On certain generalized q-Appell polynomial expansions

We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol–Bernoulli and Apostol–Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.


Introduction.
The aim of this paper is to describe how the q-umbral calculus extends in a natural way to produce q-analogues of conversion theorems and polynomial expansions for the following Appell polynomials: Hpolynomials, Apostol-Bernoulli and Apostol-Euler from the recent articles on this theme, as well as multiple variable extensions.To this aim, we use certain q-difference operators known from the book [3], and some new operators containing a factor λ from the previous work of Luo and Srivastava [8], [9], [10] on Apostol-Bernoulli polynomials.The q-Appell polynomials have been used before in [4], where their basic definition was given together with several matrix applications.The q-umbral method [3], influenced by Jordan [5] and Nørlund [12], forms the basis for the terminology and umbral method, which enables convenient q-analogues of the formulas for Appell polynomials; our formulas resemble the Appell polynomial formulas in a remarkable way.A certain q-Taylor formula plays a key role in many proofs.
This paper is organized as follows: In this section we give the general definitions.In each section, we then give the specific definitions and special values which we use there.In Section 2, we introduce two dual polynomials together with recursion formulas, symmetry relations and complementary argument theorem.
Let λ ∈ R and let E q (x) denote the q-exponential function.In Sections 3 and 4 in the spirit of Apostol, Luo and Srivastava, we introduce and discuss two dual forms of the generalized q-Apostol-Bernoulli polynomials, together with the many applications that were mentioned earlier.In Sections 5 and 6, we continue the discussion with two dual forms of the generalized q-Apostol-Euler polynomials Two of their generating functions are given below: (1) NWA,λ,ν,q (x) {ν} q !, and (2) NWA,λ,ν,q (x) {ν} q ! .This is followed by formulas which contain both kinds of these polynomials.Many of the formulas are proved by simple manipulations of the generating functions.In Section 7 we show that the many expansion formulas according to Nørlund can also be formulated for our polynomials.In Section 7 we extend the previous considerations to a more general form, named multiplicative q-Appell polynomials.More on this will come in a future paper.We now come to Section 9; in order to find q-analogues of the corresponding formulas for the generating functions, we, formally, introduce a logarithm for the q-exponential function.The calculations are valid for so-called q-real numbers.In Section 10, we briefly discuss multiple q-Appell polynomials.We now start with the definitions, compare with the book [3].Some of the notation is well known and will be skipped.

Definition 1.
Let the Gauss q-binomial coefficient be defined by Let a and b be any elements with commutative multiplication.Then the NWA q-addition is given by The JHC q-addition is the function The q-derivative is defined by ( 7) Definition 2. Let the NWA q-shift operator be given by ( 8) Definition 3. The related JHC q-shift operator is given by ( 9) E( q )(x n ) ≡ (x q 1) n .
Definition 4. For every power series f n (t), the q-Appell polynomials or Φ q polynomials of degree ν and order n have the following generating function: For x = 0 we get the Φ (n) ν,q number of degree ν and order n.Definition 5.For f n (t) of the form h(t) n , we call the q-Appell polynomial Φ q in (14) multiplicative.Theorem 1.1.We have the two q-Taylor formulas 2. The H polynomials.First, we repeat some of the definitions of certain related q-Appell polynomials for later use.These polynomials are more general forms of the polynomials we wish to study.Definition 6.The generating function for β Definition 7. The generating function for γ Definition 8.The generating function for η Definition 9.The generating function for θ We now come to the generating function of the polynomials we want to study in this section (for q = 1): The H polynomials are defined in [14, p. 532 (37)].NWA,ν,q (x) is a special case of (19): Definition 11.The generating function for H (n) JHC,ν,q (x) is a special case of (20): The polynomials in ( 23) and ( 24) are q-analogues of the generalized H polynomials.We now turn to these q-analogues.Theorem 2.1.We have H JHC,0,q = 0, H JHC,1,q = 1, (H JHC,q q 1) k + H JHC,k,q = 0, k > 1.
The following table lists some of the first H NWA,ν,q numbers.
We need not calculate the H JHC,ν,,q numbers, since we have the following symmetry relations: For ν even, H NWA,ν,q = H JHC,ν,q .
The following table lists some of the first Ward q-Bernoulli numbers.
The following three formulas express x n in terms of q-Appell polynomials.
The following complementary argument theorems extend the ones given in [3, p. 153].

