Series solutions near an irregular singular point

We study the second–order linear ODE

$$L[w]\equiv w''(z)+p(z)w'(z)+q(z)w(z)=0,$$

where $p(z)$ and $q(z)$ are meromorphic. Our goal is to understand how to construct series solutions near an irregular singular point, with particular emphasis on $z=\infty$.

At a regular singular point, Frobenius theory gives convergent power series. At an irregular singular point:

• one still has Frobenius–type formal series, but they typically diverge;
• the leading behaviour usually involves an exponential factor $e^{Q(z)}$;
• in some cases, fractional powers of $z$ enter after a ramified change of variables.

This note introduces these ideas through formal, normal, and subnormal solutions.

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1. Setup and formal Frobenius–type series

We focus on $z=\infty$ and assume that $p(z)$ and $q(z)$ are meromorphic there. Write their Laurent expansions as

$$p(z)=z^{n_1}\Bigl(a_0+\frac{a_1}{z}+\frac{a_2}{z^2}+\cdots\Bigr) =\sum_{\nu=0}^\infty a_\nu z^{n_1-\nu},$$ $$q(z)=z^{n_2}\Bigl(b_0+\frac{b_1}{z}+\frac{b_2}{z^2}+\cdots\Bigr) =\sum_{\nu=0}^\infty b_\nu z^{n_2-\nu},$$

with integers $n_1,n_2$ and $a_0,b_0$ not both zero (unless $p\equiv0$ or $q\equiv0$).

Recall that:

• If $p(z)=O(z^{-1})$ and $q(z)=O(z^{-2})$ as $z\to\infty$, then $\infty$ is an ordinary or regular singular point.
• If at least one of $$n_1>-1,\qquad n_2>-2$$ holds, the point $z=\infty$ is an irregular singular point of polar type. This is the case we study.

1.1 Frobenius–type ansatz at infinity

We first try a Frobenius–type ansatz

$$w(z)=z^\rho\sum_{k=0}^\infty c_k z^{-k},\qquad c_0\neq0.$$

Differentiate:

$$w'(z)=z^{\rho-1}\sum_{k=0}^\infty c_k(\rho-k)z^{-k},\qquad w''(z)=z^{\rho-2}\sum_{k=0}^\infty c_k(\rho-k)(\rho-k-1)z^{-k}.$$

Substitute into $L[w]=0$ and factor out $z^\rho$:

$$\begin{aligned} 0&=L[w] \\ &=z^\rho\Biggl\{ \sum_{k=0}^\infty c_k(\rho-k)(\rho-k-1) z^{-k-2} +\Bigl[\sum_{\ell=0}^\infty a_\ell z^{n_1-\ell}\Bigr] \Bigl[\sum_{k=0}^\infty c_k(\rho-k)z^{-k-1}\Bigr] \\ &\qquad\qquad +\Bigl[\sum_{\nu=0}^\infty b_\nu z^{n_2-\nu}\Bigr] \Bigl[\sum_{k=0}^\infty c_k z^{-k}\Bigr] \Biggr\}. \end{aligned}$$

Inside the braces we have a formal Laurent series in integer powers of $z$. For $w$ to be a formal solution, the coefficient of each power of $z$ must vanish.

The highest powers come from:

• the $w''$ term: at most $z^{-2}$;
• the $p w'$ term: from $a_0 c_0(\rho) z^{n_1-1}$;
• the $q w$ term: from $b_0 c_0 z^{n_2}$.

If $n_2\ge n_1$, then $z^{n_2}$ is the highest power and its coefficient is $b_0 c_0\neq0$, so no such formal series exists.

Thus a necessary condition for a Frobenius–type formal solution is

$$n_1>n_2.$$

If $p\equiv0$ (so all $a_\nu=0$), then $z^{n_2}$ is again the highest power with coefficient $b_0c_0$, and the ansatz fails unless $q\equiv0$. In that sense the “order” of $p$ is $-\infty$, and the condition $n_1>n_2$ still fails.

1.2 Determining the exponent $\rho$

Assume from now on that $n_1>n_2$ and $a_0\neq0$. The dominant power in the curly bracket is then $z^{n_1-1}$. Two subcases occur.

1. $n_1>n_2+1$.
   The exponent $n_1-1$ is strictly larger than $n_2$, so only the $p w'$ term contributes to this power. Its coefficient is $a_0 c_0 \rho$, so

   $$\rho=0.$$

2. $n_1=n_2+1$.
   Now $p w'$ and $q w$ both contribute to the power $z^{n_2}$, giving coefficient $c_0(a_0\rho+b_0)$. Hence

   $$a_0\rho+b_0=0 \quad\Rightarrow\quad \rho=-\frac{b_0}{a_0}.$$

This “indicial equation” is linear in $\rho$, in contrast with the quadratic indicial equation at a regular singular point.

