Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 27–35
On the square integrability of the q-Hermite functions
M.K. Atakishiyeva a , N.M. Atakishiyev b; ∗ , C. Villegas-Blas b
a
b
Facultad de Ciencias, UAEM, Apartado Postal 396-3, C.P. 62250 Cuernavaca, Morelos, Mexico
Instituto de Matematicas, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaca, Morelos, Mexico
Received 20 October 1997; received in revised form 21 May 1998
Abstract
2
Overlap integrals over the full real line −∞¡x¡∞ for a family of the q-Hermite functions Hn (sin x|q)e−x =2 ; 0¡
2
q= e−2 ¡1 are evaluated. In particular, an explicit form of the squared norms for these q-extensions of the Hermite
functions (or the wave functions of the linear harmonic oscillator in quantum mechanics) is obtained. The classical
Fourier–Gauss transform connects the q-Hermite functions with di erent values 0¡q¡1 and q¿1 of the parameter q.
An explicit expansion of the q-Hermite polynomials Hn (sin x|q) in terms of the Hermite polynomials Hn (x) emerges as
c 1998 Elsevier Science B.V. All rights reserved.
a by-product.
1. Introduction
The Hermite functions
√ n −1=2
2
Hn ()e− =2 ;
n () := [ 2 n!]
(1.1)
where Hn () are the classical Hermite polynomials, are of great mathematical interest as an explicit
example of an orthonormal and complete system in the Hilbert space L2 (R) of square-integrable
functions with respect to the full real line −∞¡¡∞ [1]. In mathematical physics they are known
to represent solutions of the linear harmonic oscillator problem, which plays a very important role in
quantum mechanics. In what follows, we attempt to study in detail some particular q-generalization
of the Hermite functions (1.1).
2. Overlap integrals and squared norms
Let us consider a family of q-Hermite functions
n (|q) := cn (q)Hn (sin |q)e
∗
−2 =2
;
2
0¡q = e−2 ¡1;
Corresponding author. E-mail: natig@matcuer.unam.mx.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 2 - 3
(2.1)
28
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
√
where the normalization constant
c (q) = [ (q; q)n ]−1=2 and the q-shifted factorial (q; q)n is de ned
Qn−1 n
as (z; q)0 = 1 and (z; q)n = k=0 (1 − zq k ), n = 1; 2; 3 : : : (throughout this paper, we will employ the
standard notations of q-special functions, see [2] or [3]). The continuous q-Hermite polynomials
Hn (x|q) in (2.1) are those q-extensions of the ordinary Hermite polynomials Hn (x), which satisfy
the three-term recurrence relation
Hn+1 (x|q) = 2xHn (x|q) − (1 − q n )Hn−1 (x|q);
n = 0; 1; 2; : : : ;
(2.2)
with the initial condition H0 (x|q) = 1 [4]. Their explicit form is given by the Fourier expansion
Hm (sin |q) =
m
m
X
(−1)
n=0
where
m
n
m
n
q
n
m
n
e(2n−m) ;
(2.3)
q
is the q-binomial coecient,
q
(q; q)m
m
=
:=
m−n
(q; q)n (q; q)m−n
(2.4)
:
q
The q-Hermite polynomials (2.3) are solutions of the di erence equation
eis exp −i
d
ds
+ e−is exp i
d
ds
Hn (sin s|q) = 2q−n=2 cos s Hn (sin s|q):
(2.5)
It is easy to verify that limq→1 −2n (q; q)n = 2n n!. Therefore it follows from the recurrence relation
(2.2), that
lim −n Hn (sin |q) = Hn ():
(2.6)
q→1
Thus, the normalization in (2.1) is chosen so that when the limit q → 1 is taken they coincide with
the wave functions n () of the linear harmonic oscillator in quantum mechanics, i.e.
n (|1) :=
lim
q→1
n (|q) =
n ():
As it is well known, the wave functions
Z
(2.7)
n ()
satisfy the orthogonality relation
∞
m () n () d = mn
(2.8)
−∞
and may serve as a basis in the Hilbert space L2 (R) of square-integrable functions with respect
to d.
