§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
…
►
31.5.2
►is a
polynomial of degree
, and hence a solution of (
31.2.1) that is analytic at all three finite singularities
.
These solutions are the
Heun polynomials.
…
…
►
Bernoulli Numbers and Polynomials
►The origin of the notation
,
, is not clear.
…
►
Euler Numbers and Polynomials
…
►The notations
,
, as defined in §
24.2(ii), were used in
Lucas (1891) and
Nörlund (1924).
…
§18.3 Definitions
…
►For expressions of ultraspherical, Chebyshev, and Legendre
polynomials in terms of
Jacobi polynomials, see §
18.7(i).
…For explicit power series coefficients up to
for these
polynomials and for coefficients up to
for
Jacobi and ultraspherical
polynomials see
Abramowitz and Stegun (1964, pp. 793–801).
…
►
Bessel polynomials
►Bessel
polynomials are often included among the classical OP’s.
…
§24.18 Physical Applications
►Bernoulli
polynomials appear in statistical physics (
Ordóñez and Driebe (1996)), in discussions of Casimir forces (
Li et al. (1991)), and in a study of quark-gluon plasma (
Meisinger et al. (2002)).
►Euler
polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (
Ballentine and McRae (1998)).