…

βΊthe Hermite–Darboux method (see

Whittaker and Watson (1927, pp. 570–572)) can be applied

to construct solutions of (

31.2.1) expressed in quadratures, as follows.
…

βΊHere

${\mathrm{\Xi \xa8}}_{g,N}\beta \x81\u2018(\mathrm{\Xi \xbb},z)$ is a

polynomial of degree

$g$ in

$\mathrm{\Xi \xbb}$ and of degree

$N={m}_{0}+{m}_{1}+{m}_{2}+{m}_{3}$ in

$z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation.
…(This

$\mathrm{\Xi \xbd}$ is unrelated

to the

$\mathrm{\Xi \xbd}$ in §

31.6.)
…

βΊFor

$\mathrm{\pi \x9d\x90\xa6}=({m}_{0},0,0,0)$, these solutions reduce

to Hermite’s solutions (

Whittaker and Watson (1927, §23.7)) of the

Lamé equation in its algebraic form.
…When

$\mathrm{\Xi \xbb}=-4\beta \x81\u2019q$ approaches the ends of the gaps, the solution (

31.8.2) becomes the corresponding Heun

polynomial.
…

…

βΊAll derivatives are denoted by differentials, not by primes.

βΊThe main functions treated in this chapter are the eigenvalues

${a}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m}\beta \x81\u2018\left({k}^{2}\right)$,

${a}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+1}\beta \x81\u2018\left({k}^{2}\right)$,

${b}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+1}\beta \x81\u2018\left({k}^{2}\right)$,

${b}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+2}\beta \x81\u2018\left({k}^{2}\right)$, the

Lamé functions

${\mathrm{\pi \x9d\x90\u0388\pi \x9d\x91\x90}}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x90\u0388\pi \x9d\x91\x90}}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+1}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x90\u0388\pi \x9d\x91}}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+1}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x90\u0388\pi \x9d\x91}}_{\mathrm{\Xi \xbd}}^{2\beta \x81\u2019m+2}\beta \x81\u2018(z,{k}^{2})$, and the

Lamé polynomials
${\mathrm{\pi \x9d\x91\u2019\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+1}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\x90\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+1}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\x91\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+1}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\pi \x9d\x91\x90\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+2}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\pi \x9d\x91\x91\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+2}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\x90\pi \x9d\x91\x91\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+2}^{m}\beta \x81\u2018(z,{k}^{2})$,

${\mathrm{\pi \x9d\x91\pi \x9d\x91\x90\pi \x9d\x91\x91\pi \x9d\x90\u0388}}_{2\beta \x81\u2019n+3}^{m}\beta \x81\u2018(z,{k}^{2})$.
The notation for the eigenvalues and functions is due

to Erdélyi et al. (1955, §15.5.1) and that for the

polynomials is due

to Arscott (1964b, §9.3.2).
…

βΊThe

relation to the

Lamé functions

${L}_{c\beta \x81\u2019\mathrm{\Xi \xbd}}^{(m)}$,

${L}_{s\beta \x81\u2019\mathrm{\Xi \xbd}}^{(m)}$of

Jansen (1977) is given by
…The

relation to the

Lamé functions

${\mathrm{Ec}}_{\mathrm{\Xi \xbd}}^{m}$,

${\mathrm{Es}}_{\mathrm{\Xi \xbd}}^{m}$ of

Ince (1940b) is given by
…

…

βΊIn regard

to orthogonal

polynomials on the unit circle, we now discuss monic

polynomials, Verblunsky’s Theorem, and SzegΕ’s theorem.
…

βΊ
Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions
Over the preceding two months,
the subscript parameters of the Ferrers and Legendre functions,
${\mathrm{\pi \x9d\x96\u2015}}_{n},{\mathrm{\pi \x9d\x96\xb0}}_{n},{P}_{n},{Q}_{n},{\mathrm{\pi \x9d\x91\u0388}}_{n}$
and the Laguerre polynomial, ${L}_{n}$,
were incorrectly displayed as superscripts.
*Reported by Roy Hughes on 2022-05-23*

…

βΊ
Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function
The Gegenbauer function
${C}_{\mathrm{\Xi \pm}}^{(\mathrm{\Xi \xbb})}\beta \x81\u2018\left(z\right)$, was labeled inadvertently as
the ultraspherical (Gegenbauer) polynomial
${C}_{n}^{(\mathrm{\Xi \xbb})}\beta \x81\u2018\left(z\right)$. In order to resolve this inconsistency,
this function now links correctly to its definition.
This change affects Gegenbauer functions which appear in
§§14.3(iv), 15.9(iii).

…

βΊ
Subsection 33.14(iv)
Just below (33.14.9), the constraint described in the text
“$$ when $$,” was removed.
In Equation (33.14.13), the constraint ${\mathrm{{\rm O}\u0385}}_{1},{\mathrm{{\rm O}\u0385}}_{2}>0$ was added.
In the line immediately below (33.14.13), it was clarified
that $s\beta \x81\u2018(\mathrm{{\rm O}\u0385},\mathrm{\beta \x84\x93};r)$ is $\mathrm{exp}\beta \x81\u2018\left(-r/n\right)$ times a polynomial
in $r/n$, instead of simply a polynomial in $r$.
In Equation (33.14.14), a second equality was added which relates
${\mathrm{{\rm O}\x95}}_{n,\mathrm{\beta \x84\x93}}\beta \x81\u2019(r)$
to Laguerre polynomials.
A sentence was added immediately below (33.14.15) indicating that
the functions ${\mathrm{{\rm O}\x95}}_{n,\mathrm{\beta \x84\x93}}$, $n=\mathrm{\beta \x84\x93},\mathrm{\beta \x84\x93}+1,\mathrm{\dots}$, do not form a complete orthonormal system.

…

βΊ
Chapter 25 Zeta and Related Functions
A number of additions and changes have been made to the metadata
to reflect new and changed references as well as to how some equations have been derived.

…

######
§29.14 Orthogonality

βΊLamé polynomials are orthogonal in two ways.
First, the orthogonality

relations (

29.3.19) apply; see §

29.12(i).
…is orthogonal and complete with respect

to the inner product
…

βΊEach of the following seven systems is orthogonal and complete with respect

to the inner product (

29.14.2):
…