On Integral and Finite Fourier Transforms of Continuous Q-Hermite

Home , Hermite polynomials

On integral and ﬁnite Fourier transforms

of continuous q-Hermite polynomials

M.K. Atakishiyeva1 and N.M. Atakishiyev2

1Facultad de Ciencias, Universidad Aut´onomadel Estado de Morelos,

C.P. 62250 Cuernavaca, Morelos, M´exico

2Instituto de Matem´aticas, Unidad Cuernavaca,

Universidad Nacional Aut´onomade M´exico,

A.P. 273-3, Admon.3, Cuernavaca, Morelos, 62251 M´exico

We give an overview of the remarkably simple transformation properties of the

continuous q-Hermite polynomials Hn(x| q) of Rogers with respect to the classical

Fourier integral transform. The behavior of the q-Hermite polynomials under the

ﬁnite Fourier transform and an explicit form of the q-extended eigenfunctions of the

ﬁnite Fourier transform, deﬁned in terms of these polynomials, are also discussed.

PACS numbers: 02.30.Nw, 02.30.Gp

I. INTRODUCTION

Integrals transforms are an extensively used tool in solving various boundary value problems and integral equations. Probably the most important and more frequently encountered among all of them is the Fourier integral transform

1 Z ∞ (F f)(x) ≡ √ e i x y f(y) dy = fe(x) , (1.1) 2π −∞ which interrelates two square-integrable functions on the full real line x ∈ (−∞, ∞), f(x) and fe(x). Recently, it has become clear that the integral transform (1.1) may also help us in revealing close relations between certain q-special functions (see [1] and references therein). The objective of this work is to give an overview of the striking harmony of the q-Hermite 2

polynomials Hn(x| q) of Rogers with respect to the classical Fourier integral transform (1.1), which is clearly q-independent. As was observed by Mehta [2], Dahlquist [3] and Matveev [4], the Fourier integral transform (1.1) always entails the certain ﬁnite Fourier transform.

So we shall also discuss how the q-Hermite polynomials Hn(x| q) behave under the ﬁnite Fourier transform (see [5, 6] and references therein). The foregoing exposition is divided into four parts. In the ﬁrst of them we have collected

those known facts about the continuous q-Hermite polynomials Hn(x| q) of Rogers, which are needed for deriving the Fourier integral transform for the q-Hermite polynomials in the second part and the finite Fourier transform for the same polynomials in the third part. In the final part we show that the Fourier transform (1.1) actually intertwines two q-difference equations for the q-Hermite polynomials Hn(x| q), which correspond to the distinct set of values for the deformation parameter q : 0 < q < 1 and 1 < q < ∞. Throughout this exposition we employ standard notations of the theory of special functions (see, for example, [7] or [8]).

II. CONTINUOUS q-HERMITE POLYNOMIALS

The continuous q-Hermite polynomials Hn(x| q) of Rogers are those q-extensions of the ordinary Hermite polynomials Hn(x), which are generated by the three-term recurrence relation

n Hn+1(x| q) = 2x Hn(x| q) − (1 − q ) Hn−1(x| q) ,

0 < q < 1 , n = 0, 1, 2, ... , (2.1)

with the initial condition H0(x| q) = 1. The explicit form of Hn(x| q) is exhibited by their Fourier expansion

  Xn  n    i(n−2k) θ Hn(x| q) =   e , x = cos θ , (2.2) k=0 k q 3

   n    where   is the q-binomial coeﬃcient, k q

" # " # n (q; q) n := n = . (2.3) k q (q; q)k (q; q)n−k n − k q

Although these polynomials were introduced back in 1894 by Rogers [9], their orthogonality property on the ﬁnite interval x ∈ [−1, 1]

1 Z 1 w(x| q) δ H (x| q) H (x| q)√ dx = mn (2.4) m n 2 n+1 2π −1 1 − x (q ; q)∞ was derived by Allaway [10] only in 1980. The weight function in the orthogonality relation (2.4) can be expressed in terms of the classical theta-function

