Two-Parameter Identities for q-Appell Polynomials

Emanuele Munarini Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano Italy

              emanuele.munarini@polimi.it

Abstract

In this paper, by using the techniques of the q-exponential generating series, we extend a well-known two-parameter identity for the Appell polynomials to the q-Appell polynomials of type I and II. More precisely, we obtain two different q-analogues of such an identity. Then, we specialize these identities for some q-polynomials arising in combinatorics, in q-calculus or in the theory of orthogonal polynomials. In particular, we consider the generalized q-Bernoulli and q-Euler polynomials and then we deduce some further identities involving the Bernoulli and Euler numbers. In this way, as a byproduct, we derive the symmetric Carlitz identity for the Bernoulli numbers. Finally, we find a (non-symmetric) q-analogue of Carlitz’s identity involving the q-Bernoulli numbers of type I and II.

1 Introduction

Appell sequences form an important and interesting class containing many classical polynomials arising in physics, in numerical analysis, in the theory of orthogonal polynomials [13], in analysis [6, 10, 36], in the modern umbral calculus [32, 33, 34] and in the theory of Sheffer sequences [38, 41] or, equivalently, in the theory of Sheffer matrices (or exponential Riordan arrays) [25, 26]. The ordinary powers, the generalized Hermite polynomials, the generalized Bernoulli and Euler polynomials, and the generalized rencontres polynomials are all examples of Appell polynomials.

An Appell sequence {a_{n}(x)}_{n∈ℕ} can be characterized in several ways [5, 45]. The following statements are equivalent.

1. For every n ∈ ℕ, the polynomial a_{n}(x) has degree n and a′_{n}(x) = na_{n−1}(x).

2. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}g_{n−k} x^{k}

or, equivalently,

g_{n} = ∑_{k=0}^{n} \binom{n}{k}(−1)^{k}x^{k} a_{n−k}(x).

3. The sequence {a_{n}(x)}_{n∈ℕ} has exponential generating series

A(x; t) = ∑_{n≥0} a_{n}(t) \frac{t^{n}}{n!} = g(t) e^{xt},

where g(t) = ∑_{n≥0} g_{n} \frac{t^{n}}{n!} is an exponential series with g_{0} ≠ 0.

4. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k≥0} g_{k} \frac{𝔇^{k}}{k!} x^{n},

where 𝔇 is the derivative with respect to x.

5. The sequence {a_{n}(x)}_{n∈ℕ} satisfies the Appell identity

∑_{k=0}^{n} \binom{n}{k}a_{k}(x) y^{n−k} = a_{n}(x + y).

Furthermore, the Appell polynomials a_{n}(x) and the coefficients g_{n} are related by the following two-parameter binomial identity [45, p. 316] [36, Formula (16′)]:

∑_{k=0}^{n} \binom{n}{k}(−1)^{k}x^{k} a_{m+n−k}(x) = ∑_{k=0}^{m} \binom{m}{k}g_{n+k} x^{m−k} .  (1)

It is easy to see that this is a further characterization of the Appell polynomials. More precisely, we have that the polynomials a_{n}(x) form an Appell sequence if and only if there exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, for which identity (1) is satisfied for every m, n ∈ ℕ.

In 1967, after a paper by Sharma and Chak [37], Al-Salam [2] introduced the q-Appell polynomials (of type I) and, recently, Sadjang [35] introduced the q-Appell polynomials of type II. In this paper, we extend identity (1) to the q-Appell polynomials of type I and II. Specifically, through the techniques of the theory the q-exponential generating series, we obtain two different q-analogues of this identity. Then, we specialize these identities for some q-polynomials arising in combinatorics, in q-calculus or in the theory of orthogonal polynomials, such as the q-Hermite polynomials, the Gaussian polynomials, the Al-Salam-Carlitz polynomials, the q-permutation polynomials, the q-rencontres and q-arrangement polynomials (and some of their generalizations), the q-Bernoulli and q-Euler polynomials. In this last case, we further specialize our identities to the Bernoulli and Euler polynomials and we deduce some other identities involving the Bernoulli and Euler numbers. In particular, as a byproduct, we derive the Carlitz symmetric identity for the Bernoulli numbers [9, 25]. Finally, we find a q-analogue of Carlitz’s identity. Such an identity, however, is not symmetric and involves the q-Bernoulli numbers of type I and II.