Once $\rho$ is fixed, one equates the coefficients of the lower powers of $z$ to zero, obtaining a linear recurrence that determines $c_1,c_2,\dots$ uniquely in terms of the free parameter $c_0$. The resulting series is a formal Frobenius–type solution at $z=\infty$.

1.3 Asymptotic meaning

For an irregular singular point, the coefficients $c_k$ typically grow faster than $k!$, so the series $\sum_{k\ge0}c_k z^{-k}$ diverges for every finite $z$. However it still carries information:

A formal series

$$w(z)\sim z^\rho\sum_{k=0}^\infty c_k z^{-k}\qquad (z\to\infty)$$

is an asymptotic expansion of a genuine solution if, for each fixed $N$,

$$w(z)-z^\rho\sum_{k=0}^{N-1}c_k z^{-k} = O(z^{\rho-N}) \quad \text{as } z\to\infty$$

in some sector of the complex plane.

Truncating the divergent series after finitely many terms then provides good approximations for large $\lvert z\rvert$. Stokes phenomena arise as one crosses rays where exponentially small corrections switch on and off.

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2. Normal solutions

When the necessary condition $n_1>n_2$ fails (for example if $p\equiv0$ and $q$ grows at infinity), Frobenius–type series of the form $z^\rho\sum c_k z^{-k}$ do not exist. In many such situations solutions have the more general form

$$w(z)=e^{Q(z)}z^\rho\sum_{k=0}^\infty c_k z^{-k},$$

where the exponential factor contains a polynomial $Q(z)$ that captures the dominant growth or decay.

2.1 Prototype: first–order equations

Consider the first–order equation

$$w'(z)+R(z)w(z)=0.$$

If $R(z)\to0$ as $z\to\infty$, then $-w'/w=R(z)$ has only negative powers in its Laurent expansion, and $w$ can admit a Frobenius–type expansion.

If instead $R(z)$ has a polynomial part,

$$R(z)=A_s z^s + A_{s-1}z^{s-1}+\cdots + A_1 z + A_0 +\sum_{\ell\ge1}a_\ell z^{-\ell}, \qquad s\ge0,\ A_s\ne0,$$

then

$$w(z)=C\exp\Bigl(-\int^z R(\zeta)\,d\zeta\Bigr).$$

Splitting the integral into its polynomial and decaying parts, we obtain

$$w(z) =C\,e^{Q(z)}\,z^\rho\,\psi(z),$$

with

• $Q(z)$ a polynomial (coming from the integral of the polynomial part of $R(z)$),
• $\rho$ determined by the coefficient of $z^{-1}$ in $R(z)$,
• $\psi(z)=\exp\{\text{negative powers of }z\} =\sum_{k\ge0}c_k z^{-k}$, a Frobenius–type series.

This motivates the following definition.

Normal solution (at $z=\infty$).
A solution of a linear ODE is called normal if it can be written as

$$w(z)=e^{Q(z)}z^\rho\sum_{k=0}^\infty c_k z^{-k},\qquad c_0\neq0,$$

where $Q(z)$ is a (non‑constant) polynomial.

2.2 Reduction for second–order equations

For the second–order equation

$$w''(z)+p(z)w'(z)+q(z)w(z)=0,$$

we look for a normal solution by setting

$$w(z)=e^{Q(z)}v(z),\qquad Q(z)\ \text{polynomial}.$$

Compute

$$w'=e^{Q}(v'+Q'v),\qquad w''=e^{Q}\bigl(v''+2Q'v'+Q''v+(Q')^2v\bigr).$$

Substituting into the ODE and cancelling the common factor $e^{Q}$ we get an equation for $v$:

$$v''+p^*(z)v'+q^*(z)v=0,$$

with modified coefficients

$$p^*(z)=p(z)+2Q'(z),\qquad q^*(z)=q(z)+p(z)Q'(z)+Q''(z)+[Q'(z)]^2.$$

We now apply the formal Frobenius analysis of Section 1 to this transformed equation. In particular:

• expand $p^*,q^*$ in Laurent series at infinity,
• let $\tilde n_1,\tilde n_2$ be their orders,
• demand $\tilde n_1>\tilde n_2$ so that a Frobenius–type formal series for $v$ can exist.

This requirement determines the degree and coefficients of $Q$. Once $Q$ is fixed, the exponent $\rho$ and the coefficients $c_k$ in $v(z)=z^\rho\sum c_k z^{-k}$ follow from the same recurrence process as before.

