Intrnat. J. Mth. M. Sci.
(1980) 407-421
No.
Vol.
407
CONTRIBUTIONS TO THE THEORY OF HERMITIAN SERIES
III. MEANVALUES
EINAR HILLE
8862 La Jolla Scenic Drive N.
La Jolla, California 92037
U.S.A.
(Received March 9, 1979)
ABSTRACT.
Let f(z) be holomorphc in the strip
o < y < o <
and satisfy
the conditions for having an expansion in an Hermitian series
f(z)
f
n=O
n
hn(Z)
hn(Z)
k
y)
{-
I
e
1/2 2 n
n.
_%
e-1/2Z
2
Hn (z)
Two meanvalues
absolutely convergent in the strip.
(f
(7
-kx2
f(x+iy)
12 dz} 1/2
k
0
i
are discussed, directly using the condition on f(z) or via the Hermitian
series.
Integrals involving products
hm(x+iy) hn(x-iy)
are discussed.
They lead to expansions of the mean squared in terms of Laguerre functions
of
y2
when k
0 and in terms of Hermite functions h
suunctions are holomorphic in y.
n
(21/2iy)
when k
They are strictly increasing when
i.
lYl
The
408
EINAR HILLE
increases.
KEY WORDS AND PHRASES.
Hermitan and Lague S, Meanvalues, Ineqes.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
30A04, 30A16, 22A65.
BASIC FORMULAS.
The functions
h (z)
n
n
(-i)
(1/2
2
n
n.’)
_%
e
1/2z 2 dn (e -z 2
n
(I.i)
dz
L2(-
form a closed orthonormal system in
). If x
f(x) is such that
2
e -x %
) for some Y, 0 _<_= Y < 1/2, the%the Fourier-Hermite coefficients
L2(-
exist
I
fn
f(s) hn(s) ds
(1.2)
and f(x) has a formal Hermitian series
f(x)
f
n--0
n
h (x).
n
(1.3)
Here we consider only the case where f(x+iy) is a holomorphic function of
z
x+iy in a strip
o < y <
<
and the series is absolutely convergent.
The author has shown [i] that this will be the case if for every 8, 0 < 8 < o,
there is a finite B(8) such that
If(x+iy)
< x <
for
,
0
< B()
<= Yl
<
f
n
B.
exp
[-IxI(82 y2)1/2]
This implies and is implied by
< A(a) exp [-a(2n+l)
1/2]
for 0 < a < o and a finite positive A(e).
The function z)h (z) is a solution of the Hermite-Weber differential
n
equation
(1.4)
409
THEORY OF HERMITIAN SERIES
w" + [2n +
z23 w
i
(1.6)
0
also known as the equation of the parabolic cylinder which is moreover
satisfied by
For further properties of these
h_n_l(-iz).
and
h_n_l(iz)
functions see E. Hille [i
3, 4]
2
Here
We shall also encounter Laguerre polynomials in the discussion.
the polynomial L
(a)
e
and degree n is defined by
(u) of order
n
-u
u
e
L
(a)
n
(u)
.,
l d
n
[e
du
n
-u
un+a]
(1.7)
or explicitly
L
(e)
n
n
7.
(u)
j=0
See G. Szeg8 [5, pp. 96-98, i02].
Here we set u
(-I)
i
n
H2n (u)
21/2 y and
h2n(i
For u < 0, -1 <
2
(n+) (-u)J
n-j
(1.8)
j’.
We note that
(-4) n n. L (-)
n
.u2..
(1.9)
use (I.i) to get
1/2 y)
(n1/2") ]
[(n+l)
the function u>L
n
and its graph is concave upwards.
(c)
1/2
(-1/2)
Ln
(u)
2
(-2y2) ey
(i.i0)
is positive, strictly decreasing
Formula (i.i0) shows that
h2n(i
2
1/2
y)
has similar properties for y > 0.
Both the Hermite and the Laguerre functions are essentially special cases
of confluent hypergeometric functions, that is solutions of a second order
linear differential equation of the form
z
w" + (c
z) w’
(i.ii)
a w-- 0.
The equation is satisfied by the entire function
iFl(a,
c; z)= i +
a(a+l)
c(c+l)
j’
j=l
[
(a+j-l)
..(c+j-l)
zj
(1.12)
410
EINAR HILLE
For a
i+ we get
-n, c
iFl(-n,
I+; z)=
(n +L()(Z).n
I
(1.13)
n
The Hermite polynomials are constant multiples of
1/2; z
iFl(-n,
2)
or of
3
ZlFl(-n ;
z
2
(1.14)
according as n is even or odd.
