Multiplication formulas for q-Appell polynomials and the multiple q-power sums

In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli and Apostol-Euler polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.


Introduction. In 2006, Luo and Srivastava
found new relationships between Apostol-Bernoulli and Apostol-Euler polynomials. This was followed by the pioneering article by Luo [10], where multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, together with λ-multiple power sums were introduced. Luo also expressed these λ-multiple power sums as sums of the above polynomials. One year later, Wang and Wang [12] introduced generating functions for λ-power sums, some of the proofs use a symmetry reasoning, which lead to many four-line identities for Apostol-Bernoulli and Apostol-Euler polynomials and λ-power sums; as special cases, some of the above Luo identities were obtained.
In [5] it was proved that the q-Appell polynomials form a commutative ring; in this paper we show what this means in practice. Thus, the aim of the present paper is to find q-analogues of most of the above formulas with the aid of the multiple q-addition, the q-rational numbers, and so on. Many formulas bear a certain resemblance to the q-Taylor formula, where qrational numbers appear to the right in the function argument; this means that the alphabet is extended to Q ⊕q . In some proofs, both q-binomial coefficients and a vector binomial coefficient occur, this is connected to a vector form of the multinomial theorem, with binomial coefficients, unlike the case in [3, p. 110].
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, multiple q-Apostol-Bernoulli polynomials and q-power sums are introduced and multiplication formulas for q-Apostol-Bernoulli polynomials are proved, which are q-analogues of Luo [10].
In Section 3, multiplication formulas for q-Apostol-Euler polynomials are proved. In Section 4, formulas containing q-power sums in one dimension, q-analogues of Wang and Wang, [12] are proved. Then in Section 5, mixed formulas of the same kind are proved. Most of the proofs are similar, where different functions, previously used for the case q = 1, are used in each proof.
We now start with the definitions. Some of the notation is well-known and can be found in the book [3]. The variables i, j, k, l, m, n, ν will denote positive integers, and λ will denote complex numbers when nothing else is stated.
Let a and b be any elements with commutative multiplication. Then the NWA q-addition is given by If 0 < |q| < 1 and |z| < |1 − q| −1 , the q-exponential function is defined by The following theorem shows how Ward numbers usually appear in applications.
where each partition of k is multiplied with its number of permutations.
The semiring of Ward numbers, (N ⊕q , ⊕ q , q ) is defined as follows: denote the Ward numbers k q , k ≥ 0 together with two binary operations: ⊕ q is the usual Ward q-addition. The multiplication q is defined as follows: where ∼ denotes the equivalence in the alphabet. Furthermore, Proof. Formula (6) is proved as follows: where the number of 1s to the left is m. But this means exactly E q (n q t) m , and the result follows. Then if f (x) is the formal power series ∞ l=0 a l x l , its k'th NWA-power is given by We will later use a similar formula when q = 1 for several proofs. In order to solve systems of equations with letters as variables and Ward number coefficients, we introduce a division with a Ward number. This is equivalent to q-rational numbers with Ward numbers instead of integers.

Definition 4.
Let Q ⊕q denote the set of objects of the following type: together with a linear functional

Definition 5.
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: .
ν,q number of degree ν and order n.
Definition 6. For f n (t) of the form h(t) n , we call the q-Appell polynomial Φ q in (14) multiplicative.
Examples of multiplicative q-Appell polynomials are the two q-Appell polynomials in this article. NWA,λ,ν,q (x) are defined by Notice that the exponent n is an integer.

Remark 1.
For l = 1, formulas (16) and (17) reduce to single sums, as will be seen in section 4.
We now start rather abruptly with the theorems; we note that limits like λ → 1 and q → 1 can be taken anywhere in the paper, and also in the next one [6]; see the subsequent corollaries. Much care is needed in the proofs, since the Ward numbers need careful handling.
Proof. We use the well-known formula for a geometric sum.
The theorem follows by equating the coefficients of t ν {ν}q! .
The theorem follows by equating the coefficients of t ν {ν}q! .
Proof. We use the definition of q-Appell numbers as q-Appell polynomial at x = 0.
3. 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 NWA,ν,q (x), |t| < π.
The theorem follows by equating the coefficients of t ν {ν}q! .

Single formulas for Apostol q-power sums.
In order to keep the same notation as in [3], we make a slight change from [12, p. 309]. The following definitions are special cases of the q-power sums in section 2.
Proof. Define the following function, symmetric in i and j.
By using the formula for a geometric sequence, we can expand f q (t) in two ways: The theorem follows by equating the coefficients of t ν {ν}q! and using the symmetry in i and j of f q (t).
Moreover, we have Proof. Put i = 2 in (47) and replace x by x 2 q .
For λ = 1 and x = 0, this reduces to B NWA,n,q 1 2 q = 2 (2 q ) n − 1 B NWA,n,q . (50) Theorem 4.5. A q-analogue of [12, (22) p. 312]. Assume that i and j are either both odd, or both even, then we have Proof. Define the following symmetric function By using the formula for a geometric sequence, we can expand f q (t) in two ways: The theorem follows by equating the coefficients of t ν {ν}q! and using the symmetry in i and j of f q (t).

Remark 3.
This proves the first part of formula (32) again.

5.
Apostol q-power sums, mixed formulas. We now turn to mixed formulas, which contain polynomials of both kinds.
Proof. Define the following function By using the formula for a geometric sequence, we can expand f q (t) in two ways: By equating the coefficients of t ν {ν}q! , we obtain rows 1 and 3 of formula (56). On the other hand, we can rewrite f q (t) in the following way: By using the formula for a geometric sequence, we can expand (59) in two ways: By equating the coefficients of t ν {ν}q! , we obtain rows 2 and 4 of formula (56).
6. Discussion. As was indicated in [5], we have considered q-analogues of the currently most popular Appell polynomials, together with corresponding power sums. The beautiful symmetry of the formulas comes from the ring structure of the q-Appell polynomials. We have not considered JHC q-Appell polynomials, since we are looking for maximal symmetry in the formulas. The q-Taylor formulas have not been used in the proofs, since the generating functions were mostly used. In a further paper [6], we will find similar expansion formulas for q-Appell polynomials of arbitrary order.