\frac{A_1\left(z\right)}{A_2\left(z\right)}\right]=\mathrm{Residue}\left[\frac{A_1\left(z\right)}{A_2\left(z\right)}\right]\\ q_0=\mathrm{Constant}\left[(z-z_{\mathrm{s}}{)}^2\frac{A_0\left(z\right)}{A_2\left(z\right)}\right]=\mathrm{Residue}\left[(z-z_{\mathrm{s}})\frac{A_0\left(z\right)}{A_2\left(z\right)}\right]\end{array}$

since the residue is simply the constant coefficient of the inverse term in an expansion about the singularity. For a regular singular point at infinity, the approximation is written in terms of an inverse power $F\sim z^{-\lambda }$ with the indicial equation

$\lambda (\lambda +1)-p_{\infty }\lambda +q_{\infty }=0$

whose coefficients are given by

$p_{\infty }=\underset{z\to \infty }{lim}z\frac{A_1\left(z\right)}{A_2\left(z\right)}{q}_{\infty }=\underset{z\to \infty }{lim}{z}^2\frac{A_0\left(z\right)}{A_2\left(z\right)}$

These limits can also be described as residues at the point at infinity, but since this concept generally includes an additional negative sign it is simpler not to introduce that terminology.

A function with an irregular singular point varies rapidly near that point, and this behavior is can generally only be described by an asymptotic series that approximates the behavior to a limited extent. Asymptotic series eventually diverge, so it is best to focus on the leading behavior of the series. For an irregular singular point a finite location, the leading behavior of the asymptotic series is an exponential of a set of inverse powers multiplied by an overall positive power of the independent variable:

$F\left(z\right)\sim (z-z_{\mathrm{s}}{)}^{\lambda }exp\left[\sum _k{b}_k(z-z_{\mathrm{s}}{)}^{-k}\right]$

For an irregular singular point at infinity, the leading behavior of the asymptotic series is an exponential of a set of powers multiplied by an overall inverse power of the independent variable:

$F\left(z\right)\sim z^{-\lambda }exp\left[\sum _k{b}_k{z}^k\right]$

There is unfortunately no quick rule for determining the values of the constants λ and b_k like the indicial equation coefficients. These constants can only be determined by substituting the forms in a differential equation and zeroing out positive-power terms from highest to lowest for a singularity at infinity, or negative-power terms from most negative to least for a finite singularity.

It is possible, however, and critically informative as well, to have a rule as to what powers appear in the exponentials. My source for this rule is Special Functions: A Unified Theory Based on Singularities by Sergei Slavyanov and Wolfgang Lay. This small book is extremely densely written and not inexpensive, but is a wonderful exposition of the differential equations currently used in physics and those that will figure over the next century. Highly recommended, and well worth the effort involved to get all the details in place in one’s noggin!

The rule is given in two parts on the eighth page of the book. To understand the rule for finite singular points, first rewrite the coefficient polynomials as powers at the singular point,

$A_i=\sum _k{a}_{ik}{z}^k=\sum _k{b}_{ik}(z-z_{\mathrm{s}}{)}^k$

where the constants b_ik can be determined by expanding the binomials and equating coefficients of equal powers. Let s₂, s₁ and s₀ designate the lowest power in each rewritten polynomial, where the subscript indicates which derivative the polynomial accompanies. The existence of a singular point means that s₂ must be at least one, but the other two lowest exponents can be zero for leading constants at the singularity.

The behavior of the solution at a finite singular point is approximated by the form $F\sim e^{(z-z_{\mathrm{s}}{)}^{-k}}$. Substituting this in the differential equation, the following lowest powers appear:

$(z-z_{\mathrm{s}}{)}^{s_2-k-2}(z-z_{\mathrm{s}}{)}^{s_2-2k-2}(z-z_{\mathrm{s}}{)}^{s_1-k-1}(z-z_{\mathrm{s}}{)}^{s_0}$

The first two powers need to be balanced against the second two for approximate satisfaction of the differential equation. Since the first power is less inverse than the second by k, it can be ignored in this approximation. This means that k must have either of the two values $s_2-s_1-1\mathrm{or}\frac{s_2-s_0}{2}-1$ . When these two quantities are zero, the exponential is a constant function and the singularity is regular, so that an irregular singularity is characterized by nonzero values for these combinations.

The presence of singular points was defined above using ratios of coefficients. Inverting the ratios, one can equally describe the occurrence of a singular point as zeroes in the combinations

$\frac{A_2\left(z\right)}{A_1\left(z\right)}\frac{A_2\left(z\right)}{A_0\left(z\right)}$

The multiplicities of these zeroes are given by the differences $s_2-s_1\mathrm{and}{s}_2-s_0$ of exponents of of lowest powers at the singularity. Since a regular singularity has differences of one and two respectively, it is convenient to define the s-rank of a singular point so that a regular singularity has unit rank:

$R\left(z_{\mathrm{s}}\right)\equiv max[s_2-s_1,\frac{s_2-s_0}{2}]$

The s-rank of a finite irregular singular point indicates the largest inverse power of the independent variable that appears in the exponential, which is one less that the s-rank of the singular point.