3.
The NWA q-Apostol-Bernoulli polynomials.Throughout, we assume that λ = 0.The b polynomials are more general forms of the NWA q-Apostol-Bernoulli polynomials, which we will study in this section.

Definition 13. The polynomials b
(n) λ,ν,q (x) are defined by (43) The generating function for B NWA,ν,q (x) is a special case of (44): Definition 14.The generalized NWA q-Apostol-Bernoulli polynomials B (n) NWA,λ,ν,q (x) are defined by Assume that λ = 1.The poles in the denominator of (44) are the roots of E q (t) = λ −1 , which implies that in some cases the limit λ → 1 is not straightforward and needs some further consideration.
We have λ,ν,q (x).This leads to the following recurrence for the NWA q-Apostol-Bernoulli numbers: The following table lists some of the first B NWA,λ,ν,q numbers.
Proof.We have that Formula (63) now follows on equating the coefficients of t ν .

5.
The NWA q-Apostol-Euler polynomials.We start with some repetition from [3]: The generating function for the first q-Euler polynomials of degree ν and order n, F NWA,ν,q (x) is given by (66) The following table lists some of the first q-Euler numbers F NWA,n,q .
The e polynomials are more general forms of the NWA q-Apostol-Euler polynomials, which we will study in this section.
Definition 18.The e polynomials are defined by (67) Definition 19.The generalized NWA q-Apostol-Euler polynomials F (n) NWA,λ,ν,q (x) are defined by (68) Assume that λ = −1.The poles in the denominator of (68) are the roots of E q (t) = −λ −1 .Theorem 5.1.We have λ,ν,q (x), This leads to the following recurrence: The following table lists some of the first F NWA,λ,n,q numbers.
We observe that the limits for λ → 1 are the first q-Euler numbers.
Proof.The following computation with generating functions shows the way: Equating the coefficients of t ν and using (70) gives (85).
Proof.We have that Formula (89) now follows on equating the coefficients of t ν .
We conclude that the NWA q-Apostol-Bernoulli and NWA q-Apostol-Euler polynomials satisfy linear q-difference equations with constant coefficients.
Theorem 7.5 (A generalization of [3, 4.261]).Under the assumption that f (x) is analytic with q-Taylor expansion we can express powers of NWA,A,q and ∇ NWA,A,q operating on f (x) as powers of D q as follows.These series converge when the absolute value of x is small enough: NWA,λ,ν,q (x) {ν} q ! .
8. Multiplicative q-Appell polynomials.In this section we very briefly discuss multiplicative q-Appell polynomials with f n (t) equal to h(t) n .It turns out, by simple umbral manipulation that many of the formulas in [3,Section 4.3] are also valid for multiplicative q-Appell polynomials, and these equations will be presented in another article.Throughout, we denote the multiplicative q-Appell polynomials by Φ (n) M,ν,q (x).Definition 23.Under the assumption that the function h(t) n can be expressed analytically in R[[t]], and for f n (t) of the form h(t) n , we call the q-Appell polynomial Φ q in (14) multiplicative.
Then we have M,k,q (x 1 ⊕ q . ..⊕ q x s ) = where we assume that n j operates on x j .
Proof.It suffices to show that the logarithmic derivative of E q (x) is > 0. However, this follows from Definition 24.The q-logarithm log q (x) is the inverse function of E q (x), −∞ < x < (1 − q) −1 , 0 < q < 1.
Theorem 9.2.The q-logarithm log q (x) has the following properties (x and y have small real values, n ∈ N): (1) Its domain is ) It is strictly increasing.
( 128) Proof.We equate the generating functions in the following way: 10. Appendix: multiple q-Appell polynomials.Of course there are many ways to define multiple q-Appell polynomials; in this paper we concentrate on one of the simplest approaches in the spirit of Lee.