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_{q}2 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 ={±q^{n} : n∈Z}, Rq,+={q^{n} : 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)/2}z^{k}. (2)

2.2. Operators and elementary q-special functions.

The Jackson’sq-derivative is defined by (see [3, 5])
D_{q}f(z) =

f(z)−f(qz)

(1−q)z if z6= 0

x→0limD_{q}f(x) if z= 0.

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

∂q(f)(z) =

f(q^{−1}z) +f(−q^{−1}z)−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 tof^{0}(z) asq
tends to 1.

We state the following easily proved result:

Proposition 2.1. For alln∈N, we have
(1) ∂_{q}^{2n}f =q^{−n(n+1)} D^{2n}_{q} f_{e}

oΛ^{n}_{q} +q^{−n}^{2} D^{2n}_{q} f_{o}
oΛ^{n}_{q},
(2) ∂_{q}^{2n+1}f =q^{−(n+1)}^{2} D_{q}^{2n+1}fe

oΛ^{(n+1)}_{q} +q^{−n(n+1)} D^{2n+1}_{q} fo
oΛ^{n}_{q},
wherefe andfo are, respectively, the even and the odd parts off, andΛ^{n}_{q}
is the function defined by Λ^{n}_{q}(x) =q^{−n}x.

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(aq^{n})q^{n}, (5)

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

0

f(x)dqx= (1−q)

∞

X

n=−∞

f(q^{n})q^{n}, (6)

Z ∞

−∞

f(x)dqx= (1−q)

∞

X

n=−∞

f(q^{n})q^{n}+ (1−q)

∞

X

n=−∞

f(−q^{n})q^{n}, (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(q^{n}t)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_{q}t= 2
Z ∞

0

f(t)d_{q}t.

(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)|^{p}d_{q}x
^{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)_{k}z^{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

z^{n}

n!_{q} (12)

and

Expq(z) :=Eq((1−q)z) =

∞

X

n=0

q^{n(n−1)}^{2} z^{n}
n!q

. (13)

It follows from (1) that

Exp_{q}2(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;q^{2}) :=

∞

X

n=0

(−1)^{n}b_{2n}(x;q^{2}) (16)
and

sin(x;q^{2}) :=

∞

X

n=0

(−1)^{n}b_{2n+1}(x;q^{2}), (17)
where

bn(x;q^{2}) =q[^{n}_{2}]([^{n}_{2}]^{+1})
n!q

x^{n} (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;q^{2}) := cos(−iz;q^{2}) +isin(−iz;q^{2}) =

∞

X

n=0

bn(z;q^{2}). (19)
e(z;q^{2}) is absolutely convergent for allzin the plane, and we have lim

q→1^{−}e(z;q^{2}) =e^{z}
point-wise and uniformly on compacta. Note that we have

Lemma 2.2. (see [8])

For all λ∈C, ∂_{q}e(λz;q^{2}) =λe(λz;q^{2}). (20)
For all x∈Rq, |e(ix;q^{2})| ≤ 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;q^{2})d_{q}t, x∈Req, (22)
where

K= (q;q^{2})_{∞}

2(q^{2};q^{2})_{∞}(1−q)^{1}^{2}. (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∈L^{1}_{q}, then
Z ∞

−∞

F_{q}(f)(x)g(x)d_{q}x=
Z ∞

−∞

f(x)F_{q}(g)(x)d_{q}x.

(2) If f(u), uf(u)∈L^{1}_{q}, then

∂_{q}Fq(f)(x) =Fq(−iuf(u))(x).

(3) If f, ∂qf ∈L^{1}_{q}, then

Fq(∂qf)(x) =ixFq(f)(x).

