The Ramanujan Journal

, Volume 28, Issue 3, pp 443–461 | Cite as

The q-cosine Fourier transform and the q-heat equation

Open Access


The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula.

The second theme of this paper is an application of the q-Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the q-analogue. We construct the q-solution source, the q-heat polynomials and solve the q-analytic Cauchy problem.


Basic orthogonal polynomials and functions Basic hypergeometric integrals 

Mathematics Subject Classification (2000)

33D45 33D6043 

1 Introduction

During the last years, an intensive work was founded about the so-called q-basic theory. Taking account of the well-known Ramanujan works shown at the beginning of this century by Jackson ([9, 10]), many authors such as Askey, Gasper, Ismail, Rogers, Andrew, Koornwinder, and others (see references) have recently developed this topic.

The present article is devoted to the study of the q-analogue of the Fourier transforms and to showing how it plays a central role in solving the q-heat equation associated to the second q-derivative operator. The method used here differs from those given by T.H. Koornwinder and R.F. Swarttouw, who discovered a q-analogue of Hankel’s Fourier–Bessel via some q-analogue orthogonality relations. We note that Ph. Feinsilver [4] gave a q-Harmonic Analysis for a q-Laplace transform with inversion formula.

Without entering into a dilemma through the analysis presented here, it seems that the point of view of T.H. Koornwinder and R.F. Swarttouw [12] is more suitable for harmonic analysis. We take as definition of the q-cosine the one given by the previous authors with a simple change and we prefer to write it as a series of functions denoted as bn(x;q2). This q-cosine appears as an eigenfunction of the operator Δq. Owing to a nice paper [12], we give a product formula written with the q-Jackson integral and we study the q-translation and the q-convolution. Next we define the q-analogue of the cosine Fourier transform with the purpose to find the transformation inverse. To this end, we prove the equivalent of the so-called Riemann–Lebesgue Lemma and discover that the Titchmarsh approach holds [15].

A motivation behind this work is to state some result about the q-heat equation associated to Δq operator. We attempt to extend the heat representation theory studied in many cases ([5, 7, 14], etc.). We define the q-heat polynomials and find that they are linked to the q-Hermite polynomials [13] and constitute with the q-associated functions a biorthogonal system. We conclude by solving the q-analytic Cauchy problem related to the q-heat equation.

2 Notations and preliminaries

We begin by recalling some q-elements of quantum analysis adapting the notation used in the book of Gasper and Rahman [6]. Let a and q be real numbers such that 0<q<1, the q-shift factorial is defined by
$$ (a;q)_{0}=1,\qquad (a;q)_{n}={\prod_{k=0}^{n-1}} \bigl(1-aq^{k} \bigr),\quad n=1,2,\ldots ,\infty. $$
A basic hypergeometric series is
$${}_{r}\varphi_{s}(a_{1},\ldots ,a_{r};b_{1},\ldots ,b_{s};q,z)=\sum _{k=0}^{\infty}\frac{(a_{1},\ldots ,a_{r};q)_{k}}{(b_{1},\ldots ,b_{s},q;q)_{k}} \bigl[ (-1)^{k}q^{\binom{k}{2}} \bigr]^{1+s-r}z^{k}. $$
A function f is q-regular at zero if limn→∞f(xqn)=f(0) exists and is independent of x.
The q-derivative Dqf of a function f is defined by
$$ D_{q}f(x)=\frac{f(x)-f(qx)}{(1-q)x},\quad x\neq0. $$
The q-derivative at zero is defined by
$$D_{q}f ( 0 ) =\lim_{n\rightarrow\infty}\frac{f ( xq^{n} ) -f ( 0 ) }{xq^{n}}, $$
if it exists and does not depend on x.
We introduce the set
$$\mathbb{R}_{q}= \bigl\{q^{k};k\in\mathbb{Z} \bigr\}.$$
The q-integral of Jackson is defined by The q-integration by parts is given for suitable functions f and g by
$$ {\int_{0}^{\infty}} f(x)D_{q}g(x)\,d_{q}x= \bigl[ f(x)g(x) \bigr]_{0}^{\infty}-{\int _{0}^{\infty}} f(x)D_{q}g \bigl(q^{-1}x \bigr)\,d_{q}x. $$
The q-analogue of the Gamma function is defined as
$$ \varGamma_{q}(x)=\frac{(q;q)_{\infty}}{(q^{x};q)_{\infty}}(1-q)^{1-x}, $$
which tends to Γ(x) when q tends to 1.