We shall need the asymptotic behavior of these functions for large values
of n and fixed, non-real negative, values of z.
Here the basic results are
due to Oskar Perron [6. p.72].
LEMMA i:
For large positive values of @ and complex, non-real negative
values of z, we have the asymptotic expansion
lFl(a+n
F(c)
c; z)
e1/2Z (zn)-1/2c e2(Zn)
1/2
+
[i
2"2
j
ipj
where the powers and roots take their principal values.
replaced by its m
th
n-1/2J ],
(1.15)
If the series is
partial sum the error committed is of the order of
magnitude of the last term.
Here n
Szeg8
/
Here
.
In the Laguerre case n
+- instead
and we get (see G.
5], p. 193)
LEMMA 2:
L
+
n
() (z)
Co(Z)
Let
.
be an arbitrary real number.
-1/2
e
1/2z
1/2 p-i
(-z) -k-1/2n-1/4e 2(-nz)
j=0
1 and
Cj’s
Cj (z)n-1/2J+o (n -1/2p)
(1.16)
are independent of n and are holomorphic in the plane
cut along the positive real axis.
for z real negative.
Then
The powers must be taken real positive
The bound for the remainder holds uniformly in any
closed region having no point in common with x > 0.
The Hermitian case is discussed in Section 5.
We shall use (i.i0) a lot
411
THEORY OF HERMITIAN SERIES
and this makes it desirable to have comparison formulas for L
(k)
n
(-u) with
n
For all non-negative integers k and n and all u > 0
LENNA 3:
L
(-u)<
F (1/2) F (k+n+l)
F(k+l) F (n+)
L
n
(-1/2)(-u).
(1.17)
Formula (1.8) shows that inequality is trivially true if for
PROOF.
j
(k)
n
0, i, 2
n we have
F (1/2) F (k+n+l)
F(k+l) F(n)
with Q
(z .zs)
or equivalently
(n+k) (n+k-l)... (k+j+l)
This is true with equality for j
=< Q
(n-1/2)
(n)... (1/2+j)
0 by the choice of Q.
If this equality
is divided, the left member by
(k+l) (k+2)
(k+j+2)
and the right member by the smaller quantity
1/2(1/2+i)
(1/2+j+l)
the inequality results.
2.
FORMbAS OF FELDHEIM.
In 1940 Ervin Feldheim prod.uced a paper
on our problem.
this work.
7
which has an important bearing
As a matter of fact it serves as the point of departure of
Several of his formulas figure in earlier papers of mine, but I can
lay no claim to his formulas (20), (20’) and (51) which are basic for the
following.
-1/2
I
Formula (20’) on p. 244 reads for m > n
2
e-t Hm(t
Here we set t
x, v
+ v)
Hn(t
-v) dt
2m
n.’
vm-n Ln(m-n) (2 v2).
iy and express the upper case
lower case h’s to obtain
H’s
in terms of the
(2.1)
EINAR HILLE
412
hm(x+iy hn(x-iy)
2
dx
1/2
(iy)
m-n
L(m-n)
(_2y2)
n
Feldheim’s formula (51) on p. 243 is quite complicated.
and
two positive parameters,
integration t.
,
e
y
2
(2.2)
It involves
an arbitrary variable v and the variable of
Here we set
i, t
-
1/2
2 x,
21/2iy
v
and introduce the h’s to obtain
n-1/2
e
-x
hm(x+iy) hn(x-iy)
=(-l)n
dx
1/2
2-1/2(m+n+l) hm+n (2 1/2 iy).
[.(m,
!..
’]
m.n.
[
(2.4)
It should be noted that the right member of this formula is positive when-
m+n
ever
3.
is even, in particular for m
THE MEANVALUES.
Let us form
|*’k f; y)
where z >f(z)
{-1/2
I e-kx2
n.
If(x+iy)
12 dx} 1/2,
k
0, I
(3.1)
f(x+iy) is supposed to satisfy condition (1.4) so that f(z)
can be expanded in an Hermitian series absolutely convergent in the strip
-o < y < o < oo.
for k
Formula (1.4) ensures the existence of the meanvalues and
0 gives the estimate
L,o (f;
y) < B(8) [R
8(y)]-%
with
RB(y)
for every y, 0 _<
IYl
(82 y2)1/2
< 8 and 8, 0 < 8 <
.
this is obviously an increasing function of
(3.3)
For every fixed admissible 8
IYI"
That the bound is increasing
413
THEORY OF HERMITIAN SERIES
evidently does not imply that the meanvalue is increasing.