(4) Fq is an isomorphism ofL^{2}_{q},satisfying forf ∈L^{2}_{q}
kFq(f)kL^{2}_{q} =kfkL^{2}_{q},
and fort∈Rq,

f(t) =K Z ∞

−∞

Fq(f)(x)e(itx;q^{2})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;q^{2})f(x) =

∞

X

n=0

bn(y;q^{2})∂_{q}^{n}f(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(.;q^{2}) is a polynomial of degree n, then for all
k≥n+ 1, we have∂_{q}^{k}bn(x;q^{2}) = 0. Then

pn,q(x, y) :=n!qTy,qbn(x;q^{2}) (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}^{k}bn(x;q^{2}) =b_{n−k}(x;q^{2}), k= 0,1, ...n. (28)
T_{y,q}b_{n}(x;q^{2}) =

n

X

k=0

b_{k}(y;q^{2})b_{n−k}(x;q^{2}) = 1

n!_{q}p_{n,q}(x, y). (29)
Ty,qe(λx;q^{2}) =e(λx;q^{2})e(λy;q^{2}). (30)
The generating function of{pn,q(., .)}_{n∈N} is given by

e(λx;q^{2})e(λy;q^{2}) =

∞

X

n=0

p_{n,q}(x, y)
n!q

λ^{n}. (31)

Proof. (28) follows easily from the fact that∂_{q}b_{n}(x;q^{2}) =b_{n−1}(x;q^{2}).

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

Using (20), we have
Ty,qe(λx;q^{2}) =

∞

X

n=0

bn(y;q^{2})∂_{q}^{n}e(λx;q^{2})

=

∞

X

n=0

bn(y;q^{2})λ^{n}e(λx;q^{2})

= e(λx;q^{2})

∞

X

n=0

b_{n}(λy;q^{2}) =e(λx;q^{2})e(λy;q^{2}).

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!q^{2}

(1 +|xy|)Eq^{2}(|x|^{2})E_{q}2(|y|^{2}) (32)

and

p_{2n+1,q}(|x|,|y|)≤ (2n+ 1)!_{q}
n!q^{2}

(|x|+|y|)E_{q}2(|x|^{2})E_{q}2(|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|;q^{2})b_{2n−k}(|y|;q^{2}).

But,

2n

X

k=0

b_{k}(|x|;q^{2})b_{2n−k}(|y|;q^{2}) =

n

X

k=0

b_{2k}(|x|;q^{2})b_{2(n−k)}(|y|;q^{2})
+

n−1

X

k=0

b2k+1(|x|;q^{2})b_{2(n−k)−1}(|y|;q^{2})
and using the fact that (2k)!q ≥(k!_{q}2)^{2}, the following inequalities hold

b2k(|x|;q^{2})≤ q^{k(k−1)}|x|^{2k}
k!_{q}2

, b2k(|y|;q^{2})≤ E_{q}2(|y|^{2})
k!_{q}2

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

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

n

X

k=0

b_{2k}(|x|;q^{2})b_{2(n−k)}(|y|;q^{2}) ≤ E_{q}2(|y|^{2})
n!q^{2}

n

X

k=0

n!_{q}2q^{k(k−1)}
(n−k)!q^{2}k!q^{2}

|x|^{2k}

≤ E_{q}2(|y|^{2})(−|x|^{2};q^{2})n

n!q^{2}

≤ Eq^{2}(|x|^{2})Eq^{2}(|y|^{2})
n!_{q}2

.

(34)

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

k!_{q}2

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

b2k−1(|y|;q^{2})≤ |y|E_{q}2(|y|^{2})
k!_{q}2

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

Consequently,

n−1

X

k=0

b2k+1(|x|;q^{2})b_{2(n−k)−1}(|y|;q^{2}) ≤ |xy|E_{q}2(|y|^{2})
n!_{q}2

n

X

k=0

n!_{q}2q^{k(k−1)}
(n−k)!_{q}2k!_{q}2

|x|^{2k}

≤ |xy|E_{q}2(|x|^{2})E_{q}2(|y|^{2})
n!_{q}2

. 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|;q^{2})b_{2n+1−k}(|y|;q^{2}),

and

2n+1

X

k=0

b_{k}(|x|;q^{2})b_{2n+1−k}(|y|;q^{2}) =

n

X

k=0

b_{2k}(|x|;q^{2})b_{2(n−k)+1}(|y|;q^{2})
+

n

X

k=0

b2k+1(|x|;q^{2})b_{2(n−k)}(|y|;q^{2}).