X∞ 1/4 n n(n+1) ϑ1(z, q) := 2q (−1) q sin(2n + 1)z n=0 q1/4 ³ ´ ³ ´ ³ ´ = q2; q2 e2iz; q2 e− 2iz; q2 2 sin z ∞ ∞ ∞ in the following way

w(x| q) 2 √ = ϑ (θ, q) , x = cos θ . 2 1/8 1 1 − x q (q; q)∞

The linear generating function for the polynomials Hn(x| q) is of the form

X∞ n ³ ´ ³ ´ t i θ −i θ Hn(x| q) = eq t e eq t e , x = cos θ , (2.5) k=0 (q; q)n

where eq(z) is the q-exponential function, deﬁned as

X∞ zn 1 eq(z) := = . (2.6) n=0 (q; q)n (z; q)∞ 4

In the limit as the deformation parameter q tends to 1, the q-Hermite polynomials Hn(x| q) reduce to the ordinary Hermite polynomials Hn(x):

³q ´ 1−q Hn 2 x| q lim ³q ´n = Hn(x) . (2.7) q→1− 1−q 2

It should be also recalled that to consider the continuous q-Hermite polynomials Hn(x| q) for the values of the parameter q in the interval [1, ∞), it is customary to introduce the

−1 so-called continuous q -Hermite polynomials hn(x| q) as [11]

−n −1 hn(x| q) := i Hn(ix| q ) . (2.8)

     n   n    k(k−n)   In view of the inversion formula   = q   for the q-binomial coeﬃcient (2.3), k k q−1 q from (2.2) and (2.3) one deduces that

  Xn  n  k k(k−n)   (n−2k) χ hn(sinh χ| q) = (−1) q   e . (2.9) k=0 k q

It is important to observe that in order to be able to formulate adequately transformation

−1 properties of the q-Hermite and q -Hermite polynomials Hn(x| q) and hn(x| q) with respect to the Fourier transforms one has to employ the following nonconventional parametrization for their argument x:

π x = cos θ = sin κ s , θ = − κ s , when 0 < q < 1 ; 2

x = sinh χ = sinh κ s , χ = κ s , when 1 < q < ∞ . (2.10)

q In both cases above κ := ln q−1/2 or, equivalently, q := e−2 κ2 . Thus, formulas (2.2) and 5

(2.9) are representable as expansions

  Xn  n  n k   i(2k− n) κ s Hn( sin κs| q) = i (−1)   e , (2.11) k=0 k q

  Xn  n  k   k(k−n) (n−2k) κ s hn( sinh κs| q) = (−1)   q e , (2.12) k=0 k q which were key relations in establishing that the Fourier integral transform interrelates q- Hermite and q−1-Hermite polynomials [12]. This integral transform is given in the next section.

Finally, the continuous q-Hermite polynomials Hn( sin κs| q) satisfy the q-diﬀerence equation

h i iκs −iκ∂s −iκs iκ∂s −n/2 e e + e e Hn( sin κs| q) = 2 q cos κs Hn( sin κs| q) , (2.13)

d ±iκ∂s ±iκ∂s where ∂x ≡ dx and e are shift operators, i.e., e f(s) = f(s ± iκ). We recall that this simple nonconventional q-diﬀerence equation for the continuous q-Hermite polynomials

Hn( sin κx| q) is a direct consequence of Rogers’ linear generating function (2.5) (see, for example, [13] and references therein). To prove this assertion, act on both sides of the identity (which is obtained from (2.5) by the same change of variable cos θ = sin κs as in (2.10)) X∞ n ³ ´ ³ ´ t −iκx iκx Hn( sin κx| q) = eq i t e eq −i t e , |t| < 1 , n=0 (q; q)n

by the diﬀerence operator eiκx e−iκ∂x + e−iκx eiκ∂x . This yields

X∞ n h i t iκx −iκ∂x −iκx iκ∂x e e + e e Hn( sin κx| q) n=0 (q; q)n

³ ´ ³ ´ ³ ´ ³ ´ iκx −iκ(x−iκ) iκ(x−iκ) −iκx −iκ(x+iκ) iκ(x+iκ) = e eq i t e eq −i t e + e eq i t e eq −i t e . (2.14) 6

