Introduction The solution of the heat equation arises as a modeling task in heat transfer and a variety of engineering, scientific, and financial applications

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 2 (2012), Pages 145-173

ON A q-ANALOGUE OF THE ONE-DIMENSIONAL HEAT

EQUATION

(COMMUNICATED BY FRANCISCO MARCELLAN )

AHMED FITOUHI & N ´EJI BETTAIBI & KAMEL MEZLINI

Abstract. In this paper, aq-analogue of the one-dimensional heat equation associated with someq-differential operators is considered and aq-analogue of the theory of the heat equation introduced by P. C. Rosenbloom and D. V.

Widder is developed.

1. Introduction

The solution of the heat equation arises as a modeling task in heat transfer and a variety of engineering, scientific, and financial applications. The best known analytic function theory associated with the heat equation is developed by P. C.

Rosenbloom and D. V. Widder in [10, 11] and it is based on the heat polynomials and associated heat functions. The radial heat equation has been investigated by Bragg [1] and more extensively by Haimo [4]. These works have been generalized by Fitouhi [2] for singular operators. For many problems, the exact solution is not available or too complicated to use. Then, a numerical method is necessary for solving the problem. It is well known that the quantum calculus provides a natural discretization of the heat equation. For this discretization, we shall replace the partial derivatives ∂

∂t and ∂

∂x by D_q2 derivative [3] and the Rubin’s∂q-derivative [8, 9] in time and in space, respectively, and we attempt to develop theq-analogue of the theory introduced by P. C. Rosenbloom and D. V. Widder. In this way, at the limit asqtends to 1, one recovers some results related to the heat equation in the continuous model.

We proclaim that, in this paper, we are not in a situation to study or discuss a numerical method, but we show by some examples and graphics that our results coincide with the classical ones whenqis near 1.

This paper is organized as follows: in Section 2, we recall some notations and useful

2000 Mathematics Subject Classification. 33D05,33D15, 33D45, 33D60, 33A30, 42A48, 33K05.

Key words and phrases. Rubin’s harmonic analysis, q-heat equation, q-heat polynomials, q-heat functions.

c

2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted December 15, 2011. Accepted May 22, 2012.

145

results. In Section 3, we define the generalized translation associated with the Ru- bin’s∂q-operator and we establish some of its properties. In Section 4, we review a few of the most basic solutions of the one-dimensional classical heat equation. Next, we introduce theq-heat equation, and we present theq-source solutionk(x, t;q) and study some of its properties. Section 5 is devoted to construct and study two basic sets of solutions of the q-heat equation: the set {v_n(x, t;q)}^∞_n=0 of q-heat polyno- mials and the q-associated functions set {w_n(x, t;q)}^∞_n=0. In particular, we show that the q-heat polynomials and theq-associated functions are closely related to the discrete q-Hermite I polynomials and the discrete q-Hermite II polynomials, respectively. Furthermore, we introduce two systems of biorthogonal polynomials related to theq-source solutionk(x, t;q). In Section 6, we discuss an asymptotic es- timations for the functionsvn(x, t;q) andwn(x, t;q) for largen. Next, we establish some results related to the series expansion of solutions ofq-heat equation. Finally, we discuss from an analytic and a graphic point of view how these q-difference operators can be used to solve approximately the heat equation and illustrate the performance of this approach with some examples.

2. Preliminaries

For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. We refer the reader to the general references [3] and [5], for the definitions, notations and properties of theq-shifted factorials and theq-hypergeometric functions.

Throughout this paper, we assumeq∈]0,1[ and we write

Rq ={±qⁿ : n∈Z}, Rq,+={qⁿ : n∈Z} and Req =Rq∪ {0}.

2.1. Basic symbols. For a complex numbera, theq-shifted factorials are defined by:

(a;q)₀= 1; (a;q)_n=

n−1

Y

k=0

(1−aq^k), n= 1,2, ...; (a;q)_∞=

∞

Y

k=0

(1−aq^k).

We also denote

[x]q= 1−q^x

1−q , x∈C and n!q= (q;q)n

(1−q)ⁿ, n∈N. It is easy to verify that

n!_q−1=q^−n(n−1)2 n!q. (1)

Using the Gaussq-binomial coefficients (see [3] ) n

k

q

= n!q

k!q(n−k)!q

,

theq-binomial theorem is given by (−z;q)_n =

n

X

k=0

n k

q

q^k(k−1)/2z^k. (2)

2.2. Operators and elementary q-special functions.

The Jackson’sq-derivative is defined by (see [3, 5]) D_qf(z) =







f(z)−f(qz)

(1−q)z if z6= 0

x→0limD_qf(x) if z= 0.

The Rubin’sq-differential operator is defined in [8, 9] by

∂q(f)(z) =







f(q⁻¹z) +f(−q⁻¹z)−f(qz) +f(−qz)−2f(−z)

2(1−q)z if z6= 0

x→0lim∂q(f)(x) if z= 0.

(3) Note that iff is differentiable atz, then∂_q(f)(z) andD_q(f)(z) tend tof⁰(z) asq tends to 1.

We state the following easily proved result:

Proposition 2.1. For alln∈N, we have (1) ∂_q²ⁿf =q^−n(n+1) D²ⁿ_q f_e

oΛⁿ_q +q⁻ⁿ² D²ⁿ_q f_o oΛⁿ_q, (2) ∂_q²ⁿ⁺¹f =q^−(n+1)2 D_q²ⁿ⁺¹fe

oΛ⁽ⁿ⁺¹⁾_q +q^−n(n+1) D²ⁿ⁺¹_q fo oΛⁿ_q, wherefe andfo are, respectively, the even and the odd parts off, andΛⁿ_q is the function defined by Λⁿ_q(x) =q⁻ⁿx.