The rule for an irregular singular point at infinity is easier to construct. Let r₂, r₁ and r₀ designate the highest power in each polynomial coefficient, where the subscript again indicates which derivative the polynomial accompanies. The behavior of the solution at a singular point at infinity is approximated by the form $F\sim e^{z^k}$ . Substituting in the differential equation, the following highest powers of the independent variable appear:

$z^{r_2+k-2}{z}^{r_2+2k-2}{z}^{r_1+k-1}{z}^{r_0}$

The first two powers again need to be balanced against the second two for approximate satisfaction of the differential equation. Since the first power is less than the second by k, it can be ignored in this approximation, and k must have either of the two values $r_1-r_2+2-1\mathrm{or}\frac{r_0-r_2+4}{2}-1$ . The s-rank of a singular point at infinity is likewise defined so that a regular singularity has unit rank:

$R\left(\infty \right)\equiv max[r_1-r_2+2,\frac{r_0-r_2+4}{2}]$

The s-rank of a irregular singular point at infinity indicates the largest direct power of the independent variable that appears in the exponential, which is one less that the s-rank of the singular point.

It is important to note that two different sets of exponents are used in the determining s-ranks for the two types of singular points. The s-ranks of finite singular points are determined with lowest exponents of powers at the singularity, whereas s-ranks of singular points at infinity are determined with highest exponents of powers of the independent variable.

For polynomial coefficients in the differential equation with integer powers, the s-rank can clearly be either integral or half-integral. When the s-rank is integral, the polynomials in the exponential solution run over integral powers of the independent variable, from one to the s-rank minus one for a singularity at infinity, and the inverse for a finite singular point. When the s-rank is half integral, the polynomials in the exponential solution run over half-integral powers, from one-half to the s-rank minus one for a singularity at infinity, and the inverse for a finite singular point.

What makes the idea of s-rank critically informative is that two functions with the same singularity structure will be related in simple ways. They may have different locations for the singularities, or different characteristic constants at those singularities, but their behavior will be very much alike. In the reverse case, two functions with different singularity structures are essentially different from one another, and knowing the natures of their singularities is a good way of deciding when solutions to two differential equations can be related to one another or not.

As an application, consider the s-ranks of the two simplest second-order equations. For an equation with constant coefficients, a naive evaluation of the s-rank at the origin leads to a value of zero, which means that there is no singularity at the origin. The s-rank at infinity, however, is two and is thus an irregular singularity as seen above. The solution to this equation is designated { ; 2 }, meaning no singular point at the origin and an irregular singular point at infinity. The Euler equation, by contrast, has an s-rank of one at both the origin and infinity, since both singular points are regular, and is designated { 1 ; 1 } .

The singularity structure of a differential equation can change during transformations, and this helps one to understand the relationships of solutions found in the previous presentation. As a part of this understanding, one must determine which transformations alter the singularity structure of a differential equation and which do not. In this determination one need only focus on power of the independent variable in each polynomials coefficient, so that details of the accompanying constants can be ignored.

Consider how the differential equation changes for splitting off a power of the independent variable by setting $F\left(z\right)=z^{k}f\left(z\right)$ . The possible change in singularity structure is independent of the sign of the exponent k, since taking derivatives of a power always decreases the existing exponent, positive or negative. The terms that appear in the transformed equation are of the form

$A_2\frac{d^{2}f}{d{z}^2},[A_1,\frac{A_2}{z}]\frac{df}{dz},[A_0,\frac{A_1}{z},\frac{A_2}{z^2}]f$

where as stated the accompanying constants are ignored. There are now five possible values for the s-rank at a finite point, which are found by applying the first half of the s-rank formula to the second and third terms here, and the second half of the s-rank formula to the final three terms:

$R\left(z_{\mathrm{s}}\right)=max[s_2-s_1,1,\frac{s_2-s_0}{2},\frac{s_2-s_1+1}{2},1]$

If the initial equation did not have any finite singular points, then this transformation adds one. If the initial equation had only regular finite singular points this transformation does not alter that structure, since the fourth value is smaller than the first in that case. Likewise, if the initial equation had irregular finite singular points this transformation does not alter that structure for the same reason.