3 q-Trigonometric functions

We define the q-cosine as
$$ \cos \bigl(x;q^{2} \bigr)={_{1}\phi_{1}} \bigl(0;q;q^{2},(1-q)^{2}x^{2} \bigr)=\sum _{n=0}^{\infty }(-1)^{n}b_{n} \bigl(x;q^{2} \bigr), $$
where we have put
$$ b_{n} \bigl(x;q^{2} \bigr)=b_{n} \bigl(1;q^{2} \bigr)x^{2n}=q^{n(n-1)} \frac{(1-q)^{2n}}{(q;q)_{2n}}x^{2n}. $$
In the same way, the q-sine is given by
$$\sin \bigl(x;q^{2} \bigr)= ( 1-q ) x{_{1} \phi_{1}} \bigl(0;q^{3};q^{2},(1-q)^{2}x^{2} \bigr)=\sum_{n=0}^{\infty}(-1)^{n}c_{n} \bigl(x;q^{2} \bigr), $$
$$c_{n} \bigl(x;q^{2} \bigr)=c_{n} \bigl(1;q^{2} \bigr)x^{2n+1}=\frac{q^{n(n-1)}(1-q)^{2n+1}}{(q;q)_{2n+1}}x^{2n+1}. $$
These q-trigonometric functions differ and should not be confused with the functions cosq and sinq considered in [6, p. 23]; but coincide with the one given in [12] and [15] with a minor change of variable. Furthermore, we have

Proposition 3.1

The following statements hold:
  1. 1.
    $$b_{n} \bigl(0,q^{2} \bigr)=\delta_{n,0},\qquad \varDelta_{q}b_{n} \bigl(x;q^{2} \bigr)=b_{n-1}\bigl(x;q^{2} \bigr),\quad n\geq1; $$
  2. 2.
    $$\bigl \vert b_{n} \bigl(x;q^{2} \bigr) \bigr \vert \leq \frac{x^{2n}}{ ( 2n ) !}, $$
$$ \varDelta_{q}u ( x ) = \bigl( D_{q}^{2}u \bigr) \bigl( q^{-1}x \bigr) . $$


We only prove Part 2 since Part 1 is deduced from the definition of Δq.

The coefficients bn(1;q2), defined by (6), can be written as where we have put q=et, t>0.
Since the functions
$$f(t)=\frac{e^{-jt}-e^{-(j+1)t}}{1-e^{-(j+1)t}}\quad \mbox{and}\quad g(t)=\frac {e^{-jt}-e^{-(j+1)t}}{1-e^{-(2j+2)t}}, $$
decrease on ]0,∞[, we obtain
$$b_{n} \bigl(1;q^{2} \bigr)\leq{\frac{1}{{(2n)!}}}. $$
As a consequence of the previous proposition, we can show that for λ∈ℂ the function
$$\cos \bigl(\lambda x;q^{2} \bigr)=\sum_{0}^{\infty}(-1)^{n}b_{n} \bigl(x;q^{2} \bigr){\lambda}^{2n},$$
is the unique analytic solution of the q-differential equation
$$ \varDelta_{q}u(x)=-{\lambda}^{2}u(x), $$
$$ u(0,q)=1,\qquad ( D_{q}u ) ( 0 ) =0. $$

Proposition 3.2

Forx∈ℝqand\(\frac{\operatorname{Log}(1-q)}{\operatorname{Log}(q)}\in\mathbb{Z}\), we have
  1. 1.
    $$\bigl \vert \cos \bigl(x,q^{2} \bigr) \bigr \vert \leq \frac{1}{(q;q^{2})_{\infty}^{2}}; $$
  2. 2.
    $$\lim_{x\rightarrow\infty}\cos \bigl(x,q^{2} \bigr)=0; $$
  3. 3.
    $$\bigl \vert \sin \bigl(x,q^{2} \bigr) \bigr \vert \leq \frac{1}{(q;q^{2})_{\infty}^{2}}; $$
  4. 4.
    $$\lim _{x\rightarrow\infty}\sin \bigl(x,q^{2} \bigr)=0. $$


To prove Parts 1 and 2, we use the properties of 1ϕ1 given in [12] and their connection to the q-cosine. We obtain
$$ \bigl|\cos \bigl(q^{1+n};q^{2} \bigr)\bigr|\leq{\frac{1}{{(q;q^{2})_{\infty}^{2}}}}\left\{\begin{array}{l@{\quad }l} 1& \mbox{if}\ n\geq0,\\ q^{n^{2}} & \mbox{if}\ n\leq0. \end{array}\right.$$
hence Parts 1 and 2 follow. A similar argument shows Parts 3 and 4. □

Now we try to find a product formula for the q-cosine functions. We begin by proving the following result.

Proposition 3.3

For realsxandy, y≠0, we haveNote that this formula can be expressed in terms of1φ1as follows