Theq-Jackson integral (see [3]) is defined by Z b

a

f(x)dqx= Z b

0

f(x)dqx− Z a

0

f(x)dqx, (4)

where

Z a 0

f(x)dqx=a(1−q)

∞

X

n=−∞

f(aqⁿ)qⁿ, (5)

and from 0 to +∞and from−∞to +∞are defined by Z ∞

0

f(x)dqx= (1−q)

∞

X

n=−∞

f(qⁿ)qⁿ, (6)

Z ∞

−∞

f(x)dqx= (1−q)

∞

X

n=−∞

f(qⁿ)qⁿ+ (1−q)

∞

X

n=−∞

f(−qⁿ)qⁿ, (7) provided the sums converge absolutely.

Note that whenf is continuous on [0, a], it can be shown that

q→1lim Z a

0

f(x)dqx= Z a

0

f(x)dx. (8)

The following results hold by direct computation.

Lemma 2.1.

(1) If Z ∞

−∞

f(x)dqxexists, then (a) for all integer n,

Z ∞

−∞

f(qⁿt)dqt=q⁻ⁿ Z ∞

−∞

f(t)dqt.

(b) iff is odd, then Z ∞

−∞

f(t)dqt= 0.

(c) iff is even, then Z ∞

−∞

f(t)d_qt= 2 Z ∞

0

f(t)d_qt.

(2) If Z ∞

−∞

(∂qf)(t)g(t)dqt exists, then Z ∞

−∞

(∂qf)(t)g(t)dqt=− Z ∞

−∞

f(t)(∂qg)(t)dqt. (9) Notation. Using the Jackson’sq-integral, we we denote byL^p_q =L^p_q(Rq), p >0, the space of all complex functions defined onR^q induced by the norm

kfk_p,q= Z ∞

−∞

|f(x)|^pd_qx ¹_p

.

Twoq-analogues of the exponential function are given by ( see [3]) Eq(z) :=

∞

X

k=0

q^k(k−1)2 (q;q)k

z^k= (−z;q)∞, (10)

eq(z) :=

∞

X

k=0

1

(q;q)_kz^k= 1

(z;q)_∞ |z|<1. (11) Eq is entire on C. But, for the convergence of the second series, we need |z| <1;

however, because of its product representation,eq is continuable to a meromorphic function onCand has simple poles atz=q⁻ⁿ, n∈N.

We denote by

exp_q(z) :=e_q((1−q)z) =

∞

X

n=0

zⁿ

n!_q (12)

and

Expq(z) :=Eq((1−q)z) =

∞

X

n=0

q^n(n−1)2 zⁿ n!q

. (13)

It follows from (1) that

Exp_q2(z) =exp_q⁻²(z). (14)

We have (see [3])

lim

q→1⁻exp_q(z) = lim

q→1⁻Exp_q(z) =e^z, (15) wheree^z is the classical exponential function.

Theq-trigonometric functions (see[7]) are defined onCby cos(x;q²) :=

∞

X

n=0

(−1)ⁿb_2n(x;q²) (16) and

sin(x;q²) :=

∞

X

n=0

(−1)ⁿb_2n+1(x;q²), (17) where

bn(x;q²) =q[ⁿ₂]([ⁿ₂]⁺¹) n!q

xⁿ (18)

and [x] is the integer part ofx∈R.

These two functions induce a ∂q-adapted q-analogue exponential function (see [8, 9]):

e(z;q²) := cos(−iz;q²) +isin(−iz;q²) =

∞

X

n=0

bn(z;q²). (19) e(z;q²) is absolutely convergent for allzin the plane, and we have lim

q→1⁻e(z;q²) =e^z point-wise and uniformly on compacta. Note that we have

Lemma 2.2. (see [8])

For all λ∈C, ∂_qe(λz;q²) =λe(λz;q²). (20) For all x∈Rq, |e(ix;q²)| ≤ 2

(q;q)∞

. (21)

2.3. The Fourier-Rubin transform.

In [8] and [9], R. L. Rubin defined the Fourier-Rubin transform as F_q(f)(x) :=K

Z ∞

−∞

f(t)e(−itx;q²)d_qt, x∈Req, (22) where

K= (q;q²)_∞

2(q²;q²)_∞(1−q)¹². (23) Lettingq↑1 subject to the condition

Log(1−q)

Log(q) ∈2Z, (24)

gives, at least formally, the classical Fourier transform. In the remainder of this paper, we assume that the condition (24) holds.

It was shown in [8] and [9] that the Fourier-Rubin transform F_q satisfies the fol- lowing properties:

Theorem 2.1.

(1) If f, g∈L¹_q, then Z ∞

−∞

F_q(f)(x)g(x)d_qx= Z ∞

−∞

f(x)F_q(g)(x)d_qx.

(2) If f(u), uf(u)∈L¹_q, then

∂_qFq(f)(x) =Fq(−iuf(u))(x).

(3) If f, ∂qf ∈L¹_q, then

Fq(∂qf)(x) =ixFq(f)(x).

(4) Fq is an isomorphism ofL²_q,satisfying forf ∈L²_q kFq(f)kL²_q =kfkL²_q, and fort∈Rq,

f(t) =K Z ∞

−∞

Fq(f)(x)e(itx;q²)dqx. (25)

3. The q-generalized translation operator associated with the operator∂q

Definition 3.1. Theq-translation operatorTy,q, y∈C, related to theq-differential operator ∂_q, is defined by

Ty,q(f)(x) :=e(y∂q;q²)f(x) =

∞

X

n=0

bn(y;q²)∂_qⁿf(x), (26) provided the series converges point wise.

