Let P_n(z) be a polynomial of nth order with distinct
zeros zeta_k, with k = 0, 1, ..., n-1.
Then the inverse of the polynomial is a meromorphic function of z,
with simple poles at the zeros.
n-1
[1/P_n(z)] = PI [ 1/(z - zeta_k ) ]
k=0
Then expanding the product in partial fractions gives
n-1
[1/P_n(z)] = SIGMA [R_k/(z - zeta_k ) ]
k=0
where the residue,
R_(k) := PI [1/(zeta_(k) - zeta_j) ]
j not= (k)
lim [(z - zeta_m)/P_n(z)] = R_m
z->zeta_m
The derivative of the polynomial P_n(z) can be expressed by
n-1 P_n(z)
(d/dz) P_n(z) = SIGMA ------------
k=0 (z - zeta_k)
n-1 1
= P_n(z) SIGMA ------------
k=0 (z - zeta_k)
Therefore,
n-1 1
(d/dz) log( P_n(z) ) = SIGMA ------------
k=0 (z - zeta_k)
and
d 1 d
-- [------- -- P_n(z)] =
dz P_n(z) dz
n-1 1
- SIGMA --------------
k=0 (z - zeta_k)^2
The unnormalized Hermite polynomials defined
by the Rodrigues formula [equation (9.4) and below]
satisfy (See, e. g., [Morse 1953], v1, p. 786) the following:
(d/dz) H_n(z) = 2 n H_(n-1)(z)
( (d/dz)^2 - 2z d/dz + 2n ) H_n(z) = 0
z H_n(z) = n H_(n-1)(z) + (1/2) H_(n+1)(z)
(d/dz) ( exp( -z^2 ) H_n(z) ) = -2 exp( -z^2 ) H_(n+1)(z)
infinity t^n
exp( -t^2 + 2 tz ) = SIGMA ---- H_n(z)
n=0 n!
2^n +infinity
H_n(z) = ---------- INTEGRAL (z + it)^n exp( -t^2 ) dt
sqrt( pi ) -infinity
n n!
H_n(x - y) = SIGMA --------- y^(n-j) H_j(x)
j=0 j! (n-j)!
n n!
= 2^(-n/2) SIGMA --------- H_j(x sqrt(2)) H_(n-j)(-y sqrt(2))
j=0 j! (n-j)!
n
SIGMA binom(n k) H_k(x) H_n-k(y) = 2^(n/2) H_n( (x+y)/sqrt(2) )
k=0
n
SIGMA binom(n k) H_k(x) (2y)^(n-k) = H_n( x+y )
k=0
exp -(x^2 + y^2 - 2xyz)/(1 - z^2)
---------------------------------
sqrt(1 + z^2)
infinity (z/2)^k
= exp( -(x^2 + y^2) ) SIGMA -------- H_k(x) H_k(y)
k=0 k!
From the preceding as a special case, the expansion
of the Fourier kernel in Hermite polynomials:
exp( -(1/2)(x^2 + s^2) ) infinity
exp( isx ) = ------------------------ SIGMA i^k ~H_k(x) ~H_k(s)
sqrt((2 pi)) k=0
Then also from the preceding as a special case, the expansion
of an origin centered Gaussian function in Hermite polynomials:
infinity
(1/sqrt(2)) exp(-s^2) = SIGMA ~H_k(s) ~H_k(s)
k=0
n(n-1) n(n-1)(n-2)(n-3)
H_n(z) = (2z)^n - ------ (2z)^(n-2) + ---------------- (2z)^(n-4)
1! 2!
- ... .
where the last term is
n!
(-1)^(n/2) --------
(n/2)!
if n is even and
n!
(-1)^((n-1)/2)) ---------- 2z
((n-1)/2)!
if n is odd.
In closed summation form
[n/2] 2^(n-2k) n!
H_n(z) = SIGMA (-1)^k ------------ z^(n-2k)
k=0 k! (n-2k)!