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

2n+1

X

k=0

bk(|x|;q^{2})b_{2n+1−k}(|y|;q^{2})≤(|x|+|y|)E_{q}2(|x|^{2})E_{q}2(|y|^{2})
n!_{q}2

.

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

∃ M >0 :∀n∈N,

|∂_{q}^{2n}f(0)| ≤M n!_{q}2

σ^{n} ,

|∂_{q}^{2n+1}f(0)| ≤ M n!_{q}2

σ^{n} .

(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

∂_{q}^{n}f(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

anx^{n} =

∞

X

n=0

an

b_{n}(1;q^{2})bn(x;q^{2}).

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

∂_{q}^{k}f(x) =

∞

X

n=k

a_{n}

bn(1;q^{2})b_{n−k}(x;q^{2}), (37)
from which we deduce that

∀n≥0, a_{n} =∂_{q}^{n}f(0)b_{n}(1;q^{2}). (38)

By (26), (37) and (38), we have
T_{y,q}f(x) =

∞

X

k=0

b_{k}(y;q^{2})

∞

X

n=k

an

b_{n}(1;q^{2})b_{n−k}(x;q^{2})

=

∞

X

k=0

b_{k}(y;q^{2})

∞

X

n=k

∂_{q}^{n}f(0)b_{n−k}(x;q^{2})

=

∞

X

n=0

∂_{q}^{n}f(0)

n

X

k=0

b_{k}(y;q^{2})b_{n−k}(x;q^{2}).

(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}(.;q^{2}) 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,

∂_{q}^{2n}f(0)|

(2n)!_{q} p2n,q(x, y)

≤ M

σ^{n}(1 +|xy|)E_{q}2(|x|^{2})E_{q}2(|y|^{2})
and

∂_{q}^{2n+1}f(0)|

(2n+ 1)!_{q} p2n+1,q(x, y)

≤ M

σ^{n}(|x|+|y|)E_{q}2(|x|^{2})E_{q}2(|y|^{2}).

Finally, these relations, the continuity of the functionE_{q}2 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 =∂^{2}u

∂^{2}x, (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!

[^{n}_{2}]

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/2}Hn

x

√−4πt

.

The source solution or fundamental solution of (40) is given by
k(x, t) = e^{−}^{x}^{4t}^{2}

√4πt.

The associated functionswn(x, t) are defined by
wn(x, t) =v_{n}(x,−t)k(x, t)

t^{n} , 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/2}k(x, t)Hn( x

√4πt).

We have the following biorthogonality relation Z +∞

−∞

v_{n}(x,−t)w_{m}(x, t)dx= 2^{n}n!δ_{m,n},
whereδ_{m,n}is 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:

Dq^{2},tu=∂_{q,x}^{2} u. (42)

Remark. Taking into account that lim

q→1Dq^{2},tu(t, x) =∂u

∂t(t, x) and lim

q→1∂_{q,x}^{2} u(t, x) =

∂^{2}u

∂^{2}xu(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_{q}2

− qx^{2}
t(1 +q)^{2}

, t >0, (43)

whereexp_{q}2(.) is defined by (12) and
C(t;q) =

−^{q(1−q)}_{t(1+q)},−^{qt(1+q)}_{1−q} , q;q^{2}

∞

2(1−q)

−^{q}_{t(1+q)}^{2}^{(1−q)},−^{t(1+q)}_{1−q} , q^{2};q^{2}

∞

. (44)

Since

∂q,xk(x, t;q) =−C(t;q)x
qt(1 +q)expq^{2}

− x^{2}
tq(1 +q)^{2}

(45) and

∂_{q,x}^{2} k(x, t;q) =D_{q}2,tk(x, t;q) =− C(t;q)
qt(1 +q)

1− x^{2}
t(1 +q)

exp_{q}2

− x^{2}
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_{q}2(−x^{2}) has the rapid decreasing property:

x→∞lim pn(x)e_{q}2(−x^{2}) = 0, n= 0,1,2..., (46)
for all polynomialp_{n}(x) is a of degree n.