This question will
be examined later.
I,l(f
Here (1.4) implies that
y) is more complicated.
2
[|,(f; y)-I <
2
[B(13)]2
exp 1-- x
2
Bg(y)
x-I
dx
The right member is obviously a bound decreasing function of
We can expand the integrand in powers of
function of
RB(y);
RB(y).
IYl
for fixed
The integral is an entire
as a matter of fact the Mittag-Leffler function
n
F(w)
Fa(w)
Here
4.
w
n=0
w
F(1/2n + i)
R(y).
2
is of order 2 and normal type.
THE FUNCTION h (iv).
In the first paper of this series [ 2] it was shown that h (z) is the
n
unique solution of the Volterra integral equation
w(z)
c (z)
n
+ N -1
j’z
t
2
sin
IN(z-t)] w(t) dt
O
where
N
(2n+l)
1/2 c (z)
C(n) cos (Nz
n
1/2 n)
(4.2)
while C(n) depends on the parity of n so that
C(2k)
C(2k+l)
IH2k(0) [1/2 22k
IH2k+l(0) l(4k+3
-1/2
(2k).’]
-1/2
-2
2 2k+l (2k+l)’.] -1/2
In both cases we have
C(n)
n
-
[rCk)1/2
(4.3)
Lr(+)
-2
r(k#)]
L
1/2
j
The method of successive approximations applies to (4.1) and leads to
rapidly convergent series of the form
(4.4)
(4.5)
EINAR HILLE
414
h (z)= c (z)
n
n
.
m=0
Pm(NZ) N-4m +
s (z)
n
.
m=0
Qm(NZ)
N
-4m
(4.6)
where
The factors P
1/2 n).
C(n) sin (Nz
s (z)
n
and O are polynomials in z with rational coefficients.
m
m
-
m
is
is odd and they are of degree 3m or 3m-i whichever has the right
even and
We have
parity.
P
eo(Z)
1/2 z 2
el(z)
i,
zl
For large values of
and n we have
Cn(Z).
h (z)
i-n
> 0.
iv with v
Suppose now that z
0 <
o, Ql(Z)
Qo(Z)
i,
h (iv)
n
gn(V)
gn(V)
1/2 C(n)
1
We have then (see
< C(n) e
[exp
(y3/6N)
z
3
1
z
(4.7)
3] pp. 81-82)
(4.8)
l]
where
i
If v
o(i).
o(n
6)
e
Nv
+ (-i) n
e-NV].
(4.9)
it is seen that the quantity within brackets in
(4.8) is
For our purposes a weaker result suffices:
LEMMA 4.
There is a finite quantity M(), depending only on
that for all n and 0
=< Yl =<
0 <
i-n
h (iv) < M(O) n
n
-
e
,
Nv
such
(4.10)
Combining Lemma 3 and 4 with formulas (i.i0) and (2.2) we obtain
LEMMA 5.
L
For all non-negative integers k and n and 0
2
ey
(k)(_2y2)
n
< M()
n
-1/4F (k + n +
i)
r(k+l)[ r(n .)r(n+l)
We note also (see Theorem 1.4, p. 883 of [3])
e
]1/2
__<
y <
(8n+2)1/21yl
(4.11)
THEORY OF HERMITIAN SERIES
LEMMA 6.
415
For y # 0
h (x+iy)
+ Nx
e--
i
h (iy)
n
+ 0(I/N)
(4.12)
Here the sign in the exponent is the same as that of y.
I/E < x < i/e, 0 <
uniformly with respect to x and y in the regions
IYl
<= i/.
5.
ESTIMATE OF [I, (f;
The relation holds
<
y)]2.
o
We have
If,, (f; y)
]2 -1/2
I {roT
m=0 n=0
=0
fm hm(x+iy) }{n=0[
m
n
%(x+iy) hn(x-iy)
We split this double series into three parts.
m
n, S
1
the terms where m > n and
S_I
reflection shows that
n hn(x-iy)
S_I
S
o
} dx
dx.
involving the terms where
those where n > m.
must be the complex conjugate of S
A moment’s
1
so no separate
es timate is needed.
Now by (2.2)
S
[ If nl 2
o
L
n=0
2
ey
(o)(_2y2)
n
(5.2)
Here we use (1.4) and Lemma 5 to obtain
S
which converges for
<
O
IYl
M(e)
<
-
[
exp
(5.3)
The sum of the majorant series is
0
This is a rather poor estimate.
exponent should
[-(s-lyl)(8n+2) 1/2]
n--O
(e-lyl)-2].