Since q = e−2κ2 by deﬁnition, the right hand side of (2.14) may be written as

³ ´ ³ ´ ³ ´ ³ ´ iκx 1/2 −iκx −1/2 iκx −iκx −1/2 −iκ(x+iκ) 1/2 iκ(x+iκ) e eq iq te eq −iq te + e eq iq te eq −iq te . (2.15)

Apply then the functional equation eq(qz) = (1−z) eq(z) for the q-exponential function (2.6) to the ﬁrst q-exponential factor in the ﬁrst line of (2.15) and to the second q-exponential factor in the second line of (2.15) to obtain that

h ³ ´ ³ ´i ³ ´ ³ ´ iκx −1/2 −iκx −iκx −1/2 iκx −1/2 −iκx −1/2 iκx e 1 − i q t e + e 1 + i q t e eq i q t e eq −i q t e

³ ´ ³ ´ X∞ −1/2 n −1/2 −iκx −1/2 iκx (q t) = 2 cos κx eq i q t e eq −i q t e = 2 cos κx Hn( sin κx| q) . n=0 (q; q)n One thus arrives at the identity

X∞ n h i t iκx −iκ∂x −iκx iκ∂x e e + e e Hn( sin κx| q) n=0 (q; q)n

X∞ (q−1/2t)n = 2 cos κx Hn( sin κx| q) . (2.16) n=0 (q; q)n It remains only to equate coefficients of the equal powers of the parameter t on both sides of (2.16) to obtain difference equation (2.13). In the case when the parameter q lies in the interval [ 1, ∞), the q-difference equation (2.13) takes the form

h i κs −κ∂s −κs κ∂s n/2 e e + e e hn( sinh κs| q) = 2 q cosh κs hn( sinh κs| q) . (2.17)

Valuable background material about these two diﬀerence equations (2.13) and (2.17) for the continuous q-Hermite and q−1-Hermite polynomials, respectively, can be found in [13] and [14]. 7

III. FOURIER INTEGRAL TRANSFORM

Let us brieﬂy recall some mathematical aspects of the Fourier integral transform

(1.1). The orthonormalized wave functions ψn(x) of the linear harmonic oscillator in non- relativistic quantum mechanics,

Z ∞ ψm(x) ψn(x) dx = δmn , m, n = 0, 1, 2, ... , (3.1) −∞

are explicitly given as

³ ´ q√ 2 n ψn(x) := cn Hn(ξ) exp −ξ /2 , 1/cn = π 2 n! , (3.2)

q where Hn(ξ) are the classical Hermite polynomials and ξ := m ω/h¯ x is a dimensionless coordinate (see, for example, [15]). In quantum mechanics they emerge as eigenfunctions of the Hamiltonian Hc for the linear harmonic oscillator,

Ã ! h¯ ω d2 Hc ψ (x) ≡ ξ2 − ψ (x) =h ¯ ω (n + 1/2) ψ (x) , (3.3) n 2 dξ2 n n

which is a self-adjoint diﬀerential operator of the second order. The linear harmonic oscillator

−x2/2 wave functions ψn(x) (we recall that the functions Hn(x) e are usually referred to as Hermite functions in the mathematical literature) represent an important explicit example of an orthonormal and complete system in the Hilbert space L2 of square-integrable functions on the full real line x ∈ (−∞, ∞). It is further well known that the wave functions of the

linear harmonic oscillator ψn(x) possess the simple transformation property with respect to the classical Fourier transform: they are also eigenfunctions of the Fourier integral transform, associated with the eigenvalues i n , that is,

Z ∞ 1 i x y n (F ψn)(x) ≡ √ e ψn(y) dy = i ψn(x) . (3.4) 2π −∞ 8

There are q-extensions of the Fourier integral transform (3.4), which interrelate certain q- polynomial families (see [1] and references therein). In particular, the continuous q-Hermite polynomials Hn (x| q) of Rogers possess the following transformation property [12]