(2)

x→∞lim

e_{q}2(−qx^{2})

eq^{2}(−x^{2}) = +∞. (47)

(3) Forβ >0, we have

x→∞lim

eq^{2}(−qx^{2})eq^{2}(−βx^{2})

e_{q}2(−x^{2}) = 0. (48)

Proof. (1) Observe that forn= 0,1,2, ..., we have
e_{q}2(−x^{2}) =

∞

Y

k=0

(1 +q^{2k}x^{2})^{−1}≤(1 +q^{2n}x^{2})^{−n}=O(x^{−2n}) asx→ ±∞.

(2) Forn= 0,1,2, ..., we have
e_{q}2(−qx^{2})

e_{q}2(−x^{2}) ≥

n

Y

k=0

1 +q^{2k}x^{2}
1 +q^{2k+1}x^{2},
it follows that for alln∈N,

x→∞lim

eq^{2}(−qx^{2})
e_{q}2(−x^{2}) ≥q^{−n},
which yields to the result.

(3) Since the functione_{q}2(−x^{2}) is decreasing on [0,∞[, we obtain whenβ ≥1,
e_{q}2(−βx^{2})≤e_{q}2(−x^{2}), ∀x∈R

and then

x→∞lim

e_{q}2(−qx^{2})e_{q}2(−βx^{2})
eq^{2}(−x^{2}) ≤ lim

x→∞e_{q}2(−qx^{2}) = 0.

Ifβ <1, then there existsn∈N, such that β > q^{2n} and so
e_{q}2(−βx^{2})≤e_{q}2(−q^{2n}x^{2}) = (−x^{2};q^{2})ne_{q}2(−x^{2}).

Then, by using (46), we get

x→∞lim

e_{q}2(−qx^{2})e_{q}2(−βx^{2})
e_{q}2(−x^{2}) ≤ lim

x→∞(−x^{2};q^{2})ne_{q}2(−qx^{2}) = 0.

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

0

eq^{2}(−λy^{2})y^{2n+1}dqy=(1−q)q^{−n(n+1)}(q^{2};q^{2})_{n}

λ^{n+1} (49)

and

Z ∞ 0

e_{q}2(−λy^{2})y^{2n}d_{q}y=c_{q}(λ)q^{−n}^{2}(q;q^{2})_{n}

λ^{n} , (50)

where

c_{q}(λ) = (1−q)(−qλ,−q/λ, q^{2};q^{2})_{∞}
(−λ,−q^{2}/λ, q;q^{2})∞

. (51)

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

Z ∞ 0

e_{q}2(−λy^{2})y^{2n+1}dqy= (1−q)

∞

X

k=−∞

q^{(2n+2)k}
(−λq^{2k};q^{2})_{∞}.
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_{q}2(−λy^{2})y^{2n+1}dqy=(1−q) −λq^{2n+2},−q^{−2n}/λ, q^{2};q^{2}

∞

(−λ, q^{2n+2},−q^{2}/λ;q^{2})_{∞} .
We conclude (49) by using the following two identities

−λq^{2n+2};q^{2}

∞= −q^{2}λ;q^{2}

∞

(−q^{2}λ;q^{2})_{n}
and

−q^{−2n}/λ;q^{2}

∞=q^{−n(n+1)}λ^{−n} −q^{2}λ;q^{2}

n −λ^{−1};q^{2}

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

Z ∞ 0

e_{q}2(−λy^{2})y^{2n}d_{q}y = (1−q) −λq^{2n+1},−q^{−2n+1}/λ, q^{2};q^{2}

∞

(−λ, q^{2n+1},−q^{2}/λ;q^{2})_{∞} .
Then, (50) follows by using the two following facts that

−λq^{2n+1};q^{2}

∞= −qλ;q^{2}

∞

(−qλ;q^{2})_{n}
and

−q^{−2n+1}/λ;q^{2}

∞=q^{−n(n+1)}(q/λ)^{n} −qλ;q^{2}

n −q/λ;q^{2}

∞.

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

−∞

k(x, t;q)b2n(x;q^{2})dqx = t^{n}
n!_{q}2

(53) and

Z ∞

−∞

k(y, t;q)b2n+1(|y|;q^{2})dqy=2C(t;q)t^{n+1}(1 +q)^{2n+1}n!_{q}2

q^{n+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;q^{2})n

(q;q)_{2n} = 1
(q^{2};q^{2})_{n}.