We evaluate rst the corresponding integral
Im; n (q) :=
Z
∞
−∞
m (|q) n (|q) d = In; m (q)
(2.9)
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
29
for the q-Hermite functions (2.1). Since Hn (−x|q) = (−1)n Hn (x|q) by de nition, the functions m (|q)
and n (|q) of the opposite parities (m − n = 2k + 1; k = 0; 1; 2; : : :) are orthogonal and nontrivial
integrals in (2.9) are
In; n+2k (q) = [(q; q)n (q; q)n+2k ]
Z
−1=2
∞
2
Hn (sin |q)Hn+2k (sin |q)e− d:
(2.10)
−∞
Using the Rogers linearization formula [5]
Hm (x|q)Hn (x|q) =
m
X
(q; q)n
(q; q)n−m+k
k=0
m
k
Hn−m+2k (x|q);
m 6 n;
(2.11)
q
for the q-Hermite polynomials (2.3), one can represent (2.10) as
Z ∞
n
1=2 X
2
(q n+1 ; q)2k
n
−1
√
In; n+2k (q) =
H2k+2l (sin |q)e− d:
(q; q)2k+l
l
q −∞
l=0
(2.12)
It remains only to substitute the explicit form of the q-Hermite polynomials (2.3) into (2.12) and
to evaluate the integral with respect to the variable by the aid of the well-known integral transform
Z
∞
√
2
2
d x e2xy−x = e−y :
(2.13)
−∞
The result is
In; n+2k (q) = (q
n+1
1=2
; q)2k
n
X
(−1)
k+l
l=0
2(k+l)
X
×
(−1) j
j=0
2(k + l)
j
n
l
q
2
q(k+l) =2
(q; q)2k+l
2
q j =2−j(k+l) :
(2.14)
q
The sum over j in (2.14) gives the factor (q1=2−k−l ; q)2(k+l) because of the Gauss identity
(z; q)n =
n
X
n
k
k=0
q k(k−1)=2 (−z) k :
(2.15)
q
In view of the formulas (see, for example, [2] or [3])
(z; q)n+k = (z; q)n (zq n ; q)k ;
(zq ; q)n = q
−n
−n(n+1)=2
(2.16)
n
(−z) (q=z; q)n ;
(2.17)
this factor is equal to
2
(q1=2−k−l ; q)2(k+l) = (−1) k+l q−(k+l) =2 (q1=2 ; q)2k+l :
(2.18)
Now substituting (2.18) into (2.14), yields
In; n+2k (q) = (q
n+1
1=2
; q)2k
n
X
n (q1=2 ; q)2k+l
l=0
l
q
(q; q)2k+l
:
(2.19)
30
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
Since (q; q)2k+l = (q; q)2k (q2k+1 ; q)l by (2.16) and
(q; q)n−m = (−1) m q m(m−1)=2−mn
(q; q)n
;
(q−n ; q)m
(2.20)
one can express (2.19) through the basic hypergeometric series 3 1:
In; n+2k (q) =
(q1=2 ; q)2k n+1 1=2
(q ; q)2k 3 1 (q−n ; q k+1=2 ; q k+1=2 ; q2k+1 ; q; q n ):
(q; q)2k
(2.21)
A particular case of (2.21) with k = 0,
In; n (q) =
n
X
n (q1=2 ; q)2k
k =0
k
q
(q; q)k
= 3 1 (q−n ; q1=2 ; q1=2 ; q; q; q n );
represents the squared norm of the q-Hermite function
nite and positive for all the values of q ∈ (0; 1).
(2.22)
n (|q).
As is evident from (2.22), In; n (q) is
3. Orthogonalization
It is clear that the q-Hermite functions (2.1) are linearly independent, for they are expressed
through the q-Hermite polynomials of di erent order (multiplied by the common exponential fac2
tor e− =2 ). Therefore, once the overlap integrals (2.19) for them are known, the system { n (|q)}
can be orthogonalized by the formation of suitable linear combinations. Since the subsequences
{ 2k (|q)} and { 2k+1 (|q)}; k = 0; 1; 2; : : : ; are mutually orthogonal by de nition, one needs to
form such combinations for the even and odd functions separately. In other words, if we de ne
(see [6, p. 154])
I0; 0 (q)
I0; 2 (q)
I2; 0 (q)
I2; 2 (q)
.
..
˜ (|q) =
..
.
2k
I
(q)
I
2k−2; 0
2k−2; 2 (q)
0 (|q)
2 (|q)
· · · I2k−2; 2k (q)
···
2k (|q)
···
···
..
.
I0; 2k (q)
I2; 2k (q)
..
.
I0; 0 (q)
I0; 2 (q)
I2; 2 (q)
I2; 0 (q)
..
..
−2 =2
=e
.
.
I2k−2; 0 (q)
I2k−2; 2 (q)
c0 (q)H0 (sin |q) c2 (q)H2 (sin |q)
;
···
I2k−2; 2k (q)
· · · c2k (q)H2k (sin |q)
···
···
..
.
then { ˜2k (|q)} is an orthogonal system, for (3.1) is orthogonal to
and hence to ˜2n (|q) for all n¡k.
I0; 2k (q)
I2; 2k (q)
..
.
0 (|q);
2 (|q); : : : ;
(3.1)
2k−2 (|q)
31
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
Similarly, for the odd q-Hermite functions the appropriate linear combinations are
I1; 1 (q)
I1; 3 (q)
I3; 1 (q)
I3; 3 (q)
..
.
˜ (|q) =
..
.