Formulas (3.2) and (3.3) suggest that the
be-1/2 rather than-2.
EINAR HILLE
416
For
IYl
We set k
> 0 Lemma 2 gives a slightly better result.
>
0
and get
L
(0)(_2y2)
n
e
y
2
C y
y-1/2 n-1/4
This introduces the factor
S
C(, )
<
0
-1/2
n-1/4
e
1/2y[ i + O(n-1/2)].
(gn)
(5.5)
in (5.3) and gives
y-1/2 I n-1/4 exp [n= 0
(-lyl)(8n+2) 1/2]
(5.6)
and the estimate
3
S
The series S
l
0
o
[y-1/2 (_[y[)2].
(5.7)
presents greater difficulties.
Here we set m
n+k and
note that
S
n.’ 7.
n
k=l
n=O
7.
I
n+k
f
2
n+k
Here we naturally think of Lemma 5.
(ly)k Ln (k) (-2y2)
e
y
2
It turns out to be too weak to give
absolute convergence of the series (5.8) in the whole strip
lyl
< a < o.
using
it and later also Lemma i we can prove convergence in a strip of width about
but omitting the real axis.
This result clearly indicates that the argument
has to be based on the asymptotic formula of Lemma 2.
The neighborhood of the
origin is not accessible by this method but will be taken care of by consider-
atlons of analyticity.
We use formula (1.5) to obtain
7. (1/22 n n.’) -% n.’
lSl[ < A2(a) n--0
;. 1/2 2k+n
x
[
k--’l
(k+n)
]-%
e
-a (2k+2n+l)
.
1/2 k+n
2
This we simplify and use formula (1.16).
ISll
<
B(a, ) y -%
n=l
n-1/4
e
e
-e (2n+l)
y] k
1/2
L(k)
(_2y2) ey
n
2
(5.9)
Thus
-2a(2n+l)
1/2
+ [Yl (gn) 1/2 Tn
(5.0)
417
THEORY OF HERMITIAN SERIES
where
1/2
’n
[ (k+n) ] n1/2k e’U[ (2k+2n+l)
Tn k--I1 2.k
1/2
Here the last factor is < i and may be neglected.
(k+n)
(nl) (n+2
(2n+l)
1/2]
Further
(5.2)
< i
(n+k)
so that
T
n
2 -k= i.
[
<
k:l
Hence
ISll
<
B(a, T)
ly1-1/2 [
n
which converges for O <
IYl
-1/4 exp
-
n:l
(e-lyl)(Sn+2) 1/2]
(.z3)
and has a sum satisfying (5.7).
<
,
(f; y)
o
O
[lyl
3
(=-lyl)
43.
(5.14)
which holds for every a, O < a < o and for every y, O < E
=< IYl
The constant implied by the order symbol of course depends on
6.
ESTIMATE OF [ |i, (f
y)
I
Thus
<
< O <
and
E.
.
]2.
.
With the same assumptions of f(z) as above we have by formula (2.4)
[i,l(f; y)]2
(-l)n
f
m--O n=O
m
2 -1/2(m+n+l) h
m+n
n
(i21/2y)
(6.1)
[
[
k=0 m+n=k
(-i)
n
f
m
n
]2 -1/2(k+l)
(i2
1/2y).
This series will be shown to be absolutely convergent for O
us first estimate the finite sum where
m+n
k.
By (1.5)
__<
y <
< o.
Let
418
EINAR HILLE
"
ISk I=m+n=k (-l)n fm fn < A
1/2
2
m+n m.’ n.’ )-1/2 x
(2
I
m+n=k
()
(2n+l)]}
+
exp
(2re+l)
A2()2-1/2k
k’
(k-m).’
m’
m=0
[
I
(6.2)
1/2
T
m,n
where
exp {-e
T
mn
(2m+l)
1/2 + (2n+l) 1/2]}
(6.3)
We now apply Cauchy’s inequality and find that
2
m=0
T
m.’ (k-m)
1/2k and this
k.
"
m=0
m n
Here the first sum equals 2
to m
k
<
m.
2
k.’
T
(k-m)’ m+n=k m, n
"
(6.4)
In the second sum the largest term corresponds
term is approximately
exp [-2(4k+2)
1/2]
(6.5)
so that the second sum is at most
(k+l) exp
-2 (4k+2’)
1/2]
(6.6)
Hence
ISkl < A2()
2
-1/2k
.