Proposition 4.4.

F_{q}[k(., t;q)] (x) =Kexp_{q}2 −tx^{2}
.

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;q^{2})dqy

= K

∞

X

n=0

(−1)^{n}x^{2n}
Z ∞

−∞

k(y, t;q)b_{2n}(y;q^{2})d_{q}y,

= K

∞

X

n=0

(−1)^{n}x^{2n} t^{n}
n!_{q}2

=Kexp_{q}2 −tx^{2}
.

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;q^{2})_{∞}e_{q}2(−qλ^{2})

2(1−q)(q^{2};q^{2})_{∞}e_{q}2(−λ^{2})= +∞.

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

t→0^{+}k(x, t;q) = lim

λ→+∞

(q;q^{2})∞eq^{2}(−qλ^{2})eq^{2}(−λ^{2}x^{2})
2(1−q)(q^{2};q^{2})_{∞}e_{q}2(−λ^{2}) = 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_{q}2 tz^{2}

cos(−ixz;q^{2}) =

∞

X

p=0

t^{p}z^{2p}
p!q^{2}

! _{∞}
X

k=0

q^{k(k+1)}x^{2k}z^{2k}
(2k)!q

!

=

∞

X

n=0

z^{2n}

n

X

k=0

t^{n−k}q^{k(k+1)}x^{2k}
(n−k)!_{q}2(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^{−1}andt= 10^{−4}.

and

iexp_{q}2 tz^{2}

sin(−ixz;q^{2}) =

∞

X

p=0

t^{p}z^{2p}
p!_{q}2

! _{∞}
X

k=0

q^{k(k+1)}x^{2k+1}z^{2k+1}
(2k+ 1)!_{q}

!

=

∞

X

n=0

z^{2n+1}

n

X

k=0

t^{n−k}q^{k(k+1)}x^{2k+1}
(n−k)!_{q}2(2k+ 1)!q

.

(57)

Then, using the relation

e(xz;q^{2}) = cos(−ixz;q^{2}) +isin(−ixz;q^{2}),
we obtain

exp_{q}2 tz^{2}

e(xz;q^{2}) =

∞

X

n=0

vn(x, t;q)z^{n}
n!q

, (58)

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

[^{n}_{2}]

X

k=0

t^{k}
k!_{q}2

b_{n−2k}(x;q^{2}). (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_{q}2,tv_{n}(x, t;q) = [n]_{q}[n−1]_{q}v_{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) byExpq^{2}(−tz^{2}) and next comparing the
coefficient ofz^{n}, we obtain

bn(x;q^{2}) =

[^{n}_{2}]

X

k=0

(−t)^{k}q^{k(k−1)}v_{n−2k}(x, t;q)

k!_{q}2(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;q^{2})
Z ∞

−∞

k(y, t, q)bk(y;q^{2})dqy

= n!q

[^{n}2]
X

k=0

b_{n−2k}(x;q^{2})
Z ∞

−∞

k(y, t;q)b2k(y;q^{2})dqy

= n!q

[^{n}_{2}]
X

k=0

b_{n−2k}(x;q^{2}) t^{k}
k!_{q}2

=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}

[^{n}_{2}]

X

k=0

(−1)^{k}q^{k(k−1)}x^{n−2k}
(q^{2};q^{2})_{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}(iβq^{n}x;q) (64)
and

v2n+1(x, t;q) = q^{−n}^{2}

(iβ)^{2n+1}h2n+1(iβq^{n}x;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,x}k(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}

[^{n}_{2}]

X

k=0

(−1)^{k}q^{−2nk}q^{k(2k+1)}x^{n−2k}
(q^{2};q^{2})k(q;q)n−2k

and satisfy the following Rodrigues-type formula

ω(x;q)˜hn(x;q) = (q−1)^{n}q^{−}^{n(n−1)}^{2} D^{n}_{q}[ω(x;q)],
where

ω(x;q) = 1
(−x^{2};q^{2})_{∞}.