2k+1
I2k−1; 1 (q) I2k−1; 3 (q)
1 (|q)
3 (|q)
· · · I2k−1; 2k+1 (q)
···
2k+1 (|q)
···
···
..
.
I1; 2k+1 (q)
I3; 2k+1 (q)
..
.
I1; 1 (q)
I1; 3 (q)
(q)
I3; 3 (q)
I
3; 1
.
..
−2 =2
..
=e
.
I
(q)
I
2k−1;
1
2k−1;
3 (q)
c (q)H (sin |q) c (q)H (sin |q)
1
1
3
3
A system of the functions
˜ (|q) = cn (q)H̃n (sin |q)e−2 =2 ;
n
···
···
..
.
···
· · · c2k+1
I1; 2k+1 (q)
I3; 2k+1 (q)
..
.
:
I2k−1; 2k+1 (q)
(q)H
(sin |q)
(3.2)
2k+1
n = 0; 1; 2; : : : ;
(3.3)
is thus orthogonal over the full real line −∞¡¡∞ with respect to d. The polynomials H̃n (x|q)
in (3.3) are linear combinations of the q-Hermite polynomials (2.3) of the form
H̃n (x|q) =
n
X
n; k (q)Hk (x|q):
(3.4)
k=0
From the second determinants in (3.1) and (3.2) it follows that the connection coecients n; k (q)
in (3.4) are equal to
2k; 2j (q) = (−1)
k+j+1
"
(q; q)2k
(q; q)2j
#1=2
I0; 0 (q)
I0; 2 (q) · · · I0; 2j−2 (q)
I0; 2j+2 (q) · · · I0; 2k (q)
I2; 0 (q)
I2; 2 (q) · · · I2; 2j−2 (q)
I2; 2j+2 (q) · · · I2; 2k (q)
×
;
..
..
..
..
..
..
..
.
.
.
.
.
.
.
I
2k−2; 0 (q) I2k−2; 2 (q) · · · I2k−2; 2j−2 (q) I2k−2; 2j+2 (q) · · · I2k−2; 2k (q)
2k+1; 2j+1 (q) = (−1) k+j+1
"
(q; q)2k+1
(q; q)2j+1
(3.5)
#1=2
I1; 1 (q)
I1; 3 (q) · · · I1; 2j−1 (q)
I1; 2j+3 (q) · · · I1; 2k+1 (q)
I3; 1 (q)
I3; 3 (q) · · · I3; 2j−1 (q)
I3; 2j+3 (q) · · · I3; 2k+1 (q)
×
;
..
..
..
..
..
..
..
.
.
.
.
.
.
.
I
2k−1; 1 (q) I2k−1; 3 (q) · · · I2k−1; 2j−1 (q) I2k−1; 2j+3 (q) · · · I2k−1; 2k+1 (q)
(3.6)
for n = 2k and n = 2k + 1; k = 0; 1; 2; : : : ; respectively.
32
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
4. Fourier expansion
Having established that the q-Hermite functions n (|q) are square integrable, it is natural to look
for their expansion in terms of the Hermite functions n () (or, in other words, the linear harmonic
oscillator wave functions in quantum mechanics):
n (|q) =
∞
X
Cn; k (q) k ():
(4.1)
k=0
To nd Fourier coecients Cn; k (q) of n (|q) with respect to the system { k ()}, multiply both
sides of (4.1) by m () and integrate them over the variable within in nite limits with the help
of the orthogonality (2.8). This yields
Cn; m (q) =
Z
∞
n (|q) m () d = [2
m
m!(q; q)n ]−1=2
Z
∞
2
Hn (sin |q)Hm ()e− d:
(4.2)
−∞
−∞
To evaluate the last integral in (4.2), substitute in the Fourier expansion (2.3) for Hn (sin |q)
and integrate it term by term by using the integral transform (see [6, p. 124, Eq. (23)])
Z ∞
√
2
2
(4.3)
Hn (x)e2xy−x d x = (2y)n e−y
−∞
for the Hermite polynomials Hm (). This results in
2
n
n+m m q n =8 X
k n
(−1)
(2k − n) m q k(k−n)=2 :
Cn; m (q) = √ m
k q
2 m!(q; q)n k=0
(4.4)
Reversing the order of summation in (4.4) with respect to the index k makes it evident that the
Fourier coecients Cn; m (q) are real for 0¡q¡1, namely
n
cos(n + m)=2 m n2 =8 X
n
Cn; m (q) = √ m
q
(−1) k
k
2 m!(q; q)n
k=0
q
(2k − n) m q k(k−n)=2 :
(4.5)
Note that since the both functions n (|q) and n () (see (1.1) and (2.1), respectively) contain the
2
same exponential factor e− =2 , the relations (4.1) and (4.5) are equivalent to an explicit expansion
Hn (sin |q) =
∞
X
ank (q)Hk ()
(4.6)
k=0
of the q-Hermite polynomials in terms of ordinary Hermite polynomials. The coecients of this
expansion ank (q) are real and equal to
2
n
X
k q n =8
n
cos(n + k)=2
ank (q) =
(−1)l
l
k!
l=0
q
(l − n=2) k ql(l−n)=2 :
(4.7)
As a consistency check, one may evaluate the sum over k in the right-hand side of (4.6) by
substituting in it the coecients ank (q) from (4.7) and using the generating function
∞
X
tk
k=0
k!