Remarks.

(1) Note that for suitable functionsf(x), we have lim

q→1⁻Tq,y(f)(x) =f(x+y).

(2) Since for all n ∈ N, bn(.;q²) is a polynomial of degree n, then for all k≥n+ 1, we have∂_q^kbn(x;q²) = 0. Then

pn,q(x, y) :=n!qTy,qbn(x;q²) (27) is a polynomial of degreen, that it will be called q-binomial polynomial.

Proposition 3.1. Let x, y, λ∈Candn∈N. Then,

∂_q^kbn(x;q²) =b_n−k(x;q²), k= 0,1, ...n. (28) T_y,qb_n(x;q²) =

n

X

k=0

b_k(y;q²)b_n−k(x;q²) = 1

n!_qp_n,q(x, y). (29) Ty,qe(λx;q²) =e(λx;q²)e(λy;q²). (30) The generating function of{pn,q(., .)}_n∈N is given by

e(λx;q²)e(λy;q²) =

∞

X

n=0

p_n,q(x, y) n!q

λⁿ. (31)

Proof. (28) follows easily from the fact that∂_qb_n(x;q²) =b_n−1(x;q²).

(29) is a consequence of the relations (28) and (26).

Using (20), we have Ty,qe(λx;q²) =

∞

X

n=0

bn(y;q²)∂_qⁿe(λx;q²)

=

∞

X

n=0

bn(y;q²)λⁿe(λx;q²)

= e(λx;q²)

∞

X

n=0

b_n(λy;q²) =e(λx;q²)e(λy;q²).

This proves (30). (31) follows from (30) and (27).

Lemma 3.1. Forn= 0,1,2, ..., we have p_2n,q(|x|,|y|)| ≤(2n)!_q

n!q²

(1 +|xy|)Eq²(|x|²)E_q2(|y|²) (32)

and

p_2n+1,q(|x|,|y|)≤ (2n+ 1)!_q n!q²

(|x|+|y|)E_q2(|x|²)E_q2(|y|²), (33) whereEq(.)is the q-exponential function given by (10).

Proof. From (29), we have

p2n,q(|x|,|y|) = (2n)!q 2n

X

k=0

bk(|x|;q²)b_2n−k(|y|;q²).

But,

2n

X

k=0

b_k(|x|;q²)b_2n−k(|y|;q²) =

n

X

k=0

b_2k(|x|;q²)b_2(n−k)(|y|;q²) +

n−1

X

k=0

b2k+1(|x|;q²)b_2(n−k)−1(|y|;q²) and using the fact that (2k)!q ≥(k!_q2)², the following inequalities hold

b2k(|x|;q²)≤ q^k(k−1)|x|^2k k!_q2

, b2k(|y|;q²)≤ E_q2(|y|²) k!_q2

, k= 0,1,2, ...

It follows then, by using theq-binomial theorem (2),

n

X

k=0

b_2k(|x|;q²)b_2(n−k)(|y|;q²) ≤ E_q2(|y|²) n!q²

n

X

k=0

n!_q2q^k(k−1) (n−k)!q²k!q²

|x|^2k

≤ E_q2(|y|²)(−|x|²;q²)n

n!q²

≤ Eq²(|x|²)Eq²(|y|²) n!_q2

.

(34)

Since (2k+ 1)!q ≥k!_q2(k+ 1)!_q2, the following inequalities hold b2k+1(|x|;q²)≤|x|q^k(k−1)|x|^2k

k!_q2

, k= 0,1,2, ...

b2k−1(|y|;q²)≤ |y|E_q2(|y|²) k!_q2

, k= 1,2,3, ...

Consequently,

n−1

X

k=0

b2k+1(|x|;q²)b_2(n−k)−1(|y|;q²) ≤ |xy|E_q2(|y|²) n!_q2

n

X

k=0

n!_q2q^k(k−1) (n−k)!_q2k!_q2

|x|^2k

≤ |xy|E_q2(|x|²)E_q2(|y|²) n!_q2

. This inequality together with (34) give (32).

Let us now prove the inequality (33). We have p2n+1,q(|x|,|y|) = (2n+ 1)!q

2n+1

X

k=0

bk(|x|;q²)b_2n+1−k(|y|;q²),

and

2n+1

X

k=0

b_k(|x|;q²)b_2n+1−k(|y|;q²) =

n

X

k=0

b_2k(|x|;q²)b_2(n−k)+1(|y|;q²) +

n

X

k=0

b2k+1(|x|;q²)b_2(n−k)(|y|;q²).

In the same way as in the proof of inequality (32), we obtain

2n+1

X

k=0

bk(|x|;q²)b_2n+1−k(|y|;q²)≤(|x|+|y|)E_q2(|x|²)E_q2(|y|²) n!_q2

.

Definition 3.2. For σ > 0, we denote by Eσ,q the set of all entire functions f satisfying:

∃ M >0 :∀n∈N,











|∂_q²ⁿf(0)| ≤M n!_q2

σⁿ ,

|∂_q²ⁿ⁺¹f(0)| ≤ M n!_q2

σⁿ .

(35)

Remark. In the definition of the class Eσ,q, the constantM is independent ofn, depending only on the functionf.

