Considering the most general divided difference derivative [1][2],

admitting the property that if f ( t ) = P n ( t ( s ) ) is a polynomial of degree n in t ( s ) , then D f ( t ( s ) ) = P ˜ n − 1 ( t ( s ) ) is a polynomial in t ( s ) of degree n − 1 , one is led to the following most important canonical forms for t ( s ) in order of increasing complexity:

When the function t ( s ) is given by (2)-(4), the divided difference derivative (1) leads to the ordinary differential derivative D f ( x ) = d d x f ( x ) , finite difference derivative Δ f ( t ) = f ( t + 1 ) − f ( t ) = ( e d d t − 1 ) f ( t ) and q-difference derivative (or Jackson derivative [3]) D q f ( t ) = f ( q t ) − f ( t ) q t − t = q d d t − 1 q t − t f ( t ) , respectively.

When t ( s ) is given by (5), the corresponding derivative gives

It is usually referred to as the Askey-Wilson first order divided difference operator [4] that one can write:

where x ( z ) = z + z − 1 2 , having in mind that z = q s .

The Askey-Wilson polynomials [4], P n ( x ( z ) ) , x ( z ) = z + z − 1 2 are defined by:

where the basic hypergeometric (or q-hypergeometric)series r ϕ s read:

with

while

They satisfy the following second order q-difference equation:

where

is the Askey-Wilson second order q-difference operator and

The analysis related to (2), (3), (4) and (5) are referred to as the differential, difference, q-difference and q-nonuniform difference analysis respectively.

Laplace transform of an exponential type function f ( x ) is given by

and plays a major role in pure and applied analysis, especially in solving differential equations. If we consider f ( x ) as a function of a discrete variable i.e. t ∈ ℤ , then the transformation (9) reads

It is referred to as Z transform and plays similar role in difference analysis as Laplace transform in continuous analysis, especially in solving difference equations [5].

For a given function f ( x ) on the lattice (4), we define its q-Laplace transform as the function [6]

where the intégral is the Jackson intégral [3]

and

It plays similar role in q-difference analysis as Laplace transform in continuous analysis or Z transform in difference analysis, especially in solving q-difference equations [6].

In this work, we introduce q-nonuniform Laplace transform defined for functions f ( x ( z ) ) given on the lattice (5), as

where the intégral is defined by [7][8]

and

It is expected to play similar role in q-nonuniform difference analysis as Laplace transform, Z transform and q-Laplace transform in differential, difference and q-difference analysis respectively, especially in solving q-nonuniform difference equations [9].

In the next section, we begin by introducing the concepts of q-nonuniform exponential and trigonometric functions as solutions of first or second order q-nonuniform difference equations.

Before introducing the concept of q-non nuniform Laplace Transform, we clearly first study these of q-non exponential and trigonometric functions.

Consider first the following equation

Developping it

we get

where

By the recursion

we get

or

Define the q-nonuniform exponential function

Clearly, we have the relation

This means that E q − 1 2 ( a ; z 0 ; z ) is a solution of (20) and consequently, that of (17). Moreover,

From the definition (23), we can evaluate

and

Also, one easily verifies that E q − 1 2 ( a ; z 0 ; ∞ ) is a solution of (25) for z = ∞ .

If we set z = z 0 = 1 in (26) and (27), we get

On the other side, since 0 < q < 1 , then, for a ( x ( z ) ) = a = c o n s t , z 0 , z ≠ 0 , ∞ , one can verify that

Indeed,

with

Hence

where ( u ; q ) ∞ = def ∏ i = 0 ∞ ( 1 − u q i ) . Similarly, one proves that E q − 1 2 ( a ; z 0 ; ∞ ) = 0 .

Consider now the following equation

Developping, we get the relation

where

Using the recursion

we get

or

Define the other q − nonuniform exponential function

Here also, the function E q 1 2 ( a ˜ ; z 0 ; z ) admet representations and properties similar to that of E q − 1 2 ( a ; z 0 ; z ) . In particular, E q 1 2 ( a ˜ ; z 0 ; z ) verifies the relation

This means that E q 1 2 ( a ˜ ; z 0 ; z ) is a solution of (35) and consequently, that of (32). Moreover,

Similarly, we evaluate

and

Also, one easily verifies that E q 1 2 ( a ˜ ; z 0 ; ∞ ) is a solution of (40) for z = ∞ .

Also

and

Consider now the product h ( z ) = E q − 1 2 ( a ; z 0 ; z ) ⋅ E q 1 2 ( a ˜ ; z 0 ; z ) . Define K a , a ˜ by

Simple computations give

where z 1 , z 2 and z ˜ 1 , z ˜ 2 are finite roots of p ( q 1 2 z ) = 0 and p ˜ ( q 1 2 z ) = 0 , respectively, knowing that p ˜ ( x ( z ) ) is obtained from p ( x ( z ) ) by replacing a by − a ˜ .

From (19), (23), (34) and (38), we get the following

Theorem 2.1

More generally, we have the following

Theorem 2.2If y ( x ( z ) ) and y ˜ ( x ( z ) ) are respective solutions of theequations

satifying the conditions

then

Proof.

This means that y y ˜ = c o n s t , which, by (49) gives (50).

Consider next the following definitions

1)

2)

3)

We get

where D − f ( x ( z ) ) = D f ( x ( z ) ) | z : = z q − 1 2 . Hence

Similarly

and consequently

On the other side, we have

where D + f ( x ( z ) ) = D f ( x ( z ) ) | z : = z q 1 2 . Hence

Similarly

and consequently