F(k+l)] -1/2 (k+l) 1/2 exp [-e(4k+2)
1/2]
(6.7)
Thus by (6.1), (6.7) and Lemma 4
Fh. l(f;
y)]2
2
< A ()
(2) -1/2
(k+l
k=l
)1/2
(1/2k+)
(k+l) ]%
exp
-(- ly I)
(4k+2)1/23.
(6.8)
Using the duplication formula for the Gamma function we see that this simplifies
to an expression of the form
[
k=0
Here
2-k
(1/2k + i)
exp [-
(a-ly[)(4k+2)1/23
(6.9)
419
THEORY OF HERMITIAN SERIES
G(s)
-k
2
z. r(1/2k + i)
k=0
e
s (4k+2)
1/2
(6.10)
is an entire function of s of order 2 and minimal type.
The Dirlchlet series
Compare formula (3.5) where
(6.10) converges for all finite values of s.
the majorant is of order 2 and normal type.
7.
ANALYTICITY OF THE
SUARED
MEANS.
The functions
y)]2
Y [|,o(f
y
are actually analytic functions of y.
>[I,(f; y)]2
(7.1)
To see this we consider the corre-
sponding Laguerre series in the first case, the Hermltian series in the
second.
.
In the first case we consider the series
F(z)
I [fk+n n + (-l)k fk+n f n
k:0 n:0
E
iy reduces to [. (f;
o
which for z
y)]2
z
k
L
(k)
2
-z
(2z) e
n
2
(7.2)
and represents the analytic continuation
of the mean square function in any bounded domain in the strip -o < y < o in
which the series converges uniformly with respect to z.
The variable parts of
the two series are
(iy)
respectively.
k
L(k)(-2y
2)
n
e
y
2
and
z
k
L
(k) (2z
n
2) e- z
2
(7.3)
Now Lemma 2 shows that for fixed k and n the ratio of the
absolute values of the two expressions tend to the limit
bounded away from 0 and
strip -o < y < o <
ly/zl 1/2
which is
if z is restricted to a finite domain D in the
having a positive distance from the lines y
-o, O,
.
It follows that the series (7.2) converges uniformly in D and thus represents
a holomorphic function in D.
Actually F(z) is holomorphic in the whole strip 0 _<
IYl
< o.
be concluded from an application of formula (2.1) where we set v
This may
z"
A
420
EINAR HILLE
simple calculation gives
-1/2
F(z)
valid for any z in the strip 0
I
f(t + z) f(t
__<
IYl
< o <
the maximum modulusof F(z) for the substrip
function of
.
.
z) dt
(7.4)
It follows, in particular, that
IYl -<
< o is an increasing
Similar considerations apply to the square of the second mean as we see
.I
from formulas (2.3) and (2.4) where we replace iy by z and set
G(z)
-1/2
iy reduces to
which for z
e_t
2
f(t + z) f(t
[(f; y)32.
an expansion
C [,l(f; y)
32
.
n--O
z) dt
(7.5)
While the latter function of y has
gn hn(i 21/2y)
(7.6)
h (2% z).
n
(7.7)
we have
G(z)
[ gn
n=O
By the virtue of Lemma 6 the absolute convergence of the series (7.6) for
a y
Im (z)
line.
,
< o, implies the absolute convergence of (7.7) for any z with
and the convergence is uniform in every finite subinterval of the
Thus G(z) is holomorphic in the basic strip and is actually an entire
function of z.
The properties of the maximum modulus are the same as in
the first’ case, that is an increasing function of
.
REFERENCES
i] Hille, E. Contributions to the Theory of Hermitian Series. II. The
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THEORY OF HERMITIAN SERIES
[ 2]
Hille, E. A Class of Reciprocal Functions.
427-464.
[ 3]
Hille, E.
421
Ann. Mh. {Z) 27 (1926)
Contributions to the Theory of Hermitian Series.
DaZ Mh. J.,
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4]
Hille, E.
Sur les fonctions analytiques d6finies par des series d’Hermite.
J. t. Pu66 Appl. (9) 4._0 (1961) 335-342.
[ 5]
Szeg, G.
Orthogonal Polynomials.
Vol. 2
1939.
Ame. kut. Sot., Cooqu/um
Plins
6]
Perron, O.
Uber das Verhalten einer ausgearteten hypergeometrischen
Reihe bei unbegrenztem Wachstum eines Parameters.
Mosth. 15__] (1921)68-78.
[7
Feldheim, E. Developments en serie de polynomes d’Hermite et de Laguerre
l’aide des transformations de Gauss et de Hankel. Nzde/u[. A.
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