Z ∞ 1 ixy− x2/2 n n2/4 − y2/2 √ e Hn (sin κx| q) dx = i q hn (sinh κy| q) e (3.5) 2π −∞

with respect to the classical Fourier transform (1.1). Indeed, let us evaluate the integral on the left side of (3.5) by using the Fourier expansion (2.11):

  Z n n Z ∞ x2 X  n  ∞ x2 1 ixy− i k   i[y+(2k− n)κ] x− √ e 2 Hn (sin κx| q) dx = √ (−1)   e 2 dx . (3.6) 2π −∞ 2π k=0 k −∞ q

Since the Gauss exponential function exp(−x2/2) is its own Fourier transform (that is, formula (3.4) when n = 0), the integral in the second line of (3.6) is readily evaluated and at the next step we obtain that

  Z ∞ Xn  n  1 ixy− x2/2 n k   − [ y+(2k− n) κ ]2/2 √ e Hn (sin κx| q) dx = i (−1)   e 2π −∞ k=0 k q

  Xn  n  n n2/4 −y2/2 k   k(k−n) (n−2k) κ y n n2/4 −y2/2 = i q e (−1)   q e = i q hn (sinh κy| q) e , (3.7) k=0 k q upon having employed the Fourier expansion (2.12) at the last step. This completes the proof of (3.5). Note that the Fourier integral transform (3.5) reduces to the classical result (3.4) for the

− ordinary Hermite polynomials Hn(x) in the limit as q → 1 . This property of (3.5) is a direct consequence of the limit relations for the continuous q-Hermite polynomials

−n −n lim κ Hn( sin κx| q) = lim κ hn( sinh κx| q) = Hn(x) . (3.8) q→1− q→1−

which are not hard to check, by using the recurrence relation (2.1) and deﬁnition (2.8) 9

(cf. (2.7)). This is may be the place to point out that the integral Fourier transform (3.5) interre-

−1 lates the continuous q-Hermite polynomials Hn( sin κx| q) with the q -Hermite polynomials

−1 hn( sinh κx| q). This change of q to q is the price to formulate a meaningful q-extension of the classical Fourier transform (3.4). But once the integral transform (3.5) is known it is not hard to deﬁne q-extended eigenfunctions of the Fourier integral transform (3.4) in terms

−1 of the q-Hermite polynomials Hn( sin κx| q) and the q -Hermite polynomials hn( sinh κx| q). Indeed, if one introduces functions

h i 1 2 2 2 ψ (x; q) := q−n /8 H ( sin κx| q) + qn /8 h ( sinh κx| q) e−x /2 , (3.9) n 2 n n then from (3.5) and its inverse it follows at once that

Z ∞ 1 i x y n √ e ψn(y; q) dy = i ψn(x; q) . (3.10) 2π −∞

So the ψn(x; q) are required q-extended eigenfunctions of the Fourier integral transform (3.4) (more details can be found in [6]).

IV. FINITE FOURIER TRANSFORM

The ﬁnite Fourier transform is deﬁned as an action of the operator (or the equivalent N × N unitary matrix)

1 µ2πi ¶ Φ(N) = √ exp jk , 0 ≤ j, k ≤ N − 1 . (4.1) jk N N

Functions f (l)(k), l = 0, 1, 2, ..., N − 1, which satisfy the relation

NX−1 (N) (l) (l) Φjk f (k) = λl f (j), (4.2) k=0 10

are then eigenvectors of the ﬁnite Fourier transform Φ(N), associated with the eigenvalues

(N) λl. Since the fourth power of Φ is the identity operator (or matrix), the only distinct

eigenvalues among λl’s are ±1 and ±i. The ﬁnite Fourier transform (4.1) has deep roots in classical pure mathematics and many valuable mathematical and historical details may be found in [16]-[19]. Mehta studied in [2] the eigenvalue problem (4.2) and found analytically a set of eigen-

(l) (N) vectors f (j) = Fjl of the ﬁnite Fourier transform (4.1) of the form

s  ∞ (N) X π 2 2π − N (nN+j)   Fjk := e Hk (nN + j) , (4.3) n=−∞ N

where Hk (x) is the ordinary Hermite polynomial of degree k in the variable x. These Mehta