Proposition 3.2. Let σ >1 andf be in Eσ,q. Then Ty,qf(x) =

∞

X

n=0

∂_qⁿf(0) n!q

pn,q(x, y), (36)

wherepn,q(x, y)are defined by (27).

The infinite series (36) converges locally uniformly in xandy.

Proof. First, iff is inEσ,q, then f(x) =

∞

X

n=0

anxⁿ =

∞

X

n=0

an

b_n(1;q²)bn(x;q²).

So, by (28), we have for all nonnegative integerk

∂_q^kf(x) =

∞

X

n=k

a_n

bn(1;q²)b_n−k(x;q²), (37) from which we deduce that

∀n≥0, a_n =∂_qⁿf(0)b_n(1;q²). (38)

By (26), (37) and (38), we have T_y,qf(x) =

∞

X

k=0

b_k(y;q²)

∞

X

n=k

an

b_n(1;q²)b_n−k(x;q²)

=

∞

X

k=0

b_k(y;q²)

∞

X

n=k

∂_qⁿf(0)b_n−k(x;q²)

=

∞

X

n=0

∂_qⁿf(0)

n

X

k=0

b_k(y;q²)b_n−k(x;q²).

(39)

Then, the desired conclusion follows from the relation (29).

Let us now, prove the locally uniformly convergence inxandy of the series (36).

From the definition of the functionb_n(.;q²) and the relation (29), we obtain for all x, y∈C,

|pn,q(x, y)| ≤pn,q(|x|,|y|).

Then, by using Lemma 3.1, we get, sincef ∈ Eσ,q, that there existsM >0, such that for allx, y∈Candn≥0,

∂_q²ⁿf(0)|

(2n)!_q p2n,q(x, y)

≤ M

σⁿ(1 +|xy|)E_q2(|x|²)E_q2(|y|²) and

∂_q²ⁿ⁺¹f(0)|

(2n+ 1)!_q p2n+1,q(x, y)

≤ M

σⁿ(|x|+|y|)E_q2(|x|²)E_q2(|y|²).

Finally, these relations, the continuity of the functionE_q2 and the hypothesisσ >1 prove the locally uniformly convergence inxandy of the series (36).

4. q-heat Equation and q-source solution

4.1. Heat Equation. We restrict our attention to the simplest one-dimensional heat equation onR

∂u

∂t =∂²u

∂²x, (40)

which has been the object of extensive studies. In particular, we refer the reader to the book of Widder [10]. One of the most important families of solutions of the heat equation (40) is the so-called heat polynomials defined by

vn(x, t) =n!

[ⁿ₂]

X

k=0

t^k

k!(n−2k)!x^n−2k, n= 0, 1, 2, ... (41) The heat polynomials are closely related to the Hermite polynomialsHn(x) by (see [11])

vn(x, t) = (−t)^n/2Hn

x

√−4πt

.

The source solution or fundamental solution of (40) is given by k(x, t) = e^−x4t2

√4πt.

The associated functionswn(x, t) are defined by wn(x, t) =v_n(x,−t)k(x, t)

tⁿ , n= 0, 1, 2....

They are solutions of the heat equation (40) and they are related to the Hermite polynomialsHn(x) according to

wn(x, t) =t^−n/2k(x, t)Hn( x

√4πt).

We have the following biorthogonality relation Z +∞

−∞

v_n(x,−t)w_m(x, t)dx= 2ⁿn!δ_m,n, whereδ_m,nis the Kronecker symbol.

In [10, 11] a complete study of necessary and sufficient conditions for the validity of the expansion of solutions of the heat equation (40) in terms of heat polynomials and associated functions has been developed.

4.2. The q-Heat Equation. We consider the followingq-heat equation:

Dq²,tu=∂_q,x² u. (42)

Remark. Taking into account that lim

q→1Dq²,tu(t, x) =∂u

∂t(t, x) and lim

q→1∂_q,x² u(t, x) =

∂²u

∂²xu(t, x), it is clear that (42) isq-analogue of the standard heat equation (40).

That is, equation (40) can be recovered whenqtends to 1.

Consider now, the following function k(x, t;q) =C(t;q)exp_q2

− qx² t(1 +q)²

, t >0, (43)

whereexp_q2(.) is defined by (12) and C(t;q) =

−^q(1−q)_t(1+q),−^qt(1+q)_1−q , q;q²

∞

2(1−q)

−^q_t(1+q)^2(1−q),−^t(1+q)_1−q , q²;q²

∞

. (44)

Since

∂q,xk(x, t;q) =−C(t;q)x qt(1 +q)expq²

− x² tq(1 +q)²

(45) and

∂_q,x² k(x, t;q) =D_q2,tk(x, t;q) =− C(t;q) qt(1 +q)

1− x² t(1 +q)

exp_q2

− x² tq(1 +q)²

, then k(x, t;q) is a solution of the q-heat equation (42), that it will be called the q-source solution.

For discussing its properties, we need the following preliminary results.

Proposition 4.1.

(1) The functione_q2(−x²) has the rapid decreasing property:

x→∞lim pn(x)e_q2(−x²) = 0, n= 0,1,2..., (46) for all polynomialp_n(x) is a of degree n.

(2)

x→∞lim

e_q2(−qx²)

eq²(−x²) = +∞. (47)

(3) Forβ >0, we have

x→∞lim

eq²(−qx²)eq²(−βx²)

e_q2(−x²) = 0. (48)

Proof. (1) Observe that forn= 0,1,2, ..., we have e_q2(−x²) =

∞

Y

k=0

(1 +q^2kx²)⁻¹≤(1 +q²ⁿx²)⁻ⁿ=O(x⁻²ⁿ) asx→ ±∞.