To show (11) and (12), we begin by expanding the q-cosines in series absolutely and uniformly convergent on every compact of ℝ. From the product rule of series and the fact that
$$\frac{1}{(q;q)_{2n-2k}}=\frac{(q^{2n-2k+1},q)_{\infty}}{(q;q)_{\infty}}=0,\quad k>n, $$
we obtain for y≠0
$$\cos \bigl(x;q^{2} \bigr)\cos \bigl(y;q^{2} \bigr)=\sum _{k=0}^{\infty}{\frac{q^{2k^{2}}}{{(q;q)_{2k}}}} \biggl({ \frac{x}{y}} \biggr)^{2k}\sum_{n=0}^{\infty}(-1)^{n} \frac{q^{n^{2}-n}}{(q;q)_{2n-2k}}q^{-2nk}y^{2n}. $$
On the other hand, we have
$$\frac{1}{(q;q)_{2n-2k}}=\frac{q^{-k(2k-1)+2nk}}{(q;q)_{2n}}{ \sum _{s=-k}^{s=k}} ( -1 )^{k-s}\frac{q^{\binom{k-s}{2}}}{ ( q;q )_{k-s} ( q;q )_{k+s}}q^{2ns}.$$
We deduce (11) after the interchange of summation order. To prove (12), we write
$$\cos \bigl(x;q^{2} \bigr)\cos \bigl(y;q^{2} \bigr)=I+J, $$
$$I=\sum_{s=0}^{\infty}\cos \bigl(q^{s}y;q^{2} \bigr)\sum_{k\geq s}q^{k} \biggl( \frac{x}{y}\biggr)^{2k}\frac{(-1)^{k-s}q^{\frac{(k-s)(k-s-1)}{2}}}{(q;q)_{k+s}(q;q)_{k-s}},$$
$$J=\sum_{s=-\infty}^{-1}\cos \bigl(q^{s}y;q^{2} \bigr)\sum_{k\geq-s}q^{k} \biggl( \frac{x}{y}\biggr)^{2k}\frac{(-1)^{k-s}q^{\frac{(k-s)(k-s-1)}{2}}}{(q;q)_{k-s}(q;q)_{k+s}}.$$
In I, we make the change ks into k and use the equality
$$(q;q)_{k+2s}=(q;q)_{2s} \bigl(q^{1+2s};q \bigr)_{k}, $$
to obtain
$$I=\sum_{s=0}^{\infty}q^{s} \biggl({ \frac{x}{y}} \biggr)^{2s}\frac{(q^{2s+1};q)_{\infty}}{(q;q)_{\infty}}{_{1} \phi_{1}} \bigl(0;q^{1+2s};q,q \bigl(q^{2}/y^{2} \bigr) \bigr)\cos \bigl(q^{s}y;q^{2} \bigr). $$
Now we make the change k+s into k in J and use the equalities
$$(q;q)_{k-2s}=(q;q)_{-2s} \bigl(q^{1-2s};q \bigr)_{k},\quad -s\geq1, $$
$${\frac{(k-2s)(k-2s-1)}{2}}={\frac{(k-2)(k-3)}{2}}-2sk+2s^{2}-1, $$
and This identity is easily deduced from [11]. Then we obtain
$$J=\sum_{s=-\infty}^{-1}q^{s} \bigl(x^{2}/y^{2} \bigr)^{2s}\frac{(q^{1+2s};q)_{\infty}}{(q;q)_{\infty}}{_{1} \phi_{1}} \bigl(0;q^{1+2s};q,qx^{2}/y^{2} \bigr)\cos \bigl(q^{s}y;q^{2} \bigr). $$
We add these sums to find that (12) holds. □

Remark 3.4

(1) If we replace y by qy, x by qx, and assume the proposition the hypothesis, we obtain from (12) that the following integral representation holds (2) The product formula (11) leads to
$$ \cos \bigl(x;q^{2} \bigr)\cos \bigl(y;q^{2} \bigr)=\sum _{n=0}^{\infty}b_{n} \bigl(x;q^{2} \bigr) \varDelta_{q}^{n}\cos \bigl( y;q^{2} \bigr) . $$

4 q-Translation and q-convolution

We define, for x and y in ℝq, the measure
$$ d_{q}\mu_{(x,y)}=\sum_{s=-\infty}^{\infty} \mathcal{{D}} \bigl(x,y;q^{s} \bigr)q^{s}\delta_{yq^{s}}, $$
where δu denotes the unit mass supported at u, and
$$ \mathcal{{D}} \bigl(x,y;q^{s} \bigr)= \biggl({\frac{x}{y}} \biggr)^{2s}\frac{(q({\frac{x}{y}})^{2};q)_{\infty}}{(q;q)_{\infty}}{_{1}\phi_{1}} \biggl(0;q \biggl({\frac{x}{y}}\biggr)^{2};q,q^{1+2s} \biggr). $$

Proposition 4.1

(1) Forxandyinq, we have
$$d_{q}\mu_{ ( x,y ) }=d_{q}\mu_{ ( y,x ) }. $$
(2) dqμ(x,y)is of bounded variation.
$${\int} d_{q} \mu_{ ( x,y ) }(t)=1. $$


For n,m∈ℤ, the relation (2.3) from [12] leads to
$$\mathcal{{D}} \bigl(q^{n},q^{m};q^{s} \bigr)= \mathcal{{D}} \bigl(q^{m},q^{n};q^{s+m-n} \bigr). $$
We obtain Part 1 after the change sn+m by s.
To prove Part 2, we suppose \(|{\frac{x}{y}}|\leq1\); from the formulas (2.4) in [12] we have
$$ |d_{q}\mu_{(x,y)}|_{var}\leq \biggl( \frac{|y|^{2}+q|x|^{2}}{|y|^{2}-q|x|^{2}}\biggr)\frac{(q|{\frac{x}{y}}|^{2};-q,q)_{\infty}}{(q,q)_{\infty}}. $$
Finally, from (2.8) in [12], we can show that Part 3 is true. □

We introduce the q-translation which generalizes the even translation given by \({\frac{1}{2}}(\delta_{x+y}+\delta_{x-y})\).