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,x}^{2n}k(x, t;q) =q^{n(n−2)}γ^{2n}
(q−1)^{2n}

˜h2n(γq^{−n}x;q)k(q^{−n}x, t;q)
and

∂_{q,x}^{2n+1}k(x, t;q) = q^{n(n−1)}γ^{2n+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) =q^{1}^{2}β(t, q) =
s

q(1−q)

t(1 +q). (68)

Proof. Using Proposition 2.1, we obtain

∂^{2n}_{q,x}k(x, t;q) =q^{−n(n+1)}D^{2n}_{q} [k(., t;q)]oΛ^{n}_{q}(x)
and

∂^{2n+1}_{q,x} k(x, t;q) =q^{−(n+1)}^{2}D_{q}^{2n+1}[k(., t;q)]oΛ^{n+1}_{q} (x).

But, the the fact that

k(x, t;q) =C(t;q)w(γx;q) gives

D^{n}_{q,x}[k(x, t;q)] =C(t;q)D_{q}^{n}[ω(γx;q)] =C(t;q)γ^{n}[D_{q}^{n}ω](γx;q).

So, from the Rodrigues-type formula, we get

D_{q,x}^{n} [k(x, t;q)] =γ^{n}(q−1)^{−n}q^{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,x}^{2n}k(x, t;q) =v2n(q^{1}^{2}x,−t;q^{−1})

t^{2n}(1 +q)^{2n} k(q^{−n}x, t;q)
and

∂_{q,x}^{2n+1}k(x, t;q) =−q^{−}^{1}^{2}v_{2n+1}(q^{−}^{1}^{2}x,−t;q^{−1})

t^{2n+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^{−n}h_{n}(ix;q^{−1}) n= 0, 1, 2, .... (69)
Then, forn= 0, 1, 2, ..., we have

˜h2n(γq^{−n}x;q) =i^{−2n}h2n(iγq^{−n}x;q^{−1}) =i^{−2n}h2n(iβq^{−n}q^{1}^{2}x;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^{−n}q^{−}^{1}^{2}x;q^{−1}).

Thus, since

β(t, q) =β(−t, q^{−1}), (70)

the first equality follows from (64) and the fact that
i^{−2n}q^{n(n−2)}γ^{2n}

(1−q)^{2n} = q^{n(n−1)}
(iβ)^{2n}t^{2n}(1 +q)^{2n},
and the second equality follows from (65) and the relation

i^{−(2n+1)}q^{n(n−1)}γ^{2n+1}

q(1−q)^{2n+1} = q^{n}^{2}^{−}^{1}^{2}

(iβ)^{2n+1}t^{2n+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(q^{1}^{2}x, t;q^{−1}),

˜

v_{2n+1}(x, t;q) = q^{−}^{1}^{2}v_{2n+1}(q^{−}^{1}^{2}x, 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}^{−1}v˜_{n−1}(x, t;q), n= 1, 2 , .... (72)
2) The generating function for ˜v_{n}(x, t;q)is given by

Expq^{2} tz^{2}

e(q^{−1}xz;q^{2}) =

∞

X

n=0

˜

vn(x, t;q) z^{n}
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) = q^{n(n−1)}
β^{2n}

˜h2n(γq^{−n}x;q), n= 0,1,2, ...

˜

v2n+1(x,−t;q) = q^{n}^{2}^{−}^{1}^{2}
β^{2n+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^{−2k}^{2}^{+k}(2k)!_{q} and (2k+ 1)!_{q}−1 =q^{−2k}^{2}^{−k}(2k+ 1)!_{q}. (75)
So, for allk= 0, 1, 2, ...,

b2k(q^{1}^{2}x;q^{−2}) =b2k(q^{−1}x;q^{2}), (76)
and

q^{−}^{1}^{2}b2k+1(q^{−}^{1}^{2}x;q^{−2}) =b2k+1(q^{−1}x;q^{2}), (77)
wherebn(x;q^{2}) is defined by (18).