(2) Forn= 0,1,2, ..., we have e_q2(−qx²)

e_q2(−x²) ≥

n

Y

k=0

1 +q^2kx² 1 +q^2k+1x², it follows that for alln∈N,

x→∞lim

eq²(−qx²) e_q2(−x²) ≥q⁻ⁿ, which yields to the result.

(3) Since the functione_q2(−x²) is decreasing on [0,∞[, we obtain whenβ ≥1, e_q2(−βx²)≤e_q2(−x²), ∀x∈R

and then

x→∞lim

e_q2(−qx²)e_q2(−βx²) eq²(−x²) ≤ lim

x→∞e_q2(−qx²) = 0.

Ifβ <1, then there existsn∈N, such that β > q²ⁿ and so e_q2(−βx²)≤e_q2(−q²ⁿx²) = (−x²;q²)ne_q2(−x²).

Then, by using (46), we get

x→∞lim

e_q2(−qx²)e_q2(−βx²) e_q2(−x²) ≤ lim

x→∞(−x²;q²)ne_q2(−qx²) = 0.

Proposition 4.2. Forλ >0and n= 0,1,2, ..., we have Z ∞

0

eq²(−λy²)y²ⁿ⁺¹dqy=(1−q)q^−n(n+1)(q²;q²)_n

λⁿ⁺¹ (49)

and

Z ∞ 0

e_q2(−λy²)y²ⁿd_qy=c_q(λ)q⁻ⁿ²(q;q²)_n

λⁿ , (50)

where

c_q(λ) = (1−q)(−qλ,−q/λ, q²;q²)_∞ (−λ,−q²/λ, q;q²)∞

. (51)

Proof. (1) From the definition of the Jackson’sq-integral (6) and the relation (11), we have

Z ∞ 0

e_q2(−λy²)y²ⁿ⁺¹dqy= (1−q)

∞

X

k=−∞

q^(2n+2)k (−λq^2k;q²)_∞. Then, using the Ramanujan identity

∞

X

k=−∞

z^k

(bq^k;q)_∞ =(bz, q/(bz), q;q)_∞

(b, z, q/b;q)_∞ , b6= 0, (52) we obtain

Z ∞ 0

e_q2(−λy²)y²ⁿ⁺¹dqy=(1−q) −λq²ⁿ⁺²,−q⁻²ⁿ/λ, q²;q²

∞

(−λ, q²ⁿ⁺²,−q²/λ;q²)_∞ . We conclude (49) by using the following two identities

−λq²ⁿ⁺²;q²

∞= −q²λ;q²

∞

(−q²λ;q²)_n and

−q⁻²ⁿ/λ;q²

∞=q^−n(n+1)λ⁻ⁿ −q²λ;q²

n −λ⁻¹;q²

∞. (2) A new use of the Ramanujan identity gives

Z ∞ 0

e_q2(−λy²)y²ⁿd_qy = (1−q) −λq²ⁿ⁺¹,−q⁻²ⁿ⁺¹/λ, q²;q²

∞

(−λ, q²ⁿ⁺¹,−q²/λ;q²)_∞ . Then, (50) follows by using the two following facts that

−λq²ⁿ⁺¹;q²

∞= −qλ;q²

∞

(−qλ;q²)_n and

−q⁻²ⁿ⁺¹/λ;q²

∞=q^−n(n+1)(q/λ)ⁿ −qλ;q²

n −q/λ;q²

∞.

Proposition 4.3. Fort >0 andn= 0,1,2..., we have Z ∞

−∞

k(x, t;q)b2n(x;q²)dqx = tⁿ n!_q2

(53) and

Z ∞

−∞

k(y, t;q)b2n+1(|y|;q²)dqy=2C(t;q)tⁿ⁺¹(1 +q)²ⁿ⁺¹n!_q2

qⁿ⁺¹(2n+ 1)!q

, (54)

whereC(t;q)is defined by (44).

Proof. We obtain the result by takingλ=q(1−q)

t(1 +q) in (49) and (50) and using the relation

(q;q²)n

(q;q)_2n = 1 (q²;q²)_n.

Proposition 4.4.

F_q[k(., t;q)] (x) =Kexp_q2 −tx² .

Proof. Sincek(., t;q) is even, then by using (53), we obtain Fq(k(., t;q))(x) = K

Z ∞

−∞

k(y, t;q) cos(yx;q²)dqy

= K

∞

X

n=0

(−1)ⁿx²ⁿ Z ∞

−∞

k(y, t;q)b_2n(y;q²)d_qy,

= K

∞

X

n=0

(−1)ⁿx²ⁿ tⁿ n!_q2

=Kexp_q2 −tx² .

The following result summarizes some other properties of theq-source solution.

Theorem 4.1. For allt >0 andx∈R, we have 1)k(x, t;q)>0.

2) lim

t→0⁺k(x, t;q) =







+∞ if x= 0, 0 if x6= 0.

3)

Z ∞

−∞

k(x, t;q)dqx= 1. (55)

Proof. 1) follows from the definition of theq-source solution.

2) Note, thatk(0, t;q) =C(t;q) and putλ= s

q(1−q) t(1 +q). Using (47), we obtain

lim

t→0⁺C(t;q) = lim

λ→+∞

(q;q²)_∞e_q2(−qλ²)

2(1−q)(q²;q²)_∞e_q2(−λ²)= +∞.