Let f be a function with support in ℝq, the q-translation is defined for x and y in ℝq by
$$ T_{x,q}f(y)=\int_{0}^{\infty}f(t)\,d_{q} \mu_{(x,y)}(t). $$
From the previous proposition and the q-product formula (12), we have

Proposition 4.2

Letfbe a function with compact support inq. We have
  1. (i)
    $$T_{q,y}\cos \bigl( x;q^{2} \bigr) =\cos \bigl( x;q^{2} \bigr) \cos \bigl( y;q^{2} \bigr) . $$
  2. (ii)
  3. (iii)
  4. (iv)
    The functionu(x,y)=Tq,yf(x) is a solution of the problem
From the relation
$$\varDelta_{q}^{n}(f) ( x ) =\frac{q^{(2-n)n} ( q;q )_{2n}}{ ( 1-q )^{2n}}{\sum_{k=-n}^{n}} ( -1 )^{n-k} \frac{q^{\binom{n-k}{2}}}{ ( q;q )_{n-k} ( q;q )_{n+k}}f \bigl( q^{k}x \bigr) , $$
we can write the q-translation of a function f as
$$ T_{q,y}f ( x ) ={\sum _{n=0}^{\infty}} b_{n} \bigl(y,q^{2} \bigr)\varDelta_{q,x}^{n}f ( x ) , $$
and have in the limit when q tends to 1 the classical even translation cited before.
Now we denote by \(L_{q}^{1}(\mathbb{R}_{q})\) the space of functions f defined on ℝq such that
$$\Vert f\Vert _{1,q}=\int_{-\infty}^{\infty}\bigl|f(t)\bigr|\,d_{q}t< \infty. $$
Then we are able to define the q-convolution by
$$ f\star_{q}g(x)=\frac{(1+q^{-1})^{1/2}}{\varGamma_{q^{2}}(1/2)}\int_{0}^{\infty }T_{x,q}f(y)g(y)\,d_{q}y, $$
where f and g are two functions in \(L_{q}^{1}(\mathbb{R}_{q})\). We can show that this space is an algebra.

5 q-Analogue of Fourier-cosine

In this section, we suppose \(\frac{\operatorname{Log}(1-q)}{\operatorname{Log}(q)}\in\mathbb{Z}\). The q-analogue of Fourier transform is defined for λ∈ℝq by
$$ \mathcal{F}(f) (\lambda)={\frac{(1+q^{-1})^{1/2}}{\varGamma_{q^{2}}(1/2)}}\int_{0}^{\infty}f(t) \cos \bigl(\lambda t;q^{2} \bigr)\,d_{q}t, $$
where f is a function in \(L_{q}^{1}(\mathbb{R}_{q})\).

This definition is the same (after a minor change) as that given by T.H. Koornwinder and R.F. Swarttouw (see [12]).

Proposition 5.1

For\(f, g\in L_{q}^{1}(\mathbb{R}_{q})\), the following properties hold:
  1. (1)
    $$ \bigl \vert \mathcal{F}_{q} ( f ) ( \lambda ) \bigr \vert \leq \frac{1}{ [ q ( 1-q ) ]^{\frac{1}{2} } ( q;q )_{\infty}} \Vert f\Vert_{1,q},\quad \lambda\in\mathbb{R}_{q}; $$
  2. (2)
    $$ \mathcal{F}_{q} ( \mathcal{T}_{q,x}f ) ( \lambda ) =\cos \bigl(\lambda x;q^{2} \bigr)\mathcal{F}_{q} ( f ) ( \lambda ) ,\quad \lambda\in\mathbb{R}_{q};$$
  3. (3)
    $$\mathcal{F}_{q}(f\star_{q}g)=\mathcal{F}_{q}(f) \mathcal{F}_{q}(g). $$


Part 1. The inequality (21) follows from Proposition 3.2 and the identity
$$\bigl(q;q^{2} \bigr)_{\infty} \bigl(q^{2};q^{2} \bigr)_{\infty}=(q;q)_{\infty}. $$
Part 2 is a direct consequence of the q-product formula (12).

Part 3 is obtained after the exchange of the integration order and taking into account the invariability of the q-integral by the q-translation. □

Now we focus our attention on the inversion of the linear map \(\mathcal{F}_{q}\). We proceed by looking at the q-analogue of the Riemman–Lebesgue Lemma, the localization theorem, and we show that the Titchmarsh approach holds in the q-theory.