2 q-Appell polynomials

We start by recalling some definitions [23]. For every n ∈ ℕ, we have the q-natural number [n]_{q} = 1 + q + q^{2} + ⋯ + q^{n−1} and the q-factorial number [n]_{q}! = [n]_{q}[n − 1]_{q} ⋯ [2]_{q}[1]_{q}. Then, for every n, k ∈ ℕ, the q-binomial coefficients (or Gaussian coefficients) are defined by

\binom{n}{k}_{q} = \frac{[n]_{q}!}{[k]_{q}![n − k]_{q}!} for k = 0, 1, …, n

and by 0 otherwise. A q-polynomial is a polynomial with coefficients in ℚ_{q} = ℚ(q), i.e., the field of quotients of ℚ[q], where q is an indeterminate. The algebra of the q-polynomials are denoted by ℚ_{q}[x]. The q-Pochhammer symbol is the q-polynomial defined by

(x; q)_{n} = (1 − x)(1 − qx) ⋯ (1 − q^{n−1}x) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}.  (2)

A q-exponential series is a formal series of the form f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{[n]_{q}!}. The sum and the multiplication by a scalar are defined componentwise. The product of two q-exponential series f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{[n]_{q}!} and g(t) = ∑_{n≥0} g_{n} \frac{t^{n}}{[n]_{q}!} is defined by

f(t)g(t) = ∑_{n≥0} (∑_{k=0}^{n} \binom{n}{k}_{q}f_{k}g_{n−k}) \frac{t^{n}}{[n]_{q}!}.  (3)

The q-derivative (Jackson’s derivative) 𝔇_{q} of a q-exponential generating series f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{[n]_{q}!} is defined [21, 22] by the formula

𝔇_{q}f(t) = \frac{f(qt) − f(t)}{(q − 1)t} = ∑_{n≥0} f_{n+1} \frac{t^{n}}{[n]_{q}!}.

Given two q-exponential series f(t) and g(t), we have the q-Leibniz formula

𝔇_{q}^{m}f(t)g(t) = ∑_{k=0}^{m} \binom{m}{k}_{q}𝔇_{q}^{k}f(t) · Q_{q}^{k}𝔇_{q}^{m−k}g(t),  (4)

where Q_{q} is the operator defined by Q_{q}h(t) = h(qt).

The q-exponential series (Jackson’s q-exponential) [21]

E_{q}(t) = ∑_{n≥0} \frac{t^{n}}{[n]_{q}!} = ∏_{k≥0} \frac{1}{1 + (q − 1)q^{k}t}  (5)

is the eigenfunction of the q-derivative, that is,

𝔇_{q}E_{q}(λt) = λE_{q}(t).  (6)

In particular, since 𝔇_{q}E_{q}(t) = E_{q}(t), we have the relation

E_{q}(qt) = (1 − (1 − q)t) E_{q}(t).  (7)

Consequently, for every m ∈ ℕ, we have

E_{q}(q^{m}t) = ((1 − q)t; q)_{m} E_{q}(t).  (8)

The inverse of the q-exponential series E_{q}(t) is

E_{q}(t)^{−1} = ∑_{n≥0}(−1)^{n}q^{\binom{n}{2}} \frac{t^{n}}{[n]_{q}!}.  (9)

Given two q-exponential series f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{[n]_{q}!} and g(t) = ∑_{n≥0} g_{n} \frac{t^{n}}{[n]_{q}!}, we have f(t) = E_{q}(αt)g(t) if and only if g(t) = E_{q}(αt)^{−1}f(t). This is equivalent to the q-binomial inversion theorem:

f_{n} = ∑_{k=0}^{n} \binom{n}{k}_{q}α^{k}g_{n−k}  ⟺  g_{n} = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}α^{k}f_{n−k}.

A q-Appell sequence (of type I) [2] is a sequence {a_{n}(x)}_{n∈ℕ}, where a_{n}(x) is a q-polynomial of degree n and 𝔇_{q}a_{n}(x) = [n]_{q}a_{n−1}(x) for every n ∈ ℕ. Also the q-Appell sequences can be characterized in several ways, as in the ordinary case. Indeed, the following statements are equivalent.

1. {a_{n}(x)}_{n∈ℕ} is a q-Appell sequence (of type I).

2. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}_{q}g_{n−k} x^{k}

or, equivalently,

g_{n} = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k} a_{n−k}(x).