Ifx6= 0, then by using (48), we get lim

t→0⁺k(x, t;q) = lim

λ→+∞

(q;q²)∞eq²(−qλ²)eq²(−λ²x²) 2(1−q)(q²;q²)_∞e_q2(−λ²) = 0.

3) follows from (53) by takingn= 0.

5. q-Heat Polynomials andq-Associated functions 5.1. q-Heat Polynomials.

Forz∈C, t∈Randx∈R, we have exp_q2 tz²

cos(−ixz;q²) =

∞

X

p=0

t^pz^2p p!q²

! _∞ X

k=0

q^k(k+1)x^2kz^2k (2k)!q

!

=

∞

X

n=0

z²ⁿ

n

X

k=0

t^n−kq^k(k+1)x^2k (n−k)!_q2(2k)!_q

(56)

-6 -4 -2 0 2 4 6 0.1

0.2 0.3 0.4 0.5

-3 -2 -1 0 1 2 3

0.2 0.4 0.6 0.8 1.0 1.2

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

5 10 15 20 25 30

Figure 1: Comparison of the classical heat kernelk(x, t) (solid line) and theq-heat kernelk(x, t;q) (dashed line) atq= 0.9 fort= 0.8,t= 10⁻¹andt= 10⁻⁴.

and

iexp_q2 tz²

sin(−ixz;q²) =

∞

X

p=0

t^pz^2p p!_q2

! _∞ X

k=0

q^k(k+1)x^2k+1z^2k+1 (2k+ 1)!_q

!

=

∞

X

n=0

z²ⁿ⁺¹

n

X

k=0

t^n−kq^k(k+1)x^2k+1 (n−k)!_q2(2k+ 1)!q

.

(57)

Then, using the relation

e(xz;q²) = cos(−ixz;q²) +isin(−ixz;q²), we obtain

exp_q2 tz²

e(xz;q²) =

∞

X

n=0

vn(x, t;q)zⁿ n!q

, (58)

with, for all nonnegative integern, vn(x, t;q) =n!q

[ⁿ₂]

X

k=0

t^k k!_q2

b_n−2k(x;q²). (59)

Remarks

(1) It is clear that for all nonnegative integer n, vn(., t;q) is a polynomial of degree n and when q tends to 1, vn(., t;q) reduces to the standard heat polynomial (41). So, the polynomials v_n(., t;q) will be called q-heat poly- nomials.

(2) It is easy to derive from (28)

∂q,xvn(x, t;q) = [n]qvn−1(x, t;q) (60) and

D_q2,tv_n(x, t;q) = [n]_q[n−1]_qv_n−2(x, t;q). (61) Then all theq-heat polynomialsvn(x, t;q) are solutions of theq-heat equa- tion (42).

(3) Multiplying the both sides of (58) byExpq²(−tz²) and next comparing the coefficient ofzⁿ, we obtain

bn(x;q²) =

[ⁿ₂]

X

k=0

(−t)^kq^k(k−1)v_n−2k(x, t;q)

k!_q2(n−2k)!_q , n= 0, 1, 2.... (62) Theq-heat polynomials have the followingq-integral representations.

Proposition 5.1. Fort >0,x∈Randn= 0,1,2..., we have vn(x, t;q) =

Z ∞

−∞

k(y, t;q)pn,q(x, y)dqy. (63)

Proof. Lett >0,x∈Randnbe a nonnegative integer. Then, using the parity of the functionk(., t;q) and the relation (53), we get

Z ∞

−∞

k(y, t;q)pn,q(x, y)dqy = n!q n

X

k=0

bn−k(x;q²) Z ∞

−∞

k(y, t, q)bk(y;q²)dqy

= n!q

[ⁿ2] X

k=0

b_n−2k(x;q²) Z ∞

−∞

k(y, t;q)b2k(y;q²)dqy

= n!q

[ⁿ₂] X

k=0

b_n−2k(x;q²) t^k k!_q2

=vn(x, t;q).

The following easily proved result shows that the q-heat polynomials are closely related to the discreteq-Hermite I polynomials defined by (see [6])

h_n(x;q) = (q;q)_n

[ⁿ₂]

X

k=0

(−1)^kq^k(k−1)x^n−2k (q²;q²)_k(q;q)_n−2k .

Proposition 5.2. For all nonnegative integern,t >0 andx∈Rwe have v_2n(x, t;q) =q^−n(n−1)

(iβ)²ⁿ h_2n(iβqⁿx;q) (64) and

v2n+1(x, t;q) = q⁻ⁿ²

(iβ)²ⁿ⁺¹h2n+1(iβqⁿx;q), (65) where

β=β(t, q) =

s 1−q

t(1 +q). (66)

5.2. q-Associated functions.

Definition 5.1. Forn= 0,1,2, ...andt >0, theq-associated functionsw_n(x, t;q) is the function defined by

wn(x, t;q) = (−(1 +q))ⁿ∂ⁿ_q,xk(x, t;q). (67)

Remark. It is easy to see that theq-associated functionswn(x, t;q), n= 0,1,2, ...

are solutions of theq-heat equation (42).

We recall that the discreteq-Hermite II polynomials are given by (see [6]) h˜_n(x;q) = (q;q)_n

[ⁿ₂]

X

k=0

(−1)^kq^−2nkq^k(2k+1)x^n−2k (q²;q²)k(q;q)n−2k

and satisfy the following Rodrigues-type formula

ω(x;q)˜hn(x;q) = (q−1)ⁿq^−n(n−1)2 Dⁿ_q[ω(x;q)], where

ω(x;q) = 1 (−x²;q²)_∞.

The following result gives some relations between the q-source solution and the discreteq-Hermite II polynomials.