Proposition 5.2

Letfbe a function in\(L_{q}^{1} ( \mathbb{R}_{q} ) \), then
$$\lim_{\lambda\longrightarrow\infty}\mathcal{F}_{q} ( f ) (\lambda)=0,\quad \lambda\in\mathbb{R}_{q}. $$


To prove this, first we have from Proposition 3.2
$$\bigl \vert f(x)\cos \bigl(\lambda x;q^{2} \bigr) \bigr \vert \leq \frac{1}{(q;q^{2})_{\infty}^{2}} \bigl \vert f(x) \bigr \vert \in L_{q}^{1} ( \mathbb{R}_{q} ) ,\quad x,\lambda\in\mathbb{R}_{q}. $$
And for λ∈ℝq we have
$$\lim _{\lambda\rightarrow\infty}f(x)\cos \bigl(\lambda x;q^{2} \bigr)=0,\quad \lambda \in\mathbb{R}_{q}, $$
so the result is true. □

Proposition 5.3

We have the identity
$$\int_{0}^{\infty}\frac{\sin ( x;q^{2} ) }{x}\,d_{q}x= \frac {\varGamma_{q^{2}}^{2} ( \frac{1}{2} ) }{1+q^{-1}}. $$


This is a consequence of (2.8) in [12]. □

Proposition 5.4

Letf:(0,∞)→ℂ satisfy the conditions:
  1. (1)

    \(f\in L_{q}^{1} ( \mathbb{R}_{q} )\),

  2. (2)
    Fora∈ℝq, there existsC(a)>0 such that
    $$\bigl \vert f \bigl( aq^{k} \bigr) -f ( 0 ) \bigr \vert \leq C ( a ) q^{k},\quad k=0,1,2,\ldots . $$
$$\lim _{\lambda\rightarrow+\infty}\int_{0}^{\infty}f ( x ) \frac{\sin ( \lambda x;q^{2} ) }{x}\,d_{q}x=\frac{\varGamma_{q^{2}}^{2} ( \frac{1}{2} ) }{1+q^{-1}}f ( 0 ) . $$


Indeed, the first hypothesis shows that for an arbitrary ε>0 we have for large qN,N=0,1,… , that
$$\int_{q^{-N}}^{\infty} \biggl \vert \frac{f ( x ) }{x} \biggr \vert \,d_{q}x\leq\frac{\varepsilon}{2} \bigl(q;q^{2} \bigr)_{\infty}^{2}$$
and The second hypothesis and Proposition 3.2 show that
$$\biggl \vert \frac{f ( q^{k-N} ) -f ( 0 ) }{q^{k-N}}\sin \bigl( \lambda q^{k-N};q^{2} \bigr) \biggr \vert \leq\frac{C ( N ) }{q^{-N} ( q,q^{2} )_{\infty}^{2}}. $$
Since from Proposition 3.2 we have that sin(λx;q2) tends to zero as λ tends to ∞, the proposition is then a direct consequence. □

Theorem 5.5

(The q-cosine Fourier integral theorem)

If\(f\in L_{q}^{1} ( \mathbb{R}_{q} ) \)is such that fora∈ℝqthere exist positive constantsC(a) such that
$$ \bigl|T_{x,q}f \bigl(aq^{k} \bigr)-f \bigl(q^{k} \bigr)\bigr| \leq C(a)q^{k},\quad k=0,1,\ldots, $$

6 q-Heat equation and q-heat polynomials

In this section, the two q-analogues of the elementary exponential functions are crucial and they are defined by and
$$ e \bigl(x;q^{2} \bigr)=\frac{1}{((1-q^{2})x;q^{2})_{\infty}}=\sum _{0}^{\infty}\frac{(1-q^{2})^{n}}{(q^{2};q^{2})_{n}}x^{n}, \quad |x|<\frac{1}{1-q^{2}}. $$
These functions satisfy the identity
$$e \bigl(x;q^{2} \bigr)E \bigl(-x;q^{2} \bigr)=1, $$
and have as limit, when q tends to 1, the classical exponential function.
Now we purpose to give the q-analogue of the heat equation associated to the second derivative operator (even in x)
$$ \frac{\delta^{2}u}{\delta x^{2}}=\frac{\delta u}{\delta t},\quad x\in \mathbb{R},\ t>0. $$
We consider as q-heat equation associated to the second q-derivative operator the partial q-difference equation
$$ (\varDelta_{q,x}u) (x,t)=(D_{q^{2},t}u) (x,t). $$
We take as the initial condition
$$ u(x,0)=f(x),\quad f\in L_{q}^{1} ( \mathbb{R}_{q} ) . $$

6.1 q-Solution source

To find the solution source related to the q-heat equation, we apply the Fourier method with the adapted q-Fourier cosine studied before.

$$U(\lambda,t)=\mathcal{{F}} \bigl(u(x,t) \bigr) (\lambda), $$
Eq. (28) becomes
$$D_{q^{2},t}U(\lambda,qt)=-{\lambda}^{2}U(\lambda,t), $$
and, taking into account conditions (29), we obtain
$$U(\lambda,t)=\mathcal{{F}}(f) (\lambda)e \bigl(-{\lambda}^{2}t;q^{2} \bigr). $$
The problem consists in finding the function which has e(−λ2t;q2) as its q-Fourier-cosine transform. For this end, we need the following lemma.