From the viewpoint of the Riccati equation for $w'/w$, choosing $Q(z)$ amounts to approximating the “logarithmic derivative” by a polynomial and removing that leading behaviour.

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3. Example: Weber / Hermite equation

Consider the ODE

$$\psi''(x)+\bigl(\lambda-\alpha^2 x^2\bigr)\psi(x)=0, \qquad \alpha>0,$$

the one-dimensional harmonic oscillator (after rescaling). Here $p(x)\equiv0$, and

$$q(x)=\lambda-\alpha^2x^2\sim -\alpha^2x^2\quad(x\to\infty).$$

Because $p\equiv0$, the necessary condition $n_1>n_2$ for a Frobenius–type series fails: there is no formal solution of the simple form $x^\rho\sum c_k x^{-k}$. We therefore seek a normal solution.

3.1 Choosing the exponential factor

Set

$$\psi(x)=e^{Q(x)}v(x).$$

As above, the transformed equation for $v$ is

$$v''(x)+p^*(x)v'(x)+q^*(x)v(x)=0,$$

with

$$p^*(x)=2Q'(x),\qquad q^*(x)=\lambda-\alpha^2x^2+Q''(x)+[Q'(x)]^2.$$

We want the order of $p^*$ at infinity to be higher than that of $q^*$. Since $q^*$ contains $[Q'(x)]^2$, $Q'(x)$ must be at most linear; otherwise $[Q']^2$ would dominate and make the order of $q^*$ too large. So try

$$Q(x)=a_1x+a_2x^2.$$

Then

$$Q'(x)=a_1+2a_2x,\qquad Q''(x)=2a_2,$$ $$p^*(x)=2(a_1+2a_2x),$$ $$q^*(x)=\lambda-\alpha^2x^2+2a_2+(a_1+2a_2x)^2 =\lambda+2a_2+a_1^2+4a_1a_2x+(-\alpha^2+4a_2^2)x^2.$$

For $p^*$ to have strictly higher order than $q^*$, the coefficients of $x^2$ and $x$ in $q^*$ must vanish:

$$4a_2^2-\alpha^2=0,\qquad 4a_1a_2=0.$$

Thus

$$a_2=\pm\frac{\alpha}{2},\qquad a_1=0,$$

and we can take

$$Q(x)=\pm\frac{\alpha x^2}{2}.$$

The corresponding transformed equation is

$$v''(x)\pm2\alpha x\,v'(x)+(\lambda\pm\alpha)v(x)=0.$$

For physical applications we are interested in solutions decaying at $\lvert x\rvert\to\infty$. This corresponds to

$$Q(x)=-\frac{\alpha x^2}{2},$$

for which

$$p^*(x)=-2\alpha x,\qquad q^*(x)=\lambda-\alpha,$$

and

$$v''(x)-2\alpha x\,v'(x)+(\lambda-\alpha)v(x)=0.$$

This is the Hermite (parabolic cylinder) equation.

3.2 Formal series for $v$

Seek a Frobenius–type series

$$v(x)=x^\rho\sum_{k=0}^\infty c_k x^{-k},\qquad c_0\neq0.$$

Substituting into the Hermite equation and comparing coefficients gives:

• the exponent $$\rho=\frac{\lambda-\alpha}{2\alpha};$$
• all odd coefficients vanish, $c_{2k+1}=0$;
• the even coefficients satisfy the recurrence $$c_{2k+2} =\frac{(\rho-2k)(\rho-2k-1)}{4\alpha(k+1)}\,c_{2k}, \qquad k=0,1,2,\ldots$$

From this recurrence one can check that $c_{2k}$ grows roughly like $(2k)!$, so the series $\sum c_{2k}x^{-2k}$ diverges for all finite $x$. Nevertheless

$$\psi(x)\sim e^{-\alpha x^2/2}\,x^\rho \sum_{k=0}^\infty c_{2k}x^{-2k} \quad (x\to\infty)$$

is an asymptotic expansion of a true solution that decays along suitable rays in the complex plane.

3.3 Terminating series and Hermite polynomials

If

$$\rho=n\in\{0,1,2,\dots\}, \quad\text{equivalently } \lambda=\alpha(2n+1),$$

then for some $k$ the factor $(\rho-2k)$ in the recurrence vanishes, and the series terminates: $c_{2k'}=0$ for all $k'>k$. The solution $v(x)$ becomes a polynomial of degree $n$.