Proposition 5.3. Forn= 0,1,2, ..., we have

∂_q,x²ⁿk(x, t;q) =qⁿ⁽ⁿ⁻²⁾γ²ⁿ (q−1)²ⁿ

˜h2n(γq⁻ⁿx;q)k(q⁻ⁿx, t;q) and

∂_q,x²ⁿ⁺¹k(x, t;q) = qⁿ⁽ⁿ⁻¹⁾γ²ⁿ⁺¹ q(q−1)²ⁿ⁺¹

˜h2n+1(γq⁻⁽ⁿ⁺¹⁾x;q)k(q⁻⁽ⁿ⁺¹⁾x, t;q), where

γ=γ(t, q) =q¹²β(t, q) = s

q(1−q)

t(1 +q). (68)

Proof. Using Proposition 2.1, we obtain

∂²ⁿ_q,xk(x, t;q) =q^−n(n+1)D²ⁿ_q [k(., t;q)]oΛⁿ_q(x) and

∂²ⁿ⁺¹_q,x k(x, t;q) =q^−(n+1)2D_q²ⁿ⁺¹[k(., t;q)]oΛⁿ⁺¹_q (x).

But, the the fact that

k(x, t;q) =C(t;q)w(γx;q) gives

Dⁿ_q,x[k(x, t;q)] =C(t;q)D_qⁿ[ω(γx;q)] =C(t;q)γⁿ[D_qⁿω](γx;q).

So, from the Rodrigues-type formula, we get

D_q,xⁿ [k(x, t;q)] =γⁿ(q−1)⁻ⁿq^n(n−1)2 h˜_n(γx;q)k(x, t;q),

which yields to the desired results.

Proposition 5.4. Forn= 0,1,2, ..., we have

∂_q,x²ⁿk(x, t;q) =v2n(q¹²x,−t;q⁻¹)

t²ⁿ(1 +q)²ⁿ k(q⁻ⁿx, t;q) and

∂_q,x²ⁿ⁺¹k(x, t;q) =−q⁻¹²v_2n+1(q⁻¹²x,−t;q⁻¹)

t²ⁿ⁺¹(1 +q)²ⁿ⁺¹ k(q⁻⁽ⁿ⁺¹⁾x, t;q).

Proof. It is easy to verify that the discreteq-Hermite II polynomials are related to the discreteq-Hermite I polynomials by

˜h_n(x;q) =i⁻ⁿh_n(ix;q⁻¹) n= 0, 1, 2, .... (69) Then, forn= 0, 1, 2, ..., we have

˜h2n(γq⁻ⁿx;q) =i⁻²ⁿh2n(iγq⁻ⁿx;q⁻¹) =i⁻²ⁿh2n(iβq⁻ⁿq¹²x;q⁻¹) and

h˜2n+1(γq⁻⁽ⁿ⁺¹⁾x;q) = i^−(2n+1)h2n+1(iγq⁻⁽ⁿ⁺¹⁾x;q⁻¹)

= i^−(2n+1)h_2n+1(iβq⁻ⁿq⁻¹²x;q⁻¹).

Thus, since

β(t, q) =β(−t, q⁻¹), (70)

the first equality follows from (64) and the fact that i⁻²ⁿqⁿ⁽ⁿ⁻²⁾γ²ⁿ

(1−q)²ⁿ = qⁿ⁽ⁿ⁻¹⁾ (iβ)²ⁿt²ⁿ(1 +q)²ⁿ, and the second equality follows from (65) and the relation

i^−(2n+1)qⁿ⁽ⁿ⁻¹⁾γ²ⁿ⁺¹

q(1−q)²ⁿ⁺¹ = qⁿ²⁻¹²

(iβ)²ⁿ⁺¹t²ⁿ⁺¹(1 +q)²ⁿ⁺¹.

Definition 5.2. We define the polynomialsv˜k(x, t;q),n= 0,1,2, ..., by

˜

v2n(x, t;q) = v2n(q¹²x, t;q⁻¹),

˜

v_2n+1(x, t;q) = q⁻¹²v_2n+1(q⁻¹²x, t;q⁻¹). (71) Note that

q→1lim˜v_n(x, t;q) =v_n(x, t).

The following result summarizes some properties of the polynomials ˜vn(x, t;q).

Proposition 5.5.

1) Operation of the operator ∂q on˜vn(x, t;q):

∂q,x˜vn(x, t;q) =q⁻¹[n]_q⁻¹v˜_n−1(x, t;q), n= 1, 2 , .... (72) 2) The generating function for ˜v_n(x, t;q)is given by

Expq² tz²

e(q⁻¹xz;q²) =

∞

X

n=0

˜

vn(x, t;q) zⁿ n!_q−1

. (73)

3) The ˜vn(x, t;q) polynomials are related to the discrete q-Hermite II polynomials by

˜

v2n(x,−t;q) = qⁿ⁽ⁿ⁻¹⁾ β²ⁿ

˜h2n(γq⁻ⁿx;q), n= 0,1,2, ...

˜

v2n+1(x,−t;q) = qⁿ²⁻¹² β²ⁿ⁺¹

˜h2n+1(γq⁻⁽ⁿ⁺¹⁾x;q), n= 0,1,2, ...

(74)

whereβ andγ are, respectively, defined by (66) and (68).

Proof. 1) By (1), we have

(2k)!_q−1=q^−2k2+k(2k)!_q and (2k+ 1)!_q−1 =q^−2k2−k(2k+ 1)!_q. (75) So, for allk= 0, 1, 2, ...,

b2k(q¹²x;q⁻²) =b2k(q⁻¹x;q²), (76) and

q⁻¹²b2k+1(q⁻¹²x;q⁻²) =b2k+1(q⁻¹x;q²), (77) wherebn(x;q²) is defined by (18).