Hk (s) = e2st−t
2
(4.8)
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
33
for the ordinary Hermite polynomials Hk (s). This gives indeed the explicit form (2.3) of the
q-Hermite polynomials Hn (sin |q) in the left-hand side of (4.6).
5. Fourier integral transform
Since the q-Hermite functions (2.1) belong to L2 (R), one may de ne their Fourier transforms with
the same property of the square integrability. A remarkable fact is that the classical Fourier integral
transform relates the q-Hermite functions with di erent values 0¡q¡1 and q¿1 of the parameter q.
We remind the reader that to consider the values 1¡q¡∞ of the parameter q it is convenient to
introduce [7] the q−1 -Hermite polynomials
hn (x|q) := −n Hn (x|q−1 ):
(5.1)
They satisfy the three-term recurrence relation
hn+1 (x|q) = 2xhn (x|q) + (1 − q−n )hn−1 (x|q);
n = 0; 1; 2; : : : ;
(5.2)
with the initial condition h0 (x|q) = 1. As follows from the Fourier expansion (2.3) and the inversion
formula
n
k
= q k(k−n)
q−1
n
k
(5.3)
q
for the q-binomial coecient (2.4), the explicit form of hn (x|q) is given by
hn (sinh |q) =
n
X
k
(−1) q
k(k−n)
k=o
n
k
e(n−2k) :
(5.4)
q
The q-Hermite (2.3) and the q−1 -Hermite (5.4) polynomials are related to each other by the
classical Fourier–Gauss transform [8]
2
Hn (sin |q)e
−2 =2
q n =4
= √
2
n
Z
∞
2
hn (sinh |q)e−− =2 d:
(5.5)
−∞
This means that the q−1 -Hermite functions
n (|q
−1
2
) = q n(n+1)=4 cn (q)hn (sinh |q)e− =2 ;
(5.6)
obtained from (2.1) by the change q → q−1 of the parameter q, are connected with the q-Hermite
functions (2.1) by the classical Fourier transform
(q−1=4 )n
√
n (|q) =
2
Z
∞
e− n (|q−1 ) d:
(5.7)
−∞
Their expansion in terms of the Hermite functions (1.1) has the form
n (|q
−1
)=
∞
X
k=0
Cn; k (q−1 ) k ();
(5.8)
34
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
where the Fourier coecients Cn; k (q−1 ) are equal to
Cn; k (q−1 ) = q n=4 cos(n − k)=2Cn; k (q)
(5.9)
and the Cn; k (q) are given in (4.5).
6. Relationship with the coherent states
In the study of a number of quantum-mechanical problems it turns out very useful to employ a
system of coherent states. The wave functions of coherent states for the linear harmonic oscillator
are expressed in terms of the Hermite functions (1.1) as
(; z) = h|zi = exp(−|z|2 =2)
∞
X
zn
n=0
√
n!
n ();
(6.1)
where z is the complex parameter. The q-Hermite functions (2.1) are in fact some linear combinations
of (; z) with particular values of the parameter z. Indeed, if one substitutes the explicit form of
the Fourier coecients (4.4) into (4.1), then the sum over the index k in it can be evaluated by
(6.1). Thus the required relationship is
n
√
X
n
k n
(−1)
(; 2(k − n=2)):
n (|q) =
1=2
k q
(q; q)n k=0
(6.2)
In a similar manner, from (5.8) and (5.9) it follows that the corresponding relationship for the
q−1 -Hermite functions (5.6) is
n
√
q n(n+1)=4 X
k k(k−n) n
(−1)
q
(;
2(n=2 − k)):
n (|q ) =
k q
(q; q)n1=2 k=0
−1
(6.3)
Acknowledgements
Discussions with S.L. Woronowicz and K.B. Wolf are gratefully acknowledged. This work was
partially supported by the UNAM-DGAPA project IN106595.
References
[1] N. Wiener, The Fourier Integral and Certain of its Applications, Cambridge University Press, Cambridge, 1993.
[2] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
[3] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,
Report 94-05, Delft University of Technology, 1994.
[4] R. Askey, M.E.H. Ismail, A generalization of ultraspherical polynomials, in: P. Erdos (Ed.), Studies in Pure
Mathematics, Birkhauser, Boston, MA, 1983, pp. 55 –78.
[5] L.J. Rogers, Third memoir on the expansion of certain in nite products, Proc. Lond. Math. Soc. 26 (1895) 15–32.