Lemma 6.1

Forn=0,1,2,… andt>0, we have


From (26) we find
$$\int _{0}^{\infty}e \biggl(-\frac{\lambda^{2}}{qt(1+q)^{2}},q^{2} \biggr)\lambda^{2n}\,d_{q}\lambda=(1-q)\sum_{-\infty}^{\infty}\frac{q^{ ( 2n+1 ) k}}{ ( -\frac{1-q}{1+q}\frac{q^{2k}}{qt},q^{2} )_{\infty}}\mathtt{.}$$
Secondly, the use of the well-known Ramanujan [8] identity
$$\sum _{-\infty}^{\infty} \frac{z^{k}}{ ( bq^{k},q )_{\infty }}=\frac{ ( bz,q/bz,q,q )_{\infty}}{ ( b,z,q/b,q )_{\infty}},\quad b\neq0, $$
leads to the result after minor computation. □

Proposition 6.2

$$\frac{ ( 1+q^{-1} )^{1/2}}{\varGamma_{q^{2}} ( 1/2 ) }{\int_{0}^{\infty}} e \biggl(-\frac{\lambda^{2}}{qt(1+q)^{2}},q^{2} \biggr)\cos \bigl( \lambda x,q^{2} \bigr)\,d_{q}\lambda=A \bigl( t,q^{2} \bigr) e \bigl( -tx^{2},q^{2} \bigr), $$
$$ A \bigl( t,q^{2} \bigr) = \bigl[ (1-q)q^{-1} \bigr]^{1/2}\frac{ ( -\frac{1+q}{1-q}q^{2}t,-\frac{1-q}{1+q}\frac{1}{t},q^{2} )_{\infty}}{ ( -\frac{1-q}{1+q}\frac{1}{qt},-\frac{1+q}{1-q}q^{3}t;q^{2} )_{\infty}}. $$
As an immediate consequence we are now able to define the q-source solution associated to the q-heat equation (28) by
$$ G \bigl(x,t,q^{2} \bigr)= \bigl(A \bigl( t,q^{2} \bigr) \bigr)^{-1}e \biggl(-\frac{x^{2}}{qt(1+q)^{2}};q^{2} \biggr). $$
In the same manner as in the classical heat equation theory, we put
$$ G \bigl(x,y,t;q^{2} \bigr)=T_{y,q}G \bigl(x,t;q^{2} \bigr), $$
with Ty,q being the q-translation studied in Sect. 4.
Through this approach we show that the solution of the q-Cauchy problem (28) and (29) can been written in the form of
$$ u(x,t)= \bigl(G \bigl(\cdot,t;q^{2} \bigr)\star_{q}f \bigr) (x)=\int_{0}^{\infty}G \bigl(x,y,t;q^{2} \bigr)f(y)\,d_{q}y. $$
It is natural to ask how other properties such as the positivity of G(x,t;q2) and the existence of the q-semigroup can be established.

6.2 q-Heat polynomials

Proposition 6.3

It is easy to see that, forx∈ℝ andt>0, the analytic function
$$\lambda\rightarrow e{ \bigl(-{\lambda}^{2}t;q^{2} \bigr)\cos \bigl(\lambda x;q^{2} \bigr)}, $$
is a solution of (28) and it has the expansion
$$e \bigl( -\lambda^{2}t,q^{2} \bigr) \cos \bigl(\lambda x,q^{2} \bigr)=\sum _{n=0}^{\infty} ( -1 )^{n}v_{2n}(x,t,q)\lambda^{2n},$$
$$ v_{2n}(x,t,q)=\sum _{k=0}^{n}b_{k} \bigl( x,q^{2} \bigr) \frac {(1-q^{2})^{n-k}}{(q^{2};q^{2})_{n-k}}t^{n-k}, $$
with the functionsbnbeing given by (6).
From Proposition 3.1 we deduce immediately the following properties: We note that formula (34) can be inverted:
$$ b_{n} \bigl(x;q^{2} \bigr)=\sum _{k=0}^{n}(-1)^{n-k}v_{2k}(x,t;q)q^{ ( n-k ) ( n-k-1 ) } \frac{(1-q^{2})^{n-k}}{(q^{2};q^{2})_{n-k}}t^{n-k}. $$

Proposition 6.4

Theq-heat polynomials (34) possess theq-integral representation
  1. (1)
    $$ v_{2n}(x,t;q)={\int _{0}^{\infty}} G \bigl(x,y,t,q^{2} \bigr)b_{n} \bigl(y;q^{2} \bigr)\,d_{q}y. $$
  2. (2)
    $$ b_{n} \bigl(x;q^{2} \bigr)={\int_{0}^{\infty}} G \bigl(x,y,t,q^{2} \bigr)v_{2n} \bigl(q^{-1/2}y,t;q^{-1} \bigr)\,d_{q}y. $$
In [14], the authors defined the so-called associated functions by the Appell transform. We extend this notion by defining for t>0 the q-associated functions of v2n by
$$ w_{2n}(x,t;q)= ( -1 )^{n} \varDelta_{q,y}^{n}G \bigl(x,y,t;q^{2}\bigr) \bigl\vert_{y=0}. $$
It is easy to see that
$$ w_{2n}(x,t;q)=\frac{ ( 1+q^{-1} )^{1/2}}{\varGamma_{q^{2}} ( 1/2 ) }{ \int_{0}^{\infty}} e \bigl(-t \lambda^{2},q^{2} \bigr)\lambda^{2n}\cos \bigl( \lambda x,q^{2} \bigr)\,d_{q}\lambda. $$