(N) l eigenvectors Fjl correspond to the eigenvalues λl = i , that is,

NX−1 (N) (N) k (N) Φjl Flk = i Fjk , 0 ≤ j, k ≤ N − 1 . (4.4) l=0

The eigenfunctions (4.3) are real and naturally periodic with period N. So Mehta in fact explicitly constructed the discrete analogue of the well-known continuum case (3.4) where the

2 Hermite functions Hk (x) exp (−x /2) are constant multiples of their own Fourier transforms. Once the Fourier integral transform (3.5) is known, it is natural to look for its ﬁnite analogue. We recall that in the proof of (4.4) for Mehta’s eigenvectors F (N) of the ﬁnite Fourier transform it was essential to use the simple transformation property (3.4) of the

2 Hermite functions Hk (x) exp (−x /2) under the Fourier integral transform (1.1). So in [20] it was shown that one can actually employ Mehta’s technique to prove that the q-extensions

(N) of Mehta’s eigenvectors F , given by the continuous q-Hermite polynomials Hn(x| q) of Rogers, namely

X∞ · ¸ (N) 1 2 Fmn (q) := exp − (xk(m)) Hn(sin κ xk(m)| q), (4.5) k=−∞ 2 11

X∞ · ¸ (N) −1 n 1 2 Fmn (q ) := i exp − (xk(m)) hn(sinh κ xk(m)| q) , (4.6) k=−∞ 2 q (N) 2π at the points xk (m) := N (kN + m), also enjoy simple transformation properties with respect to the ﬁnite Fourier transform (4.1). Indeed, let us evaluate ﬁrst how the F (N)(q)

(N) −1 transforms under the ﬁnite Fourier transform (4.1). Since the Fjk (q ) is periodic function of j with period N, one may write the Fourier expansion

X∞ µ ¶ (N) −1 (k) 2πi Fjk (q ) = al (q) exp lj , (4.7) l=−∞ N

with the expansion coeﬃcients

Z N (k) 1 2πi (N) − N lx −1 al (q) = e Fxk (q ) dx . N 0

(k) An explicit form of of the coeﬃcients al (q) is evaluated exactly in the same way as for ³ ´ (j) the alk in Mehta’s paper [2], except that the ordinary Hermite polynomials Hk xn there ³ ¯ ´ (j)¯ should be replaced by the continuous q-Hermite polynomials Hn sin κxn ¯ q . Needless to say that in the case under discussion one should employ the integral transform (3.5) rather than (3.4). This yields

 ¯  s ¯ 1 2 2π ¯ 2 (k) − k /4  ¯  − π l /N al (q) = √ q Hk sin κ l¯ q e . (4.8) N N ¯

The next step is to substitute (4.8) into (4.7) in order to obtain that

 s ¯  2 ¯ Ã ! q− k /4 X∞ 2π ¯ 2πi π l 2 (N) −1  ¯  Fjk (q ) = √ Hk sin κ l¯ q exp lj − N l=−∞ N ¯ N N

 s ¯  2 N−1 µ ¶ ∞ ¯ · ¸ q− k /4 X 2πi X 2π ¯ 1 ³ ´2  (N) ¯  (N) = √ exp mj Hk sin κ xn (m)¯ q exp − xn (m) N m=0 N n=−∞ N ¯ 2

NX−1 − k 2/4 (N) (N) = q Φjm Fmk (q) , (4.9) m=0 upon rearranging all terms in the sum over index l from the ﬁrst line of (4.9) with the aid 12

of the summation formula

X∞ NX−1 X∞ fl = fm+Nn ,N ≥ 2 . l=−∞ m=0 n=−∞

Thus, we have demonstrated that the ﬁnite Fourier transform (4.1) interrelates two q- extensions of the Mehta eigenvectors (4.3),

NX−1 (N) (N) k 2/4 (N) −1 Φjm Fmk (q) = q Fjk (q ) , (4.10) m=0

deﬁned by (4.5) and (4.6), respectively. Similarly, one may begin with the Fourier expansion for the q-extended Mehta eigenvectors F (N)(q)(cf. (4.7)) in order to verify that