3. The sequence {a_{n}(x)}_{n∈ℕ} has q-exponential generating series

A(x; t) = ∑_{n≥0} a_{n}(x)\frac{t^{n}}{[n]_{q}!} = g(t) E_{q}(xt),  (10)

where g(t) = ∑_{n≥0} g_{n} \frac{t^{n}}{[n]_{q}!} is a q-exponential series with g_{0} ≠ 0.

4. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k≥0} g_{k} \frac{𝔇_{q}^{k}}{[k]_{q}!} x^{n}.

Similarly, a q-Appell sequence of type II [35] is a sequence {a_{n}(x)}_{n∈ℕ}, where a_{n}(x) is a q-polynomial of degree n and 𝔇_{q}a_{n}(x) = [n]_{q}a_{n−1}(qx), for every n ∈ ℕ. Also in this case, we have the following equivalent characterizations:

1. {a_{n}(x)}_{n∈ℕ} is a q-Appell sequence of type II.

2. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}_{q}q^{\binom{k}{2}}g_{n−k} x^{k}

or, equivalently,

g_{n} = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}x^{k} a_{n−k}(x).

3. The sequence {a_{n}(x)}_{n∈ℕ} has q-exponential generating series

A(x; t) = ∑_{n≥0} a_{n}(x)\frac{t^{n}}{[n]_{q}!} = g(t) E_{q}(−xt)^{−1},  (11)

where g(t) = ∑_{n≥0} g_{n} \frac{t^{n}}{[n]_{q}!} is a q-exponential series with g_{0} ≠ 0.

4. There exists a sequence {g_{n}}_{n∈ℕ}, with g_{0} ≠ 0, such that

a_{n}(x) = ∑_{k≥0} g_{k} q^{\binom{n−k}{2}} \frac{𝔇_{q}^{k}}{[k]_{q}!} x^{n}.

3 Two-parameter identities

To obtain a q-analogue of identity (1), we apply the q-Leibniz rule to the q-exponential generating series of a q-Appell sequence. By applying such a rule in two different ways, we obtain two different extensions of identity (1). First, we have the following result:

Theorem 1. Let {a_{n}(x)}_{n∈ℕ} be the q-Appell sequence (of type I) associated with the sequence {g_{n}}_{n∈ℕ}. Then, for every m, n ∈ ℕ, we have the identity

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}a_{m+n−k}(x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}g_{n+m−k} x^{k} .  (12)

Proof. By applying the q-Leibniz rule (4) to series (10), we get

𝔇_{q}^{m}A(x; t) = 𝔇_{q}^{m}E_{q}(xt)g(t) = ∑_{k=0}^{m} \binom{m}{k}_{q}𝔇_{q}^{k}E_{q}(xt) · Q_{q}^{k}𝔇_{q}^{m−k}g(t).

Then, by identity (6) and by setting g^{(h)}(t) = 𝔇_{q}^{h}g(t), we have

𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}x^{k}E_{q}(xt) g^{(m−k)}(q^{k}t),

that is,

E_{q}(xt)^{−1}𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}x^{k} g^{(m−k)}(q^{k}t).

Recalling formulas (3) and (9) and taking the coefficients of \frac{t^{n}}{[n]_{q}!} from the first and last series, we obtain identity (12).

To state the next theorem, we need the following definition: given the q-Appell polynomials a_{n}(x) associated with the q-numbers g_{n}, we define the shifted polynomials a_{n}^{[m]}(x) as the q-Appell polynomials associated with the q-numbers g_{m+n}; namely,

a_{n}^{[m]}(x) = ∑_{k=0}^{n} \binom{n}{k}_{q}g_{m+n−k} x^{k}.

This means that their q-exponential generating series is

∑_{n≥0} a_{n}^{[m]}(x)\frac{t^{n}}{[n]_{q}!} = g^{(m)}(t) E_{q}(xt),

where g^{(m)}(t) = 𝔇_{q}^{m}g(t).

Theorem 2. Let {a_{n}(x)}_{n∈ℕ} be the q-Appell sequence (of type I) associated with the sequence {g_{n}}_{n∈ℕ}. Then, for every m, n ∈ ℕ, we have the identity

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}a_{m+n−k}(x) = ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}x^{k}a_{m−k}^{[n]}(x), (13)

where m ∧ n denotes the minimum between m and n.