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[6] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 2, McGraw-Hill,
New York, 1953.
[7] R. Askey, Continuous q-Hermite polynomials when q¿1, in: D. Stanton (Ed.), q-Series and Partitions, IMA
Volumes in Mathematics and its Applications, Springer, New York, 1989, pp. 151–158.
[8] N.M. Atakishiyev, Sh.M. Nagiyev, On the wave functions of a covariant linear oscillator, Theor. Math. Phys. 98
(1994) 162–166.
On the square integrability of the q-Hermite functions
M.K. Atakishiyeva a , N.M. Atakishiyev b; ∗ , C. Villegas-Blas b
a
b
Facultad de Ciencias, UAEM, Apartado Postal 396-3, C.P. 62250 Cuernavaca, Morelos, Mexico
Instituto de Matematicas, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaca, Morelos, Mexico
Received 20 October 1997; received in revised form 21 May 1998
Abstract
2
Overlap integrals over the full real line −∞¡x¡∞ for a family of the q-Hermite functions Hn (sin x|q)e−x =2 ; 0¡
2
q= e−2 ¡1 are evaluated. In particular, an explicit form of the squared norms for these q-extensions of the Hermite
functions (or the wave functions of the linear harmonic oscillator in quantum mechanics) is obtained. The classical
Fourier–Gauss transform connects the q-Hermite functions with di erent values 0¡q¡1 and q¿1 of the parameter q.
An explicit expansion of the q-Hermite polynomials Hn (sin x|q) in terms of the Hermite polynomials Hn (x) emerges as
c 1998 Elsevier Science B.V. All rights reserved.
a by-product.
1. Introduction
The Hermite functions
√ n −1=2
2
Hn ()e− =2 ;
n () := [ 2 n!]
(1.1)
where Hn () are the classical Hermite polynomials, are of great mathematical interest as an explicit
example of an orthonormal and complete system in the Hilbert space L2 (R) of square-integrable
functions with respect to the full real line −∞¡¡∞ [1]. In mathematical physics they are known
to represent solutions of the linear harmonic oscillator problem, which plays a very important role in
quantum mechanics. In what follows, we attempt to study in detail some particular q-generalization
of the Hermite functions (1.1).
2. Overlap integrals and squared norms
Let us consider a family of q-Hermite functions
n (|q) := cn (q)Hn (sin |q)e
∗
−2 =2
;
2
0¡q = e−2 ¡1;
Corresponding author. E-mail: natig@matcuer.unam.mx.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 4 2 - 3
(2.1)
28
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
√
where the normalization constant
c (q) = [ (q; q)n ]−1=2 and the q-shifted factorial (q; q)n is de ned
Qn−1 n
as (z; q)0 = 1 and (z; q)n = k=0 (1 − zq k ), n = 1; 2; 3 : : : (throughout this paper, we will employ the
standard notations of q-special functions, see [2] or [3]). The continuous q-Hermite polynomials
Hn (x|q) in (2.1) are those q-extensions of the ordinary Hermite polynomials Hn (x), which satisfy
the three-term recurrence relation
Hn+1 (x|q) = 2xHn (x|q) − (1 − q n )Hn−1 (x|q);
n = 0; 1; 2; : : : ;
(2.2)
with the initial condition H0 (x|q) = 1 [4]. Their explicit form is given by the Fourier expansion
Hm (sin |q) =
m
m
X
(−1)
n=0
where
m
n
m
n
q
n
m
n
e(2n−m) ;
(2.3)
q
is the q-binomial coecient,
q
(q; q)m
m
=
:=
m−n
(q; q)n (q; q)m−n
(2.4)
:
q
The q-Hermite polynomials (2.3) are solutions of the di erence equation
eis exp −i
d
ds
+ e−is exp i
d
ds
Hn (sin s|q) = 2q−n=2 cos s Hn (sin s|q):
(2.5)
It is easy to verify that limq→1 −2n (q; q)n = 2n n!. Therefore it follows from the recurrence relation
(2.2), that
lim −n Hn (sin |q) = Hn ():
(2.6)
q→1
Thus, the normalization in (2.1) is chosen so that when the limit q → 1 is taken they coincide with
the wave functions n () of the linear harmonic oscillator in quantum mechanics, i.e.
n (|1) :=
lim
q→1
n (|q) =
n ():
As it is well known, the wave functions
Z
(2.7)
n ()
satisfy the orthogonality relation
∞
m () n () d = mn
(2.8)
−∞
and may serve as a basis in the Hilbert space L2 (R) of square-integrable functions with respect
to d.
