On the complex q-Appell polynomials

The purpose of this article is to generalize the ring of q-Appell polynomials to the complex case. The formulas for q-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex q-Appell polynomials are also q-complex analytic functions, we are able to give a first example of the q-Cauchy–Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define q-complex Bernoulli and Euler polynomials. Previously, in order to obtain the q-Appell polynomial, we would make a q-addition of the corresponding q-Appell number with x. This is now replaced by a q-addition of the corresponding q-Appell number with two infinite function sequences Cν,q(x, y) and Sν,q(x, y) for the real and imaginary part of a new so-called q-complex number appearing in the generating function. Finally, we can prove q-analogues of the Cauchy–Riemann equations. This paper is organized as follows: in Section 1, we present the simplest type of q-complex numbers, which we will later use as function arguments in our new complex q-Appell polynomials. We remark that there are also other types of q-complex numbers. In Section 2, we define complex q-Appell polynomials, and show that these polynomials obey quite similar rules as q-Appell polynomials, which always appear in doublets. In Section 3, we briefly discuss pseudo-q-complex Appell polynomials, which have a slightly different generating function. 2010 Mathematics Subject Classification. Primary 11B68; Secondary 05A40, 05A30, 30B10.

This paper is organized as follows: in Section 1, we present the simplest type of q-complex numbers, which we will later use as function arguments in our new complex q-Appell polynomials. We remark that there are also other types of q-complex numbers.
In Section 2, we define complex q-Appell polynomials, and show that these polynomials obey quite similar rules as q-Appell polynomials, which always appear in doublets.
In Section 3, we briefly discuss pseudo-q-complex Appell polynomials, which have a slightly different generating function.
In Section 4, we present the two simplest examples of q-complex Appell polynomials. We remark that many other such polynomials could easily be defined, like in the other papers of the author. The special formulas, which appear in Section 2, are not repeated, for the sake of brevity.
In Section 5, we make a brief conclusion.
1. Definition of the q-complex numbers C ⊕q . Definition 1. We define the q-complex numbers C ⊕q as the set The numbers x and y are called the q-real and the q-imaginary parts of z, denoted by Re q z and Im q z. A q-real number is a q-complex number with q-imaginary part 0. A q-imaginary number is a q-complex number with q-real part 0. Definition 2. The absolute value of z ∈ C ⊕q is given by The conjugate of z ∈ C ⊕q is given by For each z ∈ C ⊕q , a ∈ R, we define a scalar multiplication az: az ≡ ax ⊕ q iay.
We will now define the corresponding operators of +, −, , ÷ by considering the set (C ⊕q , ⊕, q ). The two operations ⊕ and q are defined with the help of the following operator h. Definition 3. The bijection h : C −→ C ⊕q maps the complex number x+iy to x ⊕ q iy ∈ C ⊕q .
The multiplication and division are also given by .
We decide to have the same priority for these operations as usual, i.e., products are computed before sums (additions) etc. We also agree to sometimes abbreviate the by juxtaposition.
For purposes which will soon become evident, we limit ourselves to formal power series.

Definition 5.
We define the complex q-derivative as Powers of D ⊕ are denoted by D m ⊕ . The function f (z) is called q-holomorphic if and only if the complex qderivative D ⊕ f (z) exists. Theorem 1.1. Formula for the complex q-derivative for functions of qcomplex numbers.
Proof. We denote the fact that h is an isomorphism by . Put The result now follows by letting lim δz−→0 ⊕q . Proof.
2. Extension of q-Appell polynomials to complex q-Appell polynomials. We will now define the complex q-Appell polynomials. Throughout, we assume that z = x ⊕ q iy, where we can use both of the previously defined q-complex numbers. In the beginning, we use the numbers C ⊕q . For the notation, we refer to [4]. ν,q (x, y) have the following generating function: The sine-q-Appell polynomials A q (s, n) of degree ν and order n have the following generating function: By putting x = y = 0, we have again (see [2, 4.105]) ν,q are the q-Appell numbers. In the following, when a formula with A (c,n) ν,q (x) is given, without a similar formula with A (s,n) ν,q (x), we always assume that c ≡ c ∨ s.
It will be convenient to fix the value for n = 0 and n = 1: The following formula is the same as [2, 4.107]: [5, (3)] expresses the q-Appell polynomial of z as the sum of the cosine and sine-q-Appell polynomials.
. Then we have generating functions for q-Appell polynomial of z and z: Addition and subtraction of formulas (4) and (5) give a q-analogue of [5, p. 3]: , .
The first formula implies the symbolic equality: We collect some obvious facts about the q-complex Appell polynomials in a theorem.
ν,q (x, y) is an odd function of y.
Proof. To prove (8), use generating function (1) together with the q-Euler formula and change summation index.
The following two functions C ν,q (x, y) and S ν,q (x, y) replace q-addition of x with q-Appell numbers.

(15)
Proof. We find that On the other hand, we have which proves (14). Formula (15) is proved in a similar way.
We can now prove q-analogues of the Cauchy-Riemann equations for qcomplex Appell polynomials.

Pseudo-q-complex Appell polynomials.
Definition 8. For every power series f n (t) given by (1) and (2), the pseudocosine and pseudosine-q-Appell polynomials A (q,c,n) ν and A (q,s,n) ν of degree ν and order n have the following generating functions: Now, for convenience, we fix the value for n = 1: We have Theorem 3.1.
We get the following two q-Taylor formulas: Proof. Use formula (20).
As a prerequisite of the next section, we extend the following formulas from [2]. The two operators NWA,q and ∇ NWA,q always refer to the variable x.
A special case of A q polynomials are the θ q polynomials of order n, which are obtained by putting f n (t) = g(t)2 n (1) and (2).
By (24) we obtain We will now define the second q-Euler polynomials, a special case of the θ q polynomials.
The proofs of the following four complementary argument formulas are made with the generating function.   NWA,ν,q (1 q x, y).

Conclusion.
We have introduced a basis for a further investigation of q-complex numbers, which will appear in another paper. These numbers are used as function arguments in formal power series of one or many variables, not just polynomials. Our proof of the q-Cauchy-Riemann equations follows [1, p. 54]. It is not unlikely that the q-Cauchy-Riemann equations can be extended to higher dimensions, like in [6].