Improved q-exponential and q-trigonometric functions

The quantum calculus (q-calculus) is an old, classical branch of mathematics, which can be traced back to Euler and Gauss [11,21] with important contributions of Jackson a century ago [18,19]. In recent years there are many new developments and applications of the q-calculus in mathematical physics, especially concerning special functions [1,8,12,14] and quantum mechanics [4,5,25,13,10,23,29]. Many papers were devoted to various approaches to q-deformations of elementary functions, including exponential and trigonometric functions [2,3,7,15,22,24,26,27,28].

In this paper we propose new definitions of the q-exponential function and q-trigonometric functions. These results are motivated by recent developments in the time scales calculus, where new exponential, hyperbolic and trigonometric function have been defined [9]. The concept of time scales unifies difference and differential calculus [16]. The q-calculus can be considered as a calculus on a special time scale (see, e.g., [6]).

The functions presented in this paper have better qualitative properties than standard q-exponential and q-trigonometric functions. In order to discuss and compare these properties we begin with a short summary of the classical results, usually following the textbook [20].

In the standard approach to the q-calculus two exponential function are used:

where q is positive, z is complex, and

(2) Hence we immediately get E z q = e z 1/q . Another, more popular, form of E z q is obtained using the identity

Both exponential functions can be represented by infinite products,

From this form we easily see that e z q E -z q = 1. Moreover,

where D q (q-derivative or Jackson’s derivative) is defined by

The existence of two representations of q-exponential functions (infinite series and infinite product) is related to well known formulae for the usual exponential function (q = 1),

Two exponential functions of the quantum calculus generate two pairs of the q-trigonometric functions. Using notation of [20] we have:

Taking into account properties of q-exponential functions (see above) we easily derive properties of standard q-trigonometric functions: cos q x Cos q x + sin q x Sin q x = 1 ,

D q sin q x = cos q x , D q cos q x = -sin q x , D q Sin q x = Cos q (qx) , D q Cos q x = -Sin q (qx) .

Note that the corresponding tangents coincide: Tan q x = tan q x.

2 Improved q-exponential function New q-exponential function E z q is defined as

where e z q , E z q are standard q-exponential functions. This definition is motivated by the classical Cayley transformation

see, e.g., [9,17]. Indeed,

Theorem 1. The q-exponential function E z q is analytic in the disc |z| < R q and

for |z| < R q , where

{n} := 1 + q + . . . + q n-1

and, finally, {n}! = {1}{2} . . . {n}.

Proof: In the disc |z| < 1 both series (1) are absolutely convergent for any q ∈ R + .

Multiplying them we get

Using Gauss’s binomial formula (see, e.g., [20], formula (5.5))

n-1 j=0

we have, as a particular case,

Substituting ( 19) into ( 17) we get the formula ( 14) with {n} defined by (16). In order to obtain the radius of convergence, we compute

for q < 1 (q-1)|z| 2q

for q > 1 (20)

Then, using d’Alembert’s test, we get (for q = 1) the radius of convergence (15). Note that R 1/q = R q . In the case q = 1 all q-exponential functions coincide with e z , hence R 1 = ∞. ✷ Theorem 2. The q-exponential function E z q has the following properties:

where z ∈ C, x ∈ R and we use the notation f (z

Proof: The first equation of ( 21) is a straightforward consequence of the definition (11). Then, E z q = E z q . Hence,

The symbol {n} depends on q. In this proof it is convenient to use more precise notation {n} ≡ {n} q , {n}! ≡ {n} q !. The equation E z q = E z 1/q follows immediately from the obvious identity

Finally,

which implies the second equation of (22). ✷

The properties ( 21) are identical with analogical properties of the exponential function e z . We point out that neither e z q nor E z q satisfies (21). Instead, we have E -z q e z q = 1.

3 Improved q-trigonometric functions New q-sine and q-cosine functions are defined in a natural way:

Theorem 3. q-Trigonometric functions defined by (27) satisfy:

Proof: Properties (28) follow directly from ( 21), ( 22) (note that E ix q E -ix q = 1). ✷ Corollary 4. q-Trigonometric functions Cos q x, Sin q x are real for x ∈ R. Moreover, for any x ∈ R, we have

Theorem 5. New q-trigonometric functions can be expressed by standard q-trigonometric functions as follows:

Cos q 2x = cos q x Cos q x -sin q x Sin q x = 1 -tan 2 q x 1 + tan 2 q x , Sin q 2x = sin q x Cos q x + cos q x Sin q x = 2 tan q x 1 + tan 2 q x .

(30)

Proof: First, we compute cos q x Cos q x -sin q x Sin q x = e ix q E ix q + e -ix q E -ix q 2 = Cos q 2x , sin q x Cos q x + cos q x Sin q x = e ix q E ix qe -ix q E -ix

Then, using (9), we get Cos q 2x = cos q x Cos q x -sin q x Sin q x cos q x Cos q x + sin q x Sin q x = 1 -tan q x Tan q x 1 -tan q x Tan q

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