Writing $\xi=\sqrt{\alpha}\,x$, these polynomials are proportional to the standard Hermite polynomials $H_n(\xi)$, defined by, e.g.,

$$H_n(\xi) =(2\xi)^n-\frac{n(n-1)}{1!}(2\xi)^{n-2} +\frac{n(n-1)(n-2)(n-3)}{2!}(2\xi)^{n-4}-\cdots.$$

Thus for the discrete values $\lambda=\alpha(2n+1)$ we obtain global, square-integrable solutions

$$\psi_n(x)=e^{-\alpha x^2/2}H_n(\sqrt{\alpha}\,x),$$

the familiar eigenfunctions of the quantum harmonic oscillator. In this case the normal solution is not just asymptotic but given by a convergent finite series.

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4. Subnormal solutions

Sometimes no polynomial $Q(z)$ makes the transformed coefficients $p^*(z),q^*(z)$ satisfy the condition needed for a Frobenius–type series. However, an algebraic change of variables can improve the situation.

A typical structure is:

• after a ramified substitution $z=\zeta^s$ with $s\in\mathbb N$, the transformed equation in $\zeta$ admits a normal solution;
• when expressed back in the original variable $z$, the solution involves exponentials and series in fractional powers of $z$.

Subnormal solution.
A solution of $w''+pw'+qw=0$ is called subnormal of order $s$ if, after the change of variable $z=\zeta^s$, it becomes a normal solution in the variable $\zeta$.

4.1 Example

Consider

$$w''(z)+q(z)w(z)=0,$$

with

$$q(z)=\sum_{\ell=1}^\infty b_\ell z^{-\ell}, \qquad b_1\neq0.$$

Here $p(z)\equiv0$ and $q(z)$ decays like $b_1/z$. One checks that no choice of polynomial $Q(z)$ yields a normal solution directly.

Introduce the substitution $z=\zeta^2$ (so $\zeta=z^{1/2}$). Using $\dfrac{d}{dz}=\dfrac{1}{2\zeta}\dfrac{d}{d\zeta}$ and

$$\frac{d^2}{dz^2} =\frac{1}{4\zeta^2}\frac{d^2}{d\zeta^2} -\frac{1}{4\zeta^3}\frac{d}{d\zeta},$$

the equation becomes

$$\frac{d^2w}{d\zeta^2} - \frac1{\zeta}\frac{dw}{d\zeta} +4\zeta^2 q(\zeta^2) w =0.$$

Now

$$4\zeta^2 q(\zeta^2)=4b_1+O(\zeta^{-2}),$$

and the coefficient of $w'$ is $-1/\zeta$, whose order at infinity is higher than that of $4\zeta^2 q(\zeta^2)$. The transformed equation therefore does admit a normal solution of the form

$$w(\zeta) =\exp\!\bigl(\pm\sqrt{-b_1}\,\zeta\bigr)\, \zeta^\rho\sum_{k=0}^\infty c_k \zeta^{-k},$$

for some exponent $\rho$ and coefficients $c_k$.

Returning to $z=\zeta^2$ gives the subnormal solution

$$w(z) =\exp\!\bigl(\pm\sqrt{-b_1}\,z^{1/2}\bigr)\, z^{1/4}\sum_{k=0}^\infty c_k z^{-k/2}.$$

The presence of $e^{\pm\sqrt{-b_1}\,z^{1/2}}$ and the series in half-integer powers of $z$ are characteristic of a subnormal solution of order 2.

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5. Summary

For the second–order ODE $w''+pw'+qw=0$ at an irregular singular point of polar type (say $z=\infty$):

• A Frobenius–type formal series $w(z)=z^\rho\sum c_k z^{-k}$ can exist only if the order of $p(z)$ at infinity exceeds that of $q(z)$. When it exists, the exponent $\rho$ is determined by a linear indicial equation and the coefficients $c_k$ follow from a recurrence. The series usually diverges and must be interpreted asymptotically.

• When this condition fails, one looks for normal solutions $w(z)=e^{Q(z)}z^\rho\sum c_k z^{-k}$ with a polynomial $Q(z)$ chosen so that the transformed equation for $v=e^{-Q}w$ does admit a Frobenius–type series. The Weber/Hermite equation is a canonical example, where $Q(x)=\pm\alpha x^2/2$.

• In more complicated cases, an algebraic change of variables $z=\zeta^s$ can create a normal solution in $\zeta$. The resulting solutions in the original variable are subnormal, involving exponentials and series in fractional powers of $z$.

These constructions give a systematic way to understand the local structure of solutions near irregular singular points and provide the starting point for global questions such as Stokes phenomena, connection formulae, and quantisation conditions in quantum mechanics.