14 Jul 2025
Frobenius Method and Special Functions
Regular singular points and the Legendre, Bessel, Hermite, and Laguerre equations, generating functions, recurrence relations, and orthogonality.
Write a second-order linear equation as
\[y''+P(x)y'+Q(x)y=0.\]A point $x_0$ is ordinary if $P$ and $Q$ are analytic there. It is a regular singular point if $(x-x_0)P(x)$ and $(x-x_0)^2Q(x)$ are analytic. Otherwise it is an irregular singular point. This classification matters because a power series is guaranteed at an ordinary point, while a regular singular point generally requires a noninteger leading power.
Frobenius method
Near a regular singular point, set $z=x-x_0$ and write
\[P(x)=\sum_{j=0}^{\infty}p_jz^{j-1}, \qquad Q(x)=\sum_{j=0}^{\infty}q_jz^{j-2},\] \[y=z^r\sum_{n=0}^{\infty}a_nz^n, \qquad a_0\ne0.\]Then
\[y'=\sum_{n=0}^{\infty}(n+r)a_nz^{n+r-1}, \qquad y''=\sum_{n=0}^{\infty}(n+r)(n+r-1)a_nz^{n+r-2}.\]The coefficient of the lowest power $z^{r-2}$ is
\[a_0[r(r-1)+p_0r+q_0]=0.\]Since $a_0\ne0$, the indicial equation is
\[\boxed{r(r-1)+p_0r+q_0=0.}\]Higher powers determine $a_n$ recursively. Equal roots, or roots differing by an integer, can require a logarithmic second solution.
Bessel equation as a Frobenius construction
For
\[x^2y''+xy'+(x^2-\nu^2)y=0,\]$x=0$ is a regular singular point. Substitute $y=\sum_{n=0}^{\infty}a_nx^{n+r}$:
\[\sum_{n=0}^{\infty}[(n+r)^2-\nu^2]a_nx^{n+r} +\sum_{n=2}^{\infty}a_{n-2}x^{n+r}=0.\]The indicial equation is $r^2-\nu^2=0$. Choose the regular order $\nu\ge0$ and the root $r=\nu$. The $n=1$ equation is $(2\nu+1)a_1=0$, hence $a_1=0$, and for $n\ge2$,
\[a_n=-\frac{a_{n-2}}{(n+\nu)^2-\nu^2} =-\frac{a_{n-2}}{n(n+2\nu)}.\]Choosing $a_0=1/[2^\nu\Gamma(\nu+1)]$ gives
\[\boxed{ J_\nu(x)=\sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+\nu+1)} \left(\frac x2\right)^{2m+\nu}. }\]For noninteger $\nu$, $J_\nu$ and $J_{-\nu}$ are independent. For integer order, the second independent solution is singular at the origin, while $J_n$ remains finite.
The integer-order generating function is
\[\boxed{ \exp\!\left[\frac x2\left(t-\frac1t\right)\right] =\sum_{n=-\infty}^{\infty}J_n(x)t^n. }\]Differentiating with respect to $t$ and $x$, then matching powers of $t$, gives
\[J_{n-1}(x)+J_{n+1}(x)=\frac{2n}{x}J_n(x),\] \[J_{n-1}(x)-J_{n+1}(x)=2J_n'(x).\]For the regular order $\nu\ge0$, let $\alpha_{\nu m}$ be the $m$th positive zero of $J_\nu$. The Bessel equation in self-adjoint form,
\[\frac d{dx}\left(x\frac{dy}{dx}\right) +\left(\alpha^2x-\frac{\nu^2}{x}\right)y=0,\]gives, after multiplying two solutions by one another and subtracting,
\[(\alpha_m^2-\alpha_n^2) \int_0^1xJ_\nu(\alpha_mx)J_\nu(\alpha_nx)dx =\left[x(y_my_n'-y_ny_m')\right]_0^1=0.\]The regular behavior $J_\nu(\alpha x)\sim x^\nu$ makes the $x=0$ boundary term vanish, while both functions vanish at $x=1$. Thus distinct modes are orthogonal. Taking the coincident-root limit and using $J_\nu’(\alpha_{\nu n})=-J_{\nu+1}(\alpha_{\nu n})$ gives