[n/2]
= SIGMA (-1)^k k! (n 2k) (2k k) (2z)^(n-2k)
k=0
where [n/2] is the integral part of n/2.
The second expression uses binomial coefficients.
Relations between the various Weber-Hermite functions
[Whittaker 1927]:
H_n(z) := (-1)^n exp( z^2 ) (d/dz)^n exp( -z^2 )
D_n(z) := (-1)^n exp( (z^2)/4 ) (d/dz)^n exp( -(z^2)/2 )
= exp( -(1/2)(z/sqrt(2))^2 )/(2^(n/2)) H_n(z/sqrt(2))
~H_n(z) := H_n(z)/sqrt((2^n n!)
~D_n(z) := D_n(z)/[ (2 pi)^(1/4) sqrt(n!) ]
~D_n(z) = exp( -(1/2)(z/sqrt(2))^2 )/(2 pi)^(1/4) ~H_n(z/sqrt(2))
The orthogonality and normalization integrals for the various
Weber-Hermite functions:
+infinity
INTEGRAL exp( -z^2 ) H_n(z) H_m(z) dz = (2 pi n! m!)^(1/2) delta_(nm)
-infinity
+infinity
INTEGRAL D_n(z) D_m(z) dz = (2 pi n! m!)^(1/2) delta_(nm)
-infinity
+infinity
INTEGRAL exp( -z^2 ) ~H_n(z) ~H_m(z) dz = sqrt( pi ) delta_(nm)
-infinity
+infinity
INTEGRAL ~D_n(z) ~D_m(z) dz = delta_(nm)
-infinity
The D_n(z) satisfy the homogeneous integral equation
+infinity
lambda INTEGRAL exp( i (sx/2) ) phi(s) ds = phi(x)
-infinity
for appropriate characteristic values {lambda_n}. In fact,
lambda_n = (-i)^n (n!/sqrt(2))
and the usual Fourier kernel can be expanded in D_n(z):
infinity
exp( i (sx/2) ) = 2 sqrt( pi ) SIGMA i^n ~D_n(s) ~D_n(x)
n=0
so one can write the eigenvalue equation of the Fourier transform:
+infinity
INTEGRAL exp( i (sx/2) ) ~D_n(x) dx = sqrt((4 pi)) i^n ~D_n(s)
-infinity
Approximating the roots of H_n(z)
In [Section IX] asymptotic formulae for the roots of H_n(z), together with a spacing rule have been found which are not very good approximations for low values of n. Using these approximations as a starting point and rescaling them so that the rescaled approximations conform to the known exact constraint (9.17) on the true roots of H_n(z) yields new approximation formulae that are remarkably good. The asymptotic values from [Section IX] [Section IX] are given by
h(n, k) = - ( pi/(2 sqrt(n)) ) (k - (1/2)(n-1))
where k = 0, 1, 2, ..., (n-1), and h(n, 0) is the maximal h(n, k).
From equation (9.17), we know that for the true roots q(n, k),
(n-1)
SIGMA q^2(n, k) = (n/2)(n-1)
k = 0
but
(n-1)
SIGMA h^2(n, k) = (n/2)(n-1) pi^2 (n+1)/(24 n)
k = 0
Now define rescaled h(n, k)
h'(n, k) := h(n, k) pi sqrt([(n+1)/(24 n)]
= - sqrt((6/(n+1)) (k - (1/2)(n-1))
so that
(n-1)
SIGMA h'(n, k) = 0
k = 0
and
(n-1)
SIGMA h'^2(n, k) = (n/2)(n-1)
k = 0
Where the spacing between the h(n, k) is ( pi/(2 sqrt(n)) ),
the spacing for the rescaled h'(n, k) is then sqrt((6/(n+1))).
There is no n > 0 for which these spacings are equal, and
for all n > 0,
( pi/(2 sqrt(n)) ) > sqrt((6/(n+1))).