We evaluate rst the corresponding integral
Im; n (q) :=
Z
∞
−∞
m (|q) n (|q) d = In; m (q)
(2.9)
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
29
for the q-Hermite functions (2.1). Since Hn (−x|q) = (−1)n Hn (x|q) by de nition, the functions m (|q)
and n (|q) of the opposite parities (m − n = 2k + 1; k = 0; 1; 2; : : :) are orthogonal and nontrivial
integrals in (2.9) are
In; n+2k (q) = [(q; q)n (q; q)n+2k ]
Z
−1=2
∞
2
Hn (sin |q)Hn+2k (sin |q)e− d:
(2.10)
−∞
Using the Rogers linearization formula [5]
Hm (x|q)Hn (x|q) =
m
X
(q; q)n
(q; q)n−m+k
k=0
m
k
Hn−m+2k (x|q);
m 6 n;
(2.11)
q
for the q-Hermite polynomials (2.3), one can represent (2.10) as
Z ∞
n
1=2 X
2
(q n+1 ; q)2k
n
−1
√
In; n+2k (q) =
H2k+2l (sin |q)e− d:
(q; q)2k+l
l
q −∞
l=0
(2.12)
It remains only to substitute the explicit form of the q-Hermite polynomials (2.3) into (2.12) and
to evaluate the integral with respect to the variable by the aid of the well-known integral transform
Z
∞
√
2
2
d x e2xy−x = e−y :
(2.13)
−∞
The result is
In; n+2k (q) = (q
n+1
1=2
; q)2k
n
X
(−1)
k+l
l=0
2(k+l)
X
×
(−1) j
j=0
2(k + l)
j
n
l
q
2
q(k+l) =2
(q; q)2k+l
2
q j =2−j(k+l) :
(2.14)
q
The sum over j in (2.14) gives the factor (q1=2−k−l ; q)2(k+l) because of the Gauss identity
(z; q)n =
n
X
n
k
k=0
q k(k−1)=2 (−z) k :
(2.15)
q
In view of the formulas (see, for example, [2] or [3])
(z; q)n+k = (z; q)n (zq n ; q)k ;
(zq ; q)n = q
−n
−n(n+1)=2
(2.16)
n
(−z) (q=z; q)n ;
(2.17)
this factor is equal to
2
(q1=2−k−l ; q)2(k+l) = (−1) k+l q−(k+l) =2 (q1=2 ; q)2k+l :
(2.18)
Now substituting (2.18) into (2.14), yields
In; n+2k (q) = (q
n+1
1=2
; q)2k
n
X
n (q1=2 ; q)2k+l
l=0
l
q
(q; q)2k+l
:
(2.19)
30
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
Since (q; q)2k+l = (q; q)2k (q2k+1 ; q)l by (2.16) and
(q; q)n−m = (−1) m q m(m−1)=2−mn
(q; q)n
;
(q−n ; q)m
(2.20)
one can express (2.19) through the basic hypergeometric series 3 1:
In; n+2k (q) =
(q1=2 ; q)2k n+1 1=2
(q ; q)2k 3 1 (q−n ; q k+1=2 ; q k+1=2 ; q2k+1 ; q; q n ):
(q; q)2k
(2.21)
A particular case of (2.21) with k = 0,
In; n (q) =
n
X
n (q1=2 ; q)2k
k =0
k
q
(q; q)k
= 3 1 (q−n ; q1=2 ; q1=2 ; q; q; q n );
represents the squared norm of the q-Hermite function
nite and positive for all the values of q ∈ (0; 1).
(2.22)
n (|q).
As is evident from (2.22), In; n (q) is
3. Orthogonalization
It is clear that the q-Hermite functions (2.1) are linearly independent, for they are expressed
through the q-Hermite polynomials of di erent order (multiplied by the common exponential fac2
tor e− =2 ). Therefore, once the overlap integrals (2.19) for them are known, the system { n (|q)}
can be orthogonalized by the formation of suitable linear combinations. Since the subsequences
{ 2k (|q)} and { 2k+1 (|q)}; k = 0; 1; 2; : : : ; are mutually orthogonal by de nition, one needs to
form such combinations for the even and odd functions separately. In other words, if we de ne
(see [6, p. 154])
I0; 0 (q)
I0; 2 (q)
I2; 0 (q)
I2; 2 (q)
.
..
˜ (|q) =
..
.
2k
I
(q)
I
2k−2; 0
2k−2; 2 (q)
0 (|q)
2 (|q)
· · · I2k−2; 2k (q)
···
2k (|q)
···
···
..
.
I0; 2k (q)
I2; 2k (q)
..
.
I0; 0 (q)
I0; 2 (q)
I2; 2 (q)
I2; 0 (q)
..
..
−2 =2
=e
.
.
I2k−2; 0 (q)
I2k−2; 2 (q)
c0 (q)H0 (sin |q) c2 (q)H2 (sin |q)
;
···
I2k−2; 2k (q)
· · · c2k (q)H2k (sin |q)
···
···
..
.
then { ˜2k (|q)} is an orthogonal system, for (3.1) is orthogonal to
and hence to ˜2n (|q) for all n¡k.