NX−1 (N) (N) −1 k − k 2/4 (N) Φjm Fmk (q ) = (−1) q Fjk (q) . (4.11) m=0

Since the parameters q and N are independent here, from (4.5) and (4.6) one readily veriﬁes, by using (3.8), that the Mehta eigenvectors (4.3) are regained in the limit as q → 1−,

h i h i − n (N) (N) −n − n (N) −1 lim κ Fjk (q) = Fjk = i lim κ Fjk (q ) , q→1− q→1−

and both relations (4.10) and (4.11) reduce to (4.4). Also, the continuous limit as N → ∞ returns the Fourier integral transform (3.5) and its inverse, respectively. Observe that the ﬁnite Fourier transformations (4.10) and (4.11) interrelate, in complete analogy with the case of the Fourier integral transform (3.5) and its inverse, the q-extended Mehta eigenvectors (4.5) and (4.6), which are associated with the two distinct sets of values of the deformation parameter q: 0 < q < 1 and 1 < q < ∞. So as the next natural step, one may construct proper q-extended eigenvectors of the ﬁnite Fourier transform (4.1) [6]. Indeed, let us consider a linear combination of (4.5) and (4.6) of the form

(N) (N) (N) −1 fmn (q) := an(q) Fmn (q) + bn(q) Fmn (q ) , (4.12) 13

with some nonzero constants an(q) and bn(q), which we need to determine. Then, upon using (4.10) and (4.11), one obtains that

NX−1 h i (N) (N) (N) −1 n 2/4 (N) −1 Φkm an(q) Fmn (q) + bn(q) Fmn (q ) = q an(q) Fkn (q ) m=0

h i n − n 2/4 (N) n (N) (N) −1 + (−1) q bn(q) Fkn (q) = i an(q) Fkn (q) + bn(q) Fkn (q ) , (4.13)

−n n 2/4 provided that the constants an(q) and bn(q) are interrelated as bn(q) = i q an(q). When this relation holds for each n ∈ {0, 1,...,N−1}, the linear combinations (4.12) are

(N) n eigenvectors of the ﬁnite Fourier transform Φ , with eigenvalues λn = i .

1 − n 2/8 With the choice for the overall constant an(q) = 2 q , we write the q-extended ﬁnite Fourier eigenvector components as

h i 1 2 2 ψ(N)(q) := q− n /8 F (N)(q) + i−n q n /4 F (N)(q−1) mn 2 mn mn

X∞ h ³ ¯ ´ ³ ´i 1 − n 2/8 (N) ¯ n 2/8 (N) = q Hn sin κ xk (m)¯ q + q hn sinh κ xk (m)| q 2 k=−∞ · ¸ 1 ³ ´2 × exp − x(N)(m) , (4.14) 2 k q (N) 2π over the equally spaced set of points xk (m) = N (kN + m), which are the same as in

(4.3). With our choice of constants an(q), it is straightforward to verify that the eigenvectors

(N) −1 ψmn (q) exhibit the symmetry with respect to the exchange q ⇒ q of the form

(N) −1 n (N) ψmn (q ) = i ψmn (q) .

1 − n 2/8 The overall constant coeﬃcients an(q) were chosen above as 2 q in order to normalize (N) the q-extended eigenvectors ψmn (q) in such a way that they exhibit the natural limit property of reducing to the Mehta eigenvectors,

− n (N) (N) lim κ ψmn (q) = Fmn (q) , q→1− 14

consequence of limq→1− [an(q)] = 1/2 for all n = 0, 1, 2, ..., N − 1.

