Digital Signal Processing Reference
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where the superscripts H and J stand for the Hermite and Jacobi polynomials.
In (2.117), the following expressions for π
(H)
(E) and π
(J
n
(E) exist [128]
n
√
(E) =−
1
n!
π
(H)
n
√
2π
[iπw(E/
2)H
n
(E)−H
n
(E)]
(2.118)
(E) =−
2
α+β+1
B(α + 1,β + 1)
1−E
π
(J)
n
2
1−E
P
(α,β)
n
(E)−Q
(α,β)
n
×
2
F
1
1,β + 1; α + β + 2;
(E)
(2.119)
dx
e
−x
2
∞
√
w(z
+
) =
i
π
z
+
−x
= e
−x
2
/2
[erf(iE/
2)−1]
(2.120)
0
where z
+
is defined by (2.42) and (α,β) are the fixed parameters from the
Jacobi polynomial, w(z) is the wfunction which is related to the error
function (erf) or the probability function, B(α,β) is the betafunction and
2
F
1
(a,b; c; z) is the Gauss hypergeometric function [132, 133]. For machine
accurate and fast computations of the wfunction of a complex variable by
means of continued fractions, the algorithm of Gautschi [100] is optimal. Fur
ther, in (2.118) and (2.119),H
n
(E) and Q
(α,β
n
(E) are the Hermite and Ja
cobi polynomials of the second kind. They stem from (2.54) where the con
stants α
n
and β
n
are the same as in the conventional recursions for H
n
(E)
and P
(α,β)
(E) [133], but with a set of different initial conditions given by
n
1
(E) = 1 and Q
(α,β)
(E) = 0, Q
(α,β)
H
1
(E) = 1. The Legendre and
Chebyshev polynomials are obtained from P
(α,β
n
(x) as two special cases with
α = 0 = β and α =−1/2 = β, respectively. Therefore, we can write, e.g.,
E
(L)
0
(E) = 0,H
0
(E) = π
(L)
(E)/P
(0,0)
(E) where π
(L)
(E) for the Legendre polynomials is
n
n
n
n
available from (2.119) for α = 0 = β.
Advantageously, the integral in (2.107) can be carried out analytically for
classical polynomials. Thus, for the Hermite H
n
(x) and Jacobi P
(α,β
n
(x) poly
nomials the specification N
(H)
(E) and N
(J)
(E) can, respectively, be deduced
in the forms [128]
∞
√
1
√
−
1
√
2π
I
n
n!
e
−E
2
/2
H
n−1
(E)
N
(H)
(E) = M
1 +
π
erf(E/
2π)
(2.121)
n=1
∞
N
(J)
(E) = M
B
E
′
(α + 1,β + 1)
B(α + 1,β + 1)
−
1
2
I
n
n
W
n
(E)P
(α+1,β+1)
(E)
(2.122)
n−1
n=1
W(x)
w
n
W(x) = (1−x)
α
(1 + x)
β
W
n
(x) =
(2.123)
2
α+β+1
2n + 1
Γ(n + α + 1)Γ(n + β + 1)
n!Γ(n + α + β + 1)
w
n
=
.
(2.124)
Here, E
′
= (1 + E)/2, Γ(a) is the gamma function and B
γ
(α,β) is the incom
plete beta function [133].
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