I0; 2k (q)
I2; 2k (q)
..
.
0 (|q);
2 (|q); : : : ;
(3.1)
2k−2 (|q)
31
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
Similarly, for the odd q-Hermite functions the appropriate linear combinations are
I1; 1 (q)
I1; 3 (q)
I3; 1 (q)
I3; 3 (q)
..
.
˜ (|q) =
..
.
2k+1
I2k−1; 1 (q) I2k−1; 3 (q)
1 (|q)
3 (|q)
· · · I2k−1; 2k+1 (q)
···
2k+1 (|q)
···
···
..
.
I1; 2k+1 (q)
I3; 2k+1 (q)
..
.
I1; 1 (q)
I1; 3 (q)
(q)
I3; 3 (q)
I
3; 1
.
..
−2 =2
..
=e
.
I
(q)
I
2k−1;
1
2k−1;
3 (q)
c (q)H (sin |q) c (q)H (sin |q)
1
1
3
3
A system of the functions
˜ (|q) = cn (q)H̃n (sin |q)e−2 =2 ;
n
···
···
..
.
···
· · · c2k+1
I1; 2k+1 (q)
I3; 2k+1 (q)
..
.
:
I2k−1; 2k+1 (q)
(q)H
(sin |q)
(3.2)
2k+1
n = 0; 1; 2; : : : ;
(3.3)
is thus orthogonal over the full real line −∞¡¡∞ with respect to d. The polynomials H̃n (x|q)
in (3.3) are linear combinations of the q-Hermite polynomials (2.3) of the form
H̃n (x|q) =
n
X
n; k (q)Hk (x|q):
(3.4)
k=0
From the second determinants in (3.1) and (3.2) it follows that the connection coecients n; k (q)
in (3.4) are equal to
2k; 2j (q) = (−1)
k+j+1
"
(q; q)2k
(q; q)2j
#1=2
I0; 0 (q)
I0; 2 (q) · · · I0; 2j−2 (q)
I0; 2j+2 (q) · · · I0; 2k (q)
I2; 0 (q)
I2; 2 (q) · · · I2; 2j−2 (q)
I2; 2j+2 (q) · · · I2; 2k (q)
×
;
..
..
..
..
..
..
..
.
.
.
.
.
.
.
I
2k−2; 0 (q) I2k−2; 2 (q) · · · I2k−2; 2j−2 (q) I2k−2; 2j+2 (q) · · · I2k−2; 2k (q)
2k+1; 2j+1 (q) = (−1) k+j+1
"
(q; q)2k+1
(q; q)2j+1
(3.5)
#1=2
I1; 1 (q)
I1; 3 (q) · · · I1; 2j−1 (q)
I1; 2j+3 (q) · · · I1; 2k+1 (q)
I3; 1 (q)
I3; 3 (q) · · · I3; 2j−1 (q)
I3; 2j+3 (q) · · · I3; 2k+1 (q)
×
;
..
..
..
..
..
..
..
.
.
.
.
.
.
.
I
2k−1; 1 (q) I2k−1; 3 (q) · · · I2k−1; 2j−1 (q) I2k−1; 2j+3 (q) · · · I2k−1; 2k+1 (q)
(3.6)
for n = 2k and n = 2k + 1; k = 0; 1; 2; : : : ; respectively.
32
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
4. Fourier expansion
Having established that the q-Hermite functions n (|q) are square integrable, it is natural to look
for their expansion in terms of the Hermite functions n () (or, in other words, the linear harmonic
oscillator wave functions in quantum mechanics):
n (|q) =
∞
X
Cn; k (q) k ():
(4.1)
k=0
To nd Fourier coecients Cn; k (q) of n (|q) with respect to the system { k ()}, multiply both
sides of (4.1) by m () and integrate them over the variable within in nite limits with the help
of the orthogonality (2.8). This yields
Cn; m (q) =
Z
∞
n (|q) m () d = [2
m
m!(q; q)n ]−1=2
Z
∞
2
Hn (sin |q)Hm ()e− d:
(4.2)
−∞
−∞
To evaluate the last integral in (4.2), substitute in the Fourier expansion (2.3) for Hn (sin |q)
and integrate it term by term by using the integral transform (see [6, p. 124, Eq. (23)])
Z ∞
√
2
2
(4.3)
Hn (x)e2xy−x d x = (2y)n e−y
−∞
for the Hermite polynomials Hm (). This results in
2
n
n+m m q n =8 X
k n
(−1)
(2k − n) m q k(k−n)=2 :
Cn; m (q) = √ m
k q
2 m!(q; q)n k=0
(4.4)
Reversing the order of summation in (4.4) with respect to the index k makes it evident that the
Fourier coecients Cn; m (q) are real for 0¡q¡1, namely
n
cos(n + m)=2 m n2 =8 X
n
Cn; m (q) = √ m
q
(−1) k
k
2 m!(q; q)n
k=0
q
(2k − n) m q k(k−n)=2 :
(4.5)
Note that since the both functions n (|q) and n () (see (1.1) and (2.1), respectively) contain the
2
same exponential factor e− =2 , the relations (4.1) and (4.5) are equivalent to an explicit expansion
Hn (sin |q) =
∞
X
ank (q)Hk ()
(4.6)
k=0
of the q-Hermite polynomials in terms of ordinary Hermite polynomials. The coecients of this
expansion ank (q) are real and equal to
2
n
X
k q n =8
n
cos(n + k)=2
ank (q) =
(−1)l
l
k!