Comparing the actually numerically calculated values of the roots
with the two approximations for n=6, we find
Note in update: you can get the numerical roots of H_n(x) in
MACSYMA (maxima under Linux) to 16 decimal places by default
by:
load("specfun"):
allroots( hermite(n, x) );
specifying the value of n. Reset the value of the internal
variable fpprec to a value greater than 16 for greater precision;
16 is the default value. Under Linux, the runtime maxima
rpm package does not require separate installation of a LISP
dialect when the LISP variant is GNU Common Lisp (gcl);
the runtime packages for other dialects require installation
of that LISP dialect separately and in addition. In those
cases, I recommend clisp or cmucl, in that order. allroots();
can give inaccurate results in case of multiple roots of a
given polynomial, but that will not be the case here with
Hermite polynomials.
wch - September 15, 2005
q(n, k):
(2.350614 1.335851 0.436078 -0.436078 -1.335851 -2.350614)
h'(n, k):
(2.31455 1.38873 0.462910 -0.462910 -1.38873 -2.31455)
h(n, k):
(1.60319 0.961911 0.320637 -0.320637 -0.961911 -1.60319)
For n=12
q(n, k):
(3.88973 3.0208356 2.27914395 1.597825 0.947782 0.314240435
-0.314240435 -0.947782 -1.597825 -2.27914395 -3.0208356 -3.88973)
h'(n, k):
(3.73651 3.05715 2.37778 1.69841 1.01905 0.339683
-0.339683 -1.01905 -1.69841 -2.37778 -3.05715 -3.73651)
h(n, k):
(2.49396 2.04052 1.58707 1.13362 0.680172 0.226724
-0.226724 -0.680172 -1.13362 -1.58707 -2.04052 -2.49396)
The goodness of approximation is typical for the values of n from 2 to 12, as may be easily verified from the formula for h'(n, k) and the table of numerically extracted roots in [Section IX]. It would appear that these approximations are as good as they are because asymptotic linearity of the roots is already a dominant effect even for low values of n.
It is always true that
sqrt(6/n) > sqrt(6/(n+1)) > π/(2 sqrt(n))
The ratio of spacings is,
sqrt(6/(n+1)) / (π / (2 sqrt(n))) =
(2 sqrt(6n)) / (π sqrt(n+1)) =
(2/π) sqrt(6) sqrt( n/(n+1) ) =
(sqrt(24)/π) sqrt( n/(n+1) )
sqrt(24) < 5
(sqrt(24)/π) = 4.898979/3.14159 = 1.559395
sqrt( n/(n+1) ) = sqrt(1 + 1/n) → 1
So, asymptotically, the ratio of these approximations of
the root spacings is
(sqrt(24)/π) = 1.559395 (approximately)
NB: While for the maximal root, both h(n, 0) and h'(n, 0)
appoximate the root from below,
h(n, 0) < h'(n, 0) < q(n, 0)
hence,
h(n, n-1) > h'(n, n-1) > q(n, n-1)
since q(n, n-1) = - q(n, 0).
h(n, k) always approximates
roots from below, while h'(n, k) for k ≠ 0 seems to obey
|h'(n, k)| > |q(n, k)|
while, also for k ≠ 0,
|h(n, k)| < |q(n, k)|
approximating the positive roots from above, and the negative
roots from below.
The maximal root is (spacing) * (n-1)/2, and its so far
best approximation is h'(n, 0), and its negative is also the best
approximation of the best approximation of the minimal root.