V. SIMILARITY TRANSFORMATION BY Fb CONVERTS q INTO q−1

In this part of our presentation we shall attempt to make transparent an algebraic struc- ture of q-difference equations (2.13) and (2.17) for the continuous q-Hermite polynomials of Rogers, which originates such striking harmony of these polynomials with the q-independent classical Fourier integral transform (1.1) and, consequently, with the finite Fourier transform (4.1). So it will be shown in what follows that the Fourier transform (1.1) actually intertwines two q-difference equations (2.13) and (2.17), which correspond to the distinct set of values for the parameter q : 0 < q < 1 and 1 < q < ∞ [21]. We begin with the remark that the integral Fourier transform (1.1) in the Hilbert space L2 of square-integrable functions f(x), x ∈ (−∞, ∞), can be represented in the operator form as µ π i ¶ Fb := exp Nc , (5.1) 2

where Nc is the self-adjoint second-order number operator,

Ã ! 1 d2 Nc := ξ2 − 1 − . (5.2) 2 dξ2

These two quantum-mechanical operators, the number operator Nc and the Hamiltonian Hc

c 1 c of the linear harmonic oscillator (3.3), are interrelated in a simple way N = ¯h ω H − 1/2. Also, from (3.3) it is evident that

c N ψn(x) = n ψn(x) . (5.3)

b n One readily verifies, by using definition (5.1) and formula (5.3), that F ψn(x) = i ψn(x), that is, the eigenfunctions and eigenvalues of the unitary operator Fb are the same as of the integral Fourier transform (3.4). Observe also that by definition (5.1), the operators Fb and 15

Nc commute, that is, h i Fb, Nc ≡ Fb Nc − Nc Fb = 0 . (5.4)

Thus, a compact form of (5.4), Fb Nc Fb−1 = N,c (5.5)

represents a standard algebraic way of expressing the fact that the number operator Nc and b Fourier-transform operator F have a common set of the eigenfunctions ψn(x), n = 0, 1, 2, ... . Note also that substituting explicit form (5.2) of the number operator Nc into (5.5) and cancelling common constant factors on both sides, one arrives at the relation

h i d Fb x2 + pb 2 Fb−1 = x2 + pb 2 , pb := −i . (5.6) x x x dx

At the same time, it is straightforward to verify that

b 2 b−1 2 F x F = pbx . (5.7)

Consequently, (5.6) simply reduces to

b 2 b−1 2 F pbx F = x . (5.8)

Thus, the invariance of the number operator Nc with respect to the similarity transformation (5.5) is rooted in the fact that, ﬁrst, the Nc contains the coordinate x and momentum operator

b d 2 b 2 px := −i dx in a symmetric way and, second, the x and px transform into each other under this similarity transformation by Fb. As we shall below, these two similarity transformations

2 2 (5.7) and (5.8) of x and pbx into each other turn out to be key relations also in the case of the q-Hermite polynomials at hand. Because of the importance of the integral Fourier transform (3.5) for the q-Hermite polynomials, we derive ﬁrst its operator form. With the aid of the explicit representation for the continuous q-Hermite polynomials (2.11), it is not hard to evaluate, by using at the last 16

step (2.12), that

  n o Xn  n  n o b − x2/2 −n k   b i(n−2k) κ x − x2/2 F Hn( sin κx| q) e = i (−1)   F e e k=0 k q

    n n X  n  d n o X  n  2 −n k   (n−2k) κ b − x2/2 −n k   − [x + ( n− 2k ) κ] /2 = i (−1)   e dx F e = i (−1)   e k=0 k k=0 k q q   Xn  n  −n k   (2k−n) κ x− (n−2k)2 κ2/2 − x2/2 = i (−1)   e e k=0 k q   Xn  n  n n2/4 − x2/2 k   k(k−n) (n−2k) κ x n n2/4 − x2/2 = i q e (−1)   q e = i q hn( sinh κx| q) e . (5.9) k=0 k q Thus the desired operator form of the integral Fourier transform (3.5) is

n o b − x2/2 n n2/4 − x2/2 F Hn( sin κx| q) e = i q hn( sinh κx| q) e . (5.10)