l=0
q
(l − n=2) k ql(l−n)=2 :
(4.7)
As a consistency check, one may evaluate the sum over k in the right-hand side of (4.6) by
substituting in it the coecients ank (q) from (4.7) and using the generating function
∞
X
tk
k=0
k!
Hk (s) = e2st−t
2
(4.8)
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
33
for the ordinary Hermite polynomials Hk (s). This gives indeed the explicit form (2.3) of the
q-Hermite polynomials Hn (sin |q) in the left-hand side of (4.6).
5. Fourier integral transform
Since the q-Hermite functions (2.1) belong to L2 (R), one may de ne their Fourier transforms with
the same property of the square integrability. A remarkable fact is that the classical Fourier integral
transform relates the q-Hermite functions with di erent values 0¡q¡1 and q¿1 of the parameter q.
We remind the reader that to consider the values 1¡q¡∞ of the parameter q it is convenient to
introduce [7] the q−1 -Hermite polynomials
hn (x|q) := −n Hn (x|q−1 ):
(5.1)
They satisfy the three-term recurrence relation
hn+1 (x|q) = 2xhn (x|q) + (1 − q−n )hn−1 (x|q);
n = 0; 1; 2; : : : ;
(5.2)
with the initial condition h0 (x|q) = 1. As follows from the Fourier expansion (2.3) and the inversion
formula
n
k
= q k(k−n)
q−1
n
k
(5.3)
q
for the q-binomial coecient (2.4), the explicit form of hn (x|q) is given by
hn (sinh |q) =
n
X
k
(−1) q
k(k−n)
k=o
n
k
e(n−2k) :
(5.4)
q
The q-Hermite (2.3) and the q−1 -Hermite (5.4) polynomials are related to each other by the
classical Fourier–Gauss transform [8]
2
Hn (sin |q)e
−2 =2
q n =4
= √
2
n
Z
∞
2
hn (sinh |q)e−− =2 d:
(5.5)
−∞
This means that the q−1 -Hermite functions
n (|q
−1
2
) = q n(n+1)=4 cn (q)hn (sinh |q)e− =2 ;
(5.6)
obtained from (2.1) by the change q → q−1 of the parameter q, are connected with the q-Hermite
functions (2.1) by the classical Fourier transform
(q−1=4 )n
√
n (|q) =
2
Z
∞
e− n (|q−1 ) d:
(5.7)
−∞
Their expansion in terms of the Hermite functions (1.1) has the form
n (|q
−1
)=
∞
X
k=0
Cn; k (q−1 ) k ();
(5.8)
34
M.K. Atakishiyeva et al. / Journal of Computational and Applied Mathematics 99 (1998) 27–35
where the Fourier coecients Cn; k (q−1 ) are equal to
Cn; k (q−1 ) = q n=4 cos(n − k)=2Cn; k (q)
(5.9)
and the Cn; k (q) are given in (4.5).
6. Relationship with the coherent states
In the study of a number of quantum-mechanical problems it turns out very useful to employ a
system of coherent states. The wave functions of coherent states for the linear harmonic oscillator
are expressed in terms of the Hermite functions (1.1) as
(; z) = h|zi = exp(−|z|2 =2)
∞
X
zn
n=0
√
n!
n ();
(6.1)
where z is the complex parameter. The q-Hermite functions (2.1) are in fact some linear combinations
of (; z) with particular values of the parameter z. Indeed, if one substitutes the explicit form of
the Fourier coecients (4.4) into (4.1), then the sum over the index k in it can be evaluated by
(6.1). Thus the required relationship is
n
√
X
n
k n
(−1)
(; 2(k − n=2)):
n (|q) =
1=2
k q
(q; q)n k=0
(6.2)
In a similar manner, from (5.8) and (5.9) it follows that the corresponding relationship for the
q−1 -Hermite functions (5.6) is
n
√
q n(n+1)=4 X
k k(k−n) n
(−1)
q
(;
2(n=2 − k)):
n (|q ) =
k q
(q; q)n1=2 k=0
−1
(6.3)
Acknowledgements
Discussions with S.L. Woronowicz and K.B. Wolf are gratefully acknowledged. This work was
partially supported by the UNAM-DGAPA project IN106595.
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