h'(n, 0) = sqrt(6/(n+1)) → q(n, 0)
For all roots other than k = 0 (maximal), and k = n-1 (minimal),
the best approximation so far is by taking the average, of the
approximations from below and above:
h"(n, k) := (1/2)( h(n, k) + h'(n, k) ) =
- (1/2) [ sqrt(6/(n+1)) + π/(2 sqrt(n)) ] (k - (n-1)/2) =
Using this scheme, the roots for n=12 through n=21 are approximately:
n=12:
3.81279 3.11956 2.42632 1.73308 1.03985 0.346610
-0.346620 -1.03986 -1.73309 -2.42632 -3.11956 -3.81280
n=13:
4.00206 3.33505 2.66804 2.00103 1.33402 0.667010
0
-0.667010 -1.33402 -2.00103 -2.66804 -3.33505 -4.00206
n=14:
4.18311 3.53956 2.89600 2.25245 1.60889 0.965330 0.321780
-0.321770 -0.965330 -1.60889 -2.25244 -2.89600 -3.53955 -4.18311
n=15:
4.35691 3.73449 3.11208 2.48966 1.86725 1.24483 0.622420
0
-0.622410 -1.24483 -1.86724 -2.48966 -3.11207 -3.73449 -4.35690
n=16:
4.52423 3.92100 3.31777 2.71454 2.11131 1.50808 0.904850 0.301620
-0.301610 -0.904840 -1.50807 -2.11130 -2.71453 -3.31776 -3.92099
-4.52422
n=17:
4.68576 4.10004 3.51432 2.92860 2.34288 1.75716 1.17144 0.585720
0
-0.585720 -1.17144 -1.75716 -2.34288 -2.92860 -3.51432 -4.10004
-4.68576
n=18:
4.84202 4.27237 3.70272 3.13307 2.56342 1.99377 1.42412 0.854470
0.284820 -0.284830 -0.854480 -1.42413 -1.99378 -2.56343 -3.13308
-3.70273 -4.27238 -4.84203
n=19:
4.99351 4.43867 3.88384 3.32901 2.77417 2.21934 1.66450 1.10967 0.554830
0
-0.554840 -1.10967 -1.66451 -2.21934 -2.77418 -3.32901
-3.88385 -4.43869 -4.99352
n=20:
5.14064 4.59952 4.05840 3.51728 2.97616 2.43504 1.89392 1.35280
0.811680 0.270560 -0.270560 -0.811680 -1.35280 -1.89392 -2.43504
-2.97616 -3.51728 -4.05840 -4.59952 -5.14066
n=21:
5.28375 4.75538 4.22700 3.69862 3.17025 2.64187
2.11350 1.58512 1.05675 0.528370
0
-0.528370 -1.05675 -1.58513 -2.11350 -2.64188
-3.17025 -3.69863 -4.22700 -4.75535 -5.28375)))
NB Writing XI!(n) Q(n) XI(n) = Qd(n) out in component form.
Are the resulting formulas useful, or a special case of
known formulas for the Hermite polynomials?
Result:
(n-1)
SIGMA (d_m d_(m-1))^(-1/2) (m/2)^(1/2) [ ~H_m(q(n, k)) ~H_(m-1)(q(n, j))
m=0
+ ~H_m(q(n, j)) ~H_(m-1)(q(n, k)) ]
= q(n, k) delta_(kj)
The following gives a specific example of an analytic expression in terms of a contour integral for the sum of values of a function at the zeros of another.
Let GAMMA be a closed contour in the complex plane that encloses the zeros
zeta_k, with k = 0, 1, ..., n-1, of a P_n(z) as defined before, and let
F(z) be some function analytic within GAMMA. By Cauchy's theorem,
if lambda is within GAMMA, the value F(lambda) can be expressed by
1 F(z)
F(lambda) = -------- INTEGRAL ------------ dz
(2 pi i) GAMMA (z - lambda)
In particular, let lambda be any of the zeta_k.
Then, summing on k,
(n-1) (n-1) 1 F(z)
SIGMA F(zeta_k) = SIGMA ---------- INTEGRAL ------------ dz
k=0 k=0 (k 2 pi i) GAMMA (z - zeta_k)
1 (n-1) 1
= -------- INTEGRAL F(z) SIGMA ------------ dz
(2 pi i) GAMMA k=0 (z - zeta_k)
1
= -------- INTEGRAL F(z) (d/dz)[log P_n(z)] dz
(2 pi i) GAMMA
If P_n(z) = H_n(z), so H_n(zeta_k) = 0, then
(n-1) n H_(n-1)(z)
SIGMA F(zeta_k) = ------ INTEGRAL F(z) ---------- dz
k=0 (pi i) H_n(z)
Email me, Bill Hammel at