Similarly applying the s-rank formula for singularities at infinity gives the five values

$R\left(\infty \right)=max[r_1-r_2+2,1,\frac{r_0-r_2+4}{2},\frac{r_1-r_2+3}{2},1]$

The results at infinity are the same as a finite values for the same reasons. This transformation will add a regular singular point at infinity if there is no singular point there already, otherwise it will not change the existing singularity structure.

Now consider how the differential equation changes for splitting off an exponential by setting $F\left(z\right)=e^{z}f\left(z\right)$ . The terms that appear in the transformed equation are of the form

$A_2\frac{d^{2}f}{d{z}^2},[A_1,A_2]\frac{df}{dz},[A_0,A_1,A_2]f$

where again constants are ignored. The possible values of s-rank at finite points and infinity are

$\begin{array}{l}R\left(z_{\mathrm{s}}\right)=max[s_2-s_1,0,\frac{s_2-s_0}{2},\frac{s_2-s_1}{2},0]\\ R\left(\infty \right)=max[r_1-r_2+2,2,\frac{r_0-r_2+4}{2},\frac{r_1-r_2+4}{2},2]\end{array}$

This transformation will not alter the singularity structure at finite points in the least, but will add an irregular singularity at infinity of s-rank two unless there already is an irregular singularity there.

Contrast this with a transformation that splits off an exponential of a positive power by setting $F\left(z\right)=e^{z^k}f\left(z\right)$ . The terms that appear in the transformed equation are of the form

$A_2\frac{d^{2}f}{d{z}^2},[A_1,z^{k-1}{A}_2]\frac{df}{dz},[A_0,z^{k-1}{A}_1,z^{k-2}{A}_2,z^{2k-2}{A}_2]f$

and possible values of s-rank at finite points and infinity are

$\begin{array}{l}R\left(z_{\mathrm{s}}\right)=max[s_2-s_1,1-k,\frac{s_2-s_0}{2},\frac{s_2-s_1+1-k}{2},\frac{2-k}{2},1-k]\\ R\left(\infty \right)=max[r_1-r_2+2,k+1,\frac{r_0-r_2+4}{2},\frac{r_1-r_2+k+3}{2},\frac{k+2}{2},k+1]\end{array}$

This transformation does not alter the singularity structure at finite points, as negative values under the maximum function should be ignored. It does, however, add an irregular singularity at infinity of s-rank k+1 , unless there is already something worse there.

Changing the sign of the exponent k in this last set of values indicates that a transformation that splits off an exponential of an inverse power will not alter the singularity structure at infinity, but can be used to introduce an irregular singularity of s-rank k+1 at finite points.

The final simple transformation to consider is the change of independent variable $u=z^k$ . The terms that appear in the transformed equation are of the form

$u^{2(k-1)/k}{A}_2\left(u\right)\frac{d^{2}f}{d{u}^2},[u^{(k-1)/k}{A}_1\left(u\right),u^{(k-2)/k}{A}_2\left(u\right)]\frac{df}{du},A_0\left(u\right)f$

Possible values of s-rank at finite points and infinity are

$\begin{array}{l}R\left(z_{\mathrm{s}}\right)=max[\frac{s_2-s_1+k-1}{k},1,\frac{s_2-s_0+2k-2}{2k}]\\ R\left(\infty \right)=max[\frac{r_1-r_2+k+1}{k},1,\frac{r_0-r_2+2k+2}{2k}]\end{array}$

For an equation with regular singularities, this transformation does not alter the singularity structure: regular singularities remain regular. If however the equation does not have singularities initially or has irregular singularities, then the transformation does alter the singularity structure. It can introduce singularities at finite points or infinity, as well as change the s-rank of existing singular points.

Under an inverse power transformation, one must first move all powers of the independent variable into numerators before determining the s-rank values. It turns out that the same formulae can be used with the absolute value of the exponent and finite singularities switched with singularities at infinity. For the change of independent variable $u=z^{-k}$ , possible values of s-rank at finite points and infinity are

$\begin{array}{l}R\left(z_{\mathrm{s}}\right)=max[\frac{r_1-r_2+k+1}{k},1,\frac{r_0-r_2+2k+2}{2k}]\\ R\left(\infty \right)=max[\frac{s_2-s_1+k-1}{k},1,\frac{s_2-s_0+2k-2}{2k}]\end{array}$

Under this transformation regular singularities will again remain regular but their positions will be switched between finite and infinite, while irregular singularities can be altered along with switching between finite and infinite. The special case of changing to an inverse variable is seen to switch the locations of singularities without altering their values. This is used a check on the accuracy of a programmatic evaluation of s-rank values under transformations.

This completes the motivation for the definition of s-ranks of singular points. It will be used in the second part of this presentation to decide when differential equations with polynomial coefficients have similar general solutions and to elucidate connections between known solutions. Onward!

Uploaded 2013.07.14 — Updated 2013.07.24 analyticphysics.com