Proposition 6.5


Fort>0 andn,m∈ℕ, we have
$${\int_{0}^{\infty}} w_{2m}(x,t;q)v_{2n} \bigl(q^{1/2}x,-t;q \bigr)\,d_{q}x= ( -1 )^{m}\delta_{n,m}. $$


By (37), we have
$$\varDelta_{q}^{m}b_{n} \bigl(x;q^{2} \bigr)={\int_{0}^{\infty}} \varDelta_{q}^{m}G \bigl(x,y,t,q^{2} \bigr)v_{2n} \bigl(q^{-1/2}y,t;q^{-1} \bigr)\,d_{q}y. $$
Putting x=0, we obtain
$${\int_{0}^{\infty}} w_{2m}(y,t;q)v_{2n} \bigl(q^{-1/2}y,t;q^{-1} \bigr)\,d_{q}y= ( -1 )^{m}\delta_{n,m}. $$

6.3 Convergence of ∑n≥0αnv2n(x,t;q)

Now we establish the following estimates that will be needed later

Lemma 6.6

Forn=0,1,… and\(0< \frac{x_{0}^{2}}{t_{0}} <+\infty\), we have
$$\bigl| v_{2n}(x_{0},t_{0},q)\bigr| \geq \frac{(1-q^{2})^{n}}{(q^{2};q^{2})_{n}}\vert t_{0}\vert ^{n}\geq \frac{\vert t_{0}\vert ^{n}}{n!}. $$


Indeed, the first inequality is a consequence of b0(1;q2)=1 and the hypothesis, and the second follows from
$$\frac{1}{n!}\leq\frac{(1-q^{2})^{n}}{(q^{2};q^{2})_{n}}. $$

Corollary 6.7

Forn=0,1,… and\(0 < \frac{x_{0}^{2}}{t_{0}}< +\infty\), we have
$$\big| v_{2n}(x_{0},t_{0},q)\big| \geq Cn^{-\frac{1}{2}} \biggl( \frac{\vert t_{0}\vert e}{n} \biggr)^{n}, $$
where C is a constant depending onx0andt0.

Lemma 6.8

Forn=0,1,…,δ>0, and\(\vert \frac{x^{2}}{\delta ( 1+q ) }\vert <1\), we have
$$ \frac{(1-q^{2})^{n}}{(q^{2};q^{2})_{n}}\bigl| v_{2n} \bigl(\vert x\vert ,\vert t\vert ,q \bigr)\bigr| \leq q^{-n ( n-1 ) }\frac{ ( \delta+\vert t\vert )^{n}}{n!}e \biggl( \frac{x^{2}}{\delta ( 1+q ) };q \biggr) . $$


To show (40), we note that
$$(q;q)_{2k}= \bigl(q,q^{2};q^{2} \bigr)_{k},$$
$$\bigl(q;q^{2} \bigr)_{k}\geq(q;q)_{k}. $$
For δ>0, and by using the fact that
$$\frac{(1-q)^{k}}{(q;q)_{k}}\frac{|x|^{2}}{(\delta(1+q))^{k}}\leq q^{-\binom {k}{2}}\exp \biggl( \frac{\vert x\vert ^{2}}{\delta(1+q)} \biggr), $$
we obtain The inequalities
$$\biggl(-\frac{|t|}{\delta};q^{2} \biggr)_{n}\leq \biggl( \frac{|t|}{\delta}+1 \biggr)^{n}, $$
$$q^{\binom{n}{2}}n!\leq\frac{ ( q;q )_{n}}{ ( 1-q )^{n}}\leq n! , $$
give the result. □

By the Stirling formula, we obtain

Corollary 6.9

Forn=0,1,…,δ>0, and\(\vert \frac{x^{2}}{\delta ( 1+q ) }\vert <1\), we have
$$ v_{2n} \bigl(\vert x\vert ,\vert t\vert ,q \bigr)\leq K q^{-n ( n-1 ) } \biggl( \bigl( \delta+\vert t\vert \bigr) \frac{n}{e} \biggr)^{n}, $$
whereKis a constant dependingδ.

Theorem 6.10

Let (αn) be a sequence of real or complex numbers such that
$$\overline{\lim_{n\rightarrow\infty}} {\frac{n}{e}}q^{-2(n-1)}| \alpha_{n}|^{1/n}={\frac{1}{\sigma}}<+\infty. $$
Then the series
$$\sum_{n\geq0}\alpha_{n}v_{2n}(x,t;q), $$
converges in the strip
$$ S_{\sigma}= \bigl\{(x,t),x\in\mathbb{R},\ |t|<\sigma \bigr\}, $$
and converges uniformly in any region of this strip.