Then, from (28), we obtain for allk= 1, 2, ...,

∂_qb_2k(q¹²x;q⁻²) =q⁻¹b_2k−1(q⁻¹x;q²) =q⁻³²b_2k−1(q⁻¹²x;q⁻²) and

q⁻¹²∂_qb_2k+1(q⁻¹²x;q⁻²) =q⁻¹b_2k(q⁻¹x;q²) =q⁻¹b_2k(q¹²x;q⁻²).

Thus, (72) follows from these two equalities and the definition of ˜v_n(x, t;q).

2) By (76) and (77), we have

cos(q⁻¹z;q²) = cos(q¹²z;q⁻²) and sin(q⁻¹z;q²) =q⁻¹²sin(q⁻¹²z;q⁻²).

Then,

Exp_q2 tz²

cos(−iq⁻¹xz;q²) =exp_q⁻² tz²

cos(−iq¹²xz;q⁻²) and

Exp_q2 tz²

sin(−iq⁻¹xz;q²) =q⁻¹²exp_q⁻² tz²

sin(−iq⁻¹²xz;q⁻²).

Finally, (73) follows by replacingx,qbyq¹²x,q⁻¹in (56) and byq⁻¹²x,q⁻¹in (57).

3) Using (70), (64) and (65), we obtain v_2n(q¹²x,−t;q⁻¹) = qⁿ⁽ⁿ⁻¹⁾

(iβ)²ⁿ h_2n(iq¹²βq⁻ⁿx;q⁻¹) =qⁿ⁽ⁿ⁻¹⁾

(iβ)²ⁿ h_2n(iγq⁻ⁿx;q⁻¹) and

v2n+1(q⁻¹²x,−t;q⁻¹) = qⁿ²

(iβ)²ⁿ⁺¹h2n+1(iq⁻¹²βq⁻ⁿx;q⁻¹)

= qⁿ²

(iβ)²ⁿ⁺¹h_2n+1(iγq⁻⁽ⁿ⁺¹⁾x;q⁻¹),

which together with (69) give the result.

The following result follows easily from Proposition 5.4, and the relations (71) and (74). It shows that the q-associated functionsw_n(x, t;q) are closely related to the discreteq-Hermite II polynomials and to the polynomials ˜v_n(x,−t;q).

Proposition 5.6. Fort >0 andn= 0,1,2, ..., we have w2n(x, t;q) = ˜v2n(x,−t;q)

t²ⁿ k(q⁻ⁿx, t;q)

= qⁿ⁽ⁿ⁻¹⁾ (tβ)²ⁿ

˜h_2n(γq⁻ⁿx;q)k(q⁻ⁿx, t;q), w2n+1(x, t;q) = ˜v2n+1(x,−t;q)

t²ⁿ⁺¹ k(q⁻⁽ⁿ⁺¹⁾x, t;q)

= qⁿ²⁻¹² (tβ)²ⁿ⁺¹

˜h2n+1(γq⁻⁽ⁿ⁺¹⁾x;q)k(q⁻ⁿx, t;q),

(78)

whereβ andγ are, respectively, defined by (66) and (68).

5.3. Biorthogonal relation. To establish a biorthogonal relation between the

˜

vn(qx,−t;q) andwn(x, t;q), we need the following result.

Lemma 5.1. Fort >0 andn= 1, 2, 3, .., we have Z ∞

−∞

˜

v2n(qx,−t;q)k(x, t;q)dqx= 0. (79) Proof. Lett >0 andn≥1. Using the relation (1), we obtain

˜

v2n(qx,−t;q) = v2n(q³²x,−t;q⁻¹) = (2n)!_q⁻¹

n

X

k=0

(−t)^n−kq^−k2+2kx^2k (n−k)!_q−2(2k)!_q−1

= q⁻ⁿ²(2n)!q n

X

k=0

(−t)^n−kq^2k2−2nk+2kx^2k (n−k)!_q2(2k)!_q . The lefthand side of (79) is then equal to

C(t;q)q⁻ⁿ²(2n)!q n

X

k=0

(−t)^n−kq^2k2−2nk+2k (n−k)!_q2(2k)!_q

Z ∞

−∞

exp_q2

− qx² t(1 +q)²

x^2kdqx.

But, takingλ=q(1−q)

t(1 +q) in (50), we obtain Z ∞

−∞

exp_q2

− qx² t(1 +q)²

x^2kd_qx= 2c_q(λ)q^−k2−k(q;q²)_k(1 +q)^kt^k

(1−q)^k ,

wherec_q(λ) is defined by (51).

Then, sinceCq(t, λ) = 1

2cq(λ), theq-integral in (79) is equal to q⁻ⁿ²(2n)!q(−t)ⁿ(1−q²)ⁿ

n

X

k=0

(−1)^kq^k2−2nk+k (q²;q²)_n−k(q²;q²)_k, which is equal to

q⁻ⁿ²(2n)!_q(−t)ⁿ n!q²

n

X

k=0

n k

q²

q^k(k−1) −q^−2n+2k . Hence, from theq-binomial theorem (2), we get

Z ∞

−∞

˜

v_2n(qx,−t;q)k(x, t;q)d_qx=q⁻ⁿ²(2n)!_q(−t)ⁿ n!q²

(q⁻²ⁿ⁺²;q²)_n= 0.