It should be emphasized that (5.10) is just another form of stating the key result (3.5) for the case under investigation. But it does enable one to employ the operational calculus techniques in order to prove that the similarity transformation by Fb intertwines two q- diﬀerence equations for the continuous q-Hermite polynomials of Rogers (2.13) and (2.17). As we already know from section 2, the continuous q-Hermite and q−1 -Hermite polynomials (2.11) and (2.12) satisfy q-diﬀerence equations

h i iκx −iκ∂x −iκx iκ∂x −n/2 e e + e e Hn( sin κx| q) = 2 q cos κx Hn( sin κx| q) ,

and h i κx − κ∂x − κx κ∂x n/2 e e + e e hn( sinh κx| q) = 2 q cosh κx hn( sinh κx| q) ,

respectively. But a glance at the Fourier transforms (3.5) and (5.10) reveals that we actually 17

need q-diﬀerence equations for the functions, which are the Hn( sin κx| q) and hn( sinh κx| q) polynomials, multiplied by the same Gaussian factor e− x2/2. So, one readily deduces from diﬀerence equations (2.13) and (2.17) that the desired equations can be written as (recall

b d that px = −i dx as before)

h i − x2/2 −(2n+1)/4 − x2/2 cosh κpbx Hn( sin κx| q) e = q cos κx Hn( sin κx| q) e , (5.11)

h i − x2/2 (2n+1)/4 − x2/2 cos κpbx hn( sinh κx| q) e = q cosh κx hn( sinh κx| q) e , (5.12)

respectively. These two q-diﬀerence equations are in fact interrelated by the similarity transformation by the unitary Fourier-transform operator Fb, deﬁned in (5.1). Indeed, to prove this assertion act on both sides of equation (5.11) by the operator Fb to get

h i b − x2/2 −(2n+1)/4 b − x2/2 F cosh κpbx Hn( sin κx| q) e = q F cos κx Hn( sin κx| q) e . (5.13)

Substitute now the relation

h i − x2/2 n n2/4 b−1 − x2/2 Hn( sin κx| q) e = i q F hn( sinh κx| q) e ,

which follows from (5.10), into both sides of (5.13) and employ on the right side of it the identity

b b−1 F cos κx F = cos κpbx ,

b i a x a d b which is an easy consequence of the intertwining relation F e = e dx F. This yields

h i h i b b−1 − x2/2 −(2n+1)/4 − x2/2 F cosh κpbx F hn( sinh κx| q) e = q cos κpbx hn( sinh κx| q) e . (5.14)

Comparison of (5.12) and (5.14) shows that these two q-diﬀerence equations coincide provided that the identity

b b−1 F cosh κpbx F = cosh κx (5.15) 18

holds. But this is not hard to prove, bearing in mind the well-known series expansion

P∞ 2k b cosh z = k=0 z /(2k)! . Recall that F is a unitary operator, therefore it is not hard to verify that

b 2m b−1 2m F pbx F = x , (5.16)

where m is an arbitrary positive integer number. So, after multiplying both sides of (5.16) by the factor κ 2m/(2m)! and summing them with respect to m from zero to infinity, one arrives at the desired identity (5.15). Similarly, one may start with the q-difference equation (5.12) and show that it is reducible to the q-difference equation (5.11), provided that

b−1 b F cosh κx F = cosh κpbx . (5.17)

This identity follows at once from (5.15), already proved above. Thus, we have demonstrated that q-diﬀerence equations (5.11) and (5.12) are interrelated by the similarity transformation by the operator Fb. This algebraic property of (5.11) and (5.12) explains the simple behavior of the q-Hermite polynomials (with 0 < q < 1 and 1 < q < ∞) with respect to the integral Fourier transform in (3.5). Obviously, it is more complicated than the mere commutativity of the operators Nc and Fb in the case of the

Hermite polynomials Hn(x). But the curious fact is that although q-diﬀerence equations (5.11) and (5.12) are no longer invariant with respect to the interchange of the coordinate

b d x and momentum operator px := −i dx , those equations do exhibit simple behavior under

this interchange x ↔ pbx. Indeed, since the interchange x ↔ pbx entails the changes cos κx ↔

cos κpbx and cosh κx ↔ cosh κpbx in (5.11) and (5.12), this means that these equations convert

into each other under the interchange x ↔ pbx. In other words, exactly as in the case of the

ordinary Hermite polynomials Hn(x), the similarity transformation of q-diﬀerence equations b (5.11) and (5.12) by the operator F turns out to be equivalent to the interchange x ↔ pbx 19 in the (5.11) and (5.12).

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