\[\boxed{ \int_0^1xJ_\nu(\alpha_{\nu m}x)J_\nu(\alpha_{\nu n}x)dx =\frac12J_{\nu+1}^2(\alpha_{\nu n})\delta_{mn}. }\]Legendre functions
Legendre’s equation is
\[\frac d{dx}\left[(1-x^2)y'\right]+l(l+1)y=0, \qquad -1\le x\le1.\]Finiteness at both endpoints selects $l=0,1,2,\ldots$ and the Legendre polynomials $P_l(x)$. Their generating function is
\[\boxed{ \frac1{\sqrt{1-2xt+t^2}}=\sum_{l=0}^{\infty}P_l(x)t^l. }\]Differentiate with respect to $t$, multiply by $1-2xt+t^2$, and compare the coefficient of $t^l$ to obtain
\[\boxed{(l+1)P_{l+1}=(2l+1)xP_l-lP_{l-1}.}\]The first functions are
\[P_0=1,\quad P_1=x,\quad P_2=\frac12(3x^2-1),\quad P_3=\frac12(5x^3-3x).\]For $l\ne m$, subtracting the two self-adjoint equations and integrating gives
\[[l(l+1)-m(m+1)]\int_{-1}^{1}P_lP_mdx =\left[(1-x^2)(P_lP_m'-P_mP_l')\right]_{-1}^{1}=0.\]The norm follows from Rodrigues’ formula
\[P_l(x)=\frac1{2^ll!}\frac{d^l}{dx^l}(x^2-1)^l.\]Integrating by parts $l$ times, using $P_l^{(l)}=(2l)!/(2^ll!)$, and
\[\int_{-1}^{1}(1-x^2)^l dx =\frac{2^{2l+1}(l!)^2}{(2l+1)!},\]gives
\[\boxed{\int_{-1}^{1}P_l(x)P_m(x)dx =\frac2{2l+1}\delta_{lm}.}\]Hermite functions
Hermite’s equation is
\[y''-2xy'+2ny=0.\]Its generating function
\[\boxed{e^{2xt-t^2}=\sum_{n=0}^{\infty}H_n(x)\frac{t^n}{n!}}\]gives, by differentiating with respect to $x$ and $t$,
\[H_n'=2nH_{n-1}, \qquad H_{n+1}=2xH_n-2nH_{n-1}.\]The first functions are $H_0=1$, $H_1=2x$, $H_2=4x^2-2$, and $H_3=8x^3-12x$. To obtain their normalization, multiply two generating functions and integrate:
\[\int_{-\infty}^{\infty}e^{-x^2} e^{2xt-t^2}e^{2xu-u^2}dx =\sqrt\pi e^{2tu}.\]Comparison of the coefficient of $t^nu^m$ yields
\[\boxed{ \int_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx =\sqrt\pi\,2^nn!\,\delta_{nm}. }\]Laguerre functions
Laguerre’s equation is
\[xy''+(1-x)y'+ny=0, \qquad 0\le x<\infty.\]Its generating function is
\[\boxed{ \frac1{1-t}\exp\!\left(-\frac{xt}{1-t}\right) =\sum_{n=0}^{\infty}L_n(x)t^n. }\]Differentiation and coefficient comparison give
\[\boxed{(n+1)L_{n+1}=(2n+1-x)L_n-nL_{n-1}.}\]The first functions are
\[L_0=1,\quad L_1=1-x,\quad L_2=1-2x+\frac{x^2}{2},\quad L_3=1-3x+\frac{3x^2}{2}-\frac{x^3}{6}.\]For the norm, let $G(x,t)$ denote the generating function. Direct integration gives
\[\int_0^\infty e^{-x}G(x,t)G(x,u)dx =\frac1{1-tu}=\sum_{n=0}^{\infty}(tu)^n.\]Equating coefficients gives
\[\boxed{\int_0^\infty e^{-x}L_n(x)L_m(x)dx=\delta_{nm}.}\]
The differential-equation residuals, recurrence relations, and normalization integrals are checked in the Maxima worksheet; every printed residual is zero.
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