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

∂_{q}b_{2k}(q^{1}^{2}x;q^{−2}) =q^{−1}b_{2k−1}(q^{−1}x;q^{2}) =q^{−}^{3}^{2}b_{2k−1}(q^{−}^{1}^{2}x;q^{−2})
and

q^{−}^{1}^{2}∂_{q}b_{2k+1}(q^{−}^{1}^{2}x;q^{−2}) =q^{−1}b_{2k}(q^{−1}x;q^{2}) =q^{−1}b_{2k}(q^{1}^{2}x;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^{−1}z;q^{2}) = cos(q^{1}^{2}z;q^{−2}) and sin(q^{−1}z;q^{2}) =q^{−}^{1}^{2}sin(q^{−}^{1}^{2}z;q^{−2}).

Then,

Exp_{q}2 tz^{2}

cos(−iq^{−1}xz;q^{2}) =exp_{q}^{−2} tz^{2}

cos(−iq^{1}^{2}xz;q^{−2})
and

Exp_{q}2 tz^{2}

sin(−iq^{−1}xz;q^{2}) =q^{−}^{1}^{2}exp_{q}^{−2} tz^{2}

sin(−iq^{−}^{1}^{2}xz;q^{−2}).

Finally, (73) follows by replacingx,qbyq^{1}^{2}x,q^{−1}in (56) and byq^{−}^{1}^{2}x,q^{−1}in (57).

3) Using (70), (64) and (65), we obtain
v_{2n}(q^{1}^{2}x,−t;q^{−1}) = q^{n(n−1)}

(iβ)^{2n} h_{2n}(iq^{1}^{2}βq^{−n}x;q^{−1}) =q^{n(n−1)}

(iβ)^{2n} h_{2n}(iγq^{−n}x;q^{−1})
and

v2n+1(q^{−}^{1}^{2}x,−t;q^{−1}) = q^{n}^{2}

(iβ)^{2n+1}h2n+1(iq^{−}^{1}^{2}βq^{−n}x;q^{−1})

= q^{n}^{2}

(iβ)^{2n+1}h_{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)

t^{2n} k(q^{−n}x, t;q)

= q^{n(n−1)}
(tβ)^{2n}

˜h_{2n}(γq^{−n}x;q)k(q^{−n}x, t;q),
w2n+1(x, t;q) = ˜v2n+1(x,−t;q)

t^{2n+1} k(q^{−(n+1)}x, t;q)

= q^{n}^{2}^{−}^{1}^{2}
(tβ)^{2n+1}

˜h2n+1(γq^{−(n+1)}x;q)k(q^{−n}x, 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(q^{3}^{2}x,−t;q^{−1}) = (2n)!_{q}^{−1}

n

X

k=0

(−t)^{n−k}q^{−k}^{2}^{+2k}x^{2k}
(n−k)!_{q}−2(2k)!_{q}−1

= q^{−n}^{2}(2n)!q
n

X

k=0

(−t)^{n−k}q^{2k}^{2}^{−2nk+2k}x^{2k}
(n−k)!_{q}2(2k)!_{q} .
The lefthand side of (79) is then equal to

C(t;q)q^{−n}^{2}(2n)!q
n

X

k=0

(−t)^{n−k}q^{2k}^{2}^{−2nk+2k}
(n−k)!_{q}2(2k)!_{q}

Z ∞

−∞

exp_{q}2

− qx^{2}
t(1 +q)^{2}

x^{2k}dqx.

But, takingλ=q(1−q)

t(1 +q) in (50), we obtain Z ∞

−∞

exp_{q}2

− qx^{2}
t(1 +q)^{2}

x^{2k}d_{q}x= 2c_{q}(λ)q^{−k}^{2}^{−k}(q;q^{2})_{k}(1 +q)^{k}t^{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^{−n}^{2}(2n)!q(−t)^{n}(1−q^{2})^{n}

n

X

k=0

(−1)^{k}q^{k}^{2}^{−2nk+k}
(q^{2};q^{2})_{n−k}(q^{2};q^{2})_{k},
which is equal to

q^{−n}^{2}(2n)!_{q}(−t)^{n}
n!q^{2}

n

X

k=0

n k

q^{2}

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

Z ∞

−∞

˜

v_{2n}(qx,−t;q)k(x, t;q)d_{q}x=q^{−n}^{2}(2n)!_{q}(−t)^{n}
n!q^{2}

(q^{−2n+2};q^{2})_{n}= 0.