202
16 Hermite Polynomials
we obtain the following connection between Laguerre and Hermite polynomials of
definite parity:
H 2n (x) = (−1)
n 2
2n n! L
−1/2
n
x
2
(16.12a)
H 2n+1 (x) = (−1)
n 2
2n+1 n! x L
1/2
n
x
2
.
(16.12b)
16.2.1 The SPECFUNPHYS Class hermitepoly
The SPECFUNPHYS class hermitepoly returns the polynomial coefficients
of the Hermite polynomials. The syntax is [obj, H] = hermitepoly(n,
one) with “n” the degree of the Hermite polynomial, thus an integer scalar
and “one” optional. For ... = hermitepoly(n) all polynomial coefficients
of the Hermite polynomial of degree 0, 1, 2, · · · , n will be evaluated, and for
... = hermitepoly(n, 1) only the polynomial coefficients of the Hermite
polynomial of degree n. The output variables are “obj”, the object of the class
hermitepoly and the matrix of the polynomial coefficients “H”. The object
“obj” comes with the properties “polycoef”, a table of the polynomial coefficients,
“polynorm”, the normalization vector, “polyint”, the normalization interval (default
[−∞, +∞]), and “info” with some information. As a subclass of polymeth
hermitepoly comes with the same methods as polymeth.
16.2.2 Evaluating Hermite Polynomials
The SPECFUNPHYS Class hermitex
The SPECFUNPHYS class hermitex returns the polynomial values of the Hermite
polynomials. The syntax is [obj, Hn] = hermitex(n, x, wcom). The
input variable “n” should be an integer and x an arbitrary complex array. “wcom”
is an optional input with default value “recurrence”. For ... = hermitex(n,
x) all Hermite polynomials of degree n and lower will be evaluated at the positions
“x”, based on the recurrence formula (16.4). For ... = hermitex(n, x,
’1F1’) or “n” no positive integer value, the computation will be based on
Eq. (16.11) and only the Hermite polynomial of degree n will be computed. Please
note, for n no positive integer Eq. (16.11) will be used despite possible other
definitions in literature. The output arguments are the object “obj” and the matrix
“Hn” of the polynomial values. For degree n the corresponding values are in row
n + 1. (The degree starts with n = 0, but MATLAB indexing with 1.) For non-integer
n the first row corresponds to H 2n+1 and the second row to H 2n . The object “obj”
comes with the properties “value”, the polynomial values (obj.value is identical Hn),
“z”, the position at which the function was evaluated, “degree”, a column vector of
the polynomial degrees, and “info” with some information about the evaluation.
16 Hermite Polynomials
we obtain the following connection between Laguerre and Hermite polynomials of
definite parity:
H 2n (x) = (−1)
n 2
2n n! L
−1/2
n
x
2
(16.12a)
H 2n+1 (x) = (−1)
n 2
2n+1 n! x L
1/2
n
x
2
.
(16.12b)
16.2.1 The SPECFUNPHYS Class hermitepoly
The SPECFUNPHYS class hermitepoly returns the polynomial coefficients
of the Hermite polynomials. The syntax is [obj, H] = hermitepoly(n,
one) with “n” the degree of the Hermite polynomial, thus an integer scalar
and “one” optional. For ... = hermitepoly(n) all polynomial coefficients
of the Hermite polynomial of degree 0, 1, 2, · · · , n will be evaluated, and for
... = hermitepoly(n, 1) only the polynomial coefficients of the Hermite
polynomial of degree n. The output variables are “obj”, the object of the class
hermitepoly and the matrix of the polynomial coefficients “H”. The object
“obj” comes with the properties “polycoef”, a table of the polynomial coefficients,
“polynorm”, the normalization vector, “polyint”, the normalization interval (default
[−∞, +∞]), and “info” with some information. As a subclass of polymeth
hermitepoly comes with the same methods as polymeth.
16.2.2 Evaluating Hermite Polynomials
The SPECFUNPHYS Class hermitex
The SPECFUNPHYS class hermitex returns the polynomial values of the Hermite
polynomials. The syntax is [obj, Hn] = hermitex(n, x, wcom). The
input variable “n” should be an integer and x an arbitrary complex array. “wcom”
is an optional input with default value “recurrence”. For ... = hermitex(n,
x) all Hermite polynomials of degree n and lower will be evaluated at the positions
“x”, based on the recurrence formula (16.4). For ... = hermitex(n, x,
’1F1’) or “n” no positive integer value, the computation will be based on
Eq. (16.11) and only the Hermite polynomial of degree n will be computed. Please
note, for n no positive integer Eq. (16.11) will be used despite possible other
definitions in literature. The output arguments are the object “obj” and the matrix
“Hn” of the polynomial values. For degree n the corresponding values are in row
n + 1. (The degree starts with n = 0, but MATLAB indexing with 1.) For non-integer
n the first row corresponds to H 2n+1 and the second row to H 2n . The object “obj”
comes with the properties “value”, the polynomial values (obj.value is identical Hn),
“z”, the position at which the function was evaluated, “degree”, a column vector of
the polynomial degrees, and “info” with some information about the evaluation.