To prove the theorem, we adopt the same approach as in [14] by taking account of the q-equivalent estimation (41).


If we write u(x,t) as the sum of the previous series, then this function satisfies the q-heat equation (28) and
$$u(x,0)=\sum_{n=0}^{\infty}\alpha_{n}b_{n} \bigl(x;q^{2} \bigr), $$
where the bn(x;q2) is given by (6).

6.4 Analytic Cauchy problem related to the q-heat equation

Lemma 6.11

Under the hypothesis of Theorem 6.10 and putting
$$ u ( x,t ) =\sum_{n\geq0}\alpha_{n}v_{2n}(x,t;q), $$
u(x;t) is an analytic function of two variables x and t in the stripSσgiven by (42) and satisfies theq-heat equation (28). Furthermore, the coefficientsαnare given by
$$ \alpha_{n}= \varDelta_{q}^{n}u ( x,t ) \bigl\vert_{ ( x,t ) = ( 0,0 ) }. $$


To show this, we note that the theorem gives that u(x,t) is analytic in the whole strip Sσ. Now for a fixed integer p the series
$$\sum_{n\geq0}\alpha_{n+p}v_{2n}(x,t;q) $$
converges uniformly in any compact region of Sσ. To prove (44), it suffices to see that for integers n and p we have
$$\bigl(\varDelta_{q,x}^{n}v_{2p}(x,t;q)\bigr)\bigl|_{(0,0)}= \delta_{n,p}, $$
where δn,p is the Kronecker symbol. □

Finally the following statement is established.

Theorem 6.12

Under the hypothesis of Lemma 6.11, the functionu(x,t) given by (43) has theq-Maclaurin expansion
$$u(x,t)=\sum_{m,p\geq0}\beta_{m,p} \frac{(1-q^{2})^{m}}{(q^{2};q^{2})_{m}}x^{2p}t^{m}, $$
$$ \beta_{m,p}=\alpha_{m+p}b_{p} \bigl( 1,q^{2} \bigr) . $$
If forx∈ℝ and |t|<σthen function
$$u(x,t)=\sum_{m,p}\beta_{m,p} \frac{(1-q^{2})^{m}}{(q^{2};q^{2})_{m}}x^{2p}t^{m}, $$
satisfies theq-heat equation (28) with the coefficientsβm,pgiven by (44), thenu(x,t) can be extended to an analytic function in the stripSσand we have
$$u(x,t)=\sum_{n\geq0}\alpha_{n}v_{2n}(x,t;q). $$


Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Regional Conference Series in Math., vol. 66. Amer. Math. Soc., Providence (1986) Google Scholar
  2. 2.
    Askey, R., Ismail, M.E.H.: A Generalization of Ultraspherical Polynomials. In: Erdos, P. (ed.) Studies in Pure Mathematics. Birkhäuser, Basel (1983) Google Scholar
  3. 3.
    Baley, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935). Reprinted by Hafner Publishing Company (1972) Google Scholar
  4. 4.
    Feinsilver, Ph.: Elements of q-harmonic analysis. J. Math. Anal. Appl. 141, 509–526 (1989) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fitouhi, A.: Heat “polynomials” for a singular differential operator on (0,∞). Constr. Approx. 5, 241–270 (1989) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  7. 7.
    Haimo, D.T.: Expansion of generalized heat polynomials and their appell transform. J. Math. Mech. 15, 735–758 (1966) MathSciNetMATHGoogle Scholar
  8. 8.
    Ismail, M.E.H.: A simple proof of Ramanujan’s 1 ψ 1 sum. Proc. Am. Math. Soc. 63, 185–186 (1977) MATHGoogle Scholar
  9. 9.
    Jackson, F.H.: On q-Functions and a Certain Difference Operator. Transactions of the Royal Society of London, vol. 46, pp. 253–281 (1908) Google Scholar
  10. 10.
    Jackson, F.H.: On a q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910) MATHGoogle Scholar
  11. 11.
    Koornwinder, T.H.: q-Special functions, a tutorial, Mathematical preprint series, Report 94-08, Univer. Amsterdam, The Netherlands Google Scholar
  12. 12.
    Koornwinder, T.H., Swarttouw, R.F.: On q-analogues of the Hankel and Fourier transform. Trans. Am. Math. Soc. 333, 445–461 (1992) MathSciNetMATHGoogle Scholar
  13. 13.
    Moak, D.S.: The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81, 21–47 (1981) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rosenbloom, P.C., Widder, D.V.: Expansions in terms of heat polynomials and associated functions. Trans. Amer. Math. Soc. 92, 220–266 (1959) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Swarttouw, R.F.: The Hahn–Exton q-Bessel function, Thesis Google Scholar
  16. 16.
    Titchmarsh, E.C.: Introduction to The Theory of Fourier Integrals, 2nd edn. Oxford University Press, Oxford (1937) Google Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Sciences of Tunis University El-ManarTunisTunisia
  2. 2.Department of MathematicsCollege of Sciences, King Saud UniversityRiyadhSaudi Arabia

Personalised recommendations