Proof. By applying the q-Leibniz rule (4) to series (10), we have

𝔇_{q}^{m}A(x; t) = 𝔇_{q}^{m}g(t)E_{q}(xt) = ∑_{k=0}^{m} \binom{m}{k}_{q}𝔇_{q}^{k}g(t) · Q_{q}^{k}𝔇_{q}^{m−k}E_{q}(xt).

Then, by identity (6), we have

𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}g^{(k)}(t) x^{m−k}E_{q}(q^{k}xt)

and, by Eq. (8), we have

𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}g^{(k)}(t) x^{m−k}((1 − q)xt; q)_{k}E_{q}(xt),

that is,

E_{q}(xt)^{−1}𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}g^{(k)}(t) x^{m−k}((1 − q)xt; q)_{k}.

Hence, by Eq. (2), we have

E_{q}(xt)^{−1}𝔇_{q}^{m}A(x; t) = ∑_{k=0}^{m} \binom{m}{k}_{q}g^{(k)}(t) x^{m−k} ∑_{i=0}^{k} \binom{k}{i}_{q}(q − 1)^{i}q^{\binom{i}{2}}x^{i}t^{i}

= ∑_{i,k≥0}^{m} \binom{m}{k}_{q}\binom{k}{i}_{q}(q − 1)^{i}q^{\binom{i}{2}}x^{i+m−k}t^{i}g^{(k)}(t)

= ∑_{i,k≥0}^{m} \binom{m}{i}_{q}\binom{m − i}{k − i}_{q}(q − 1)^{i}q^{\binom{i}{2}}x^{i+m−k} ∑_{n≥0} \binom{n}{i}_{q}[i]_{q}!g_{n+k−i} \frac{t^{n}}{[n]_{q}!}

= ∑_{n≥0} ( ∑_{i,k≥0} \binom{m}{i}_{q}\binom{m − i}{k − i}_{q}\binom{n}{i}_{q}[i]_{q}!(q − 1)^{i}q^{\binom{i}{2}}g_{n+k−i}x^{m−k+i}) \frac{t^{n}}{[n]_{q}!}

= ∑_{n≥0} (∑_{i≥0} \binom{m}{i}_{q}\binom{n}{i}_{q}[i]_{q}!(q − 1)^{i}q^{\binom{i}{2}}x^{i} ∑_{k=i}^{m} \binom{m − i}{k − i}_{q}g_{n+k−i}x^{m−k}) \frac{t^{n}}{[n]_{q}!}

= ∑_{n≥0} (∑_{i≥0} \binom{m}{i}_{q}\binom{n}{i}_{q}[i]_{q}!(q − 1)^{i}q^{\binom{i}{2}}x^{i} ∑_{k=0}^{m−i} \binom{m − i}{k}_{q}g_{n+k}x^{m−i−k}) \frac{t^{n}}{[n]_{q}!}

= ∑_{n≥0} (∑_{i≥0} \binom{m}{i}_{q}\binom{n}{i}_{q}[i]_{q}!(q − 1)^{i}q^{\binom{i}{2}}x^{i} ∑_{k=0}^{m−i} \binom{m − i}{k}_{q}g_{n+m−i−k}x^{k}) \frac{t^{n}}{[n]_{q}!}

= ∑_{n≥0} (∑_{i≥0} \binom{m}{i}_{q}\binom{n}{i}_{q}[i]_{q}!(q − 1)^{i}q^{\binom{i}{2}}x^{i}a_{m−i}^{[n]}(x)) \frac{t^{n}}{[n]_{q}!}.

Taking the coefficients of \frac{t^{n}}{[n]_{q}!} in the first and last series, we obtain identity (13).

Given a sequence {g_{n}}_{n∈ℕ}, we denote by a_{n}(x) the associated q-Appell polynomials of type I and by a_{n}^{∗}(x) the associated q-Appell polynomials of type II. To establish a relation between these two sequences, we define the umbral map ψ : ℚ_{q}[x] → ℚ_{q}[x] by setting

ψ(x^{n}) = q^{\binom{n}{2}}x^{n}

and by extending it by linearity. Then, we immediately have the following first result:

Lemma 3. The umbral map ψ transforms the q-Appell polynomials of type I into the q-Appell polynomials of type II, that is, for every n ∈ ℕ, we have

ψ(a_{n}(x)) = a_{n}^{∗}(x).

This result can be extended as follows.

Lemma 4. For every m ∈ ℕ and for every polynomial p(x), we have

ψ(x^{m}p(x)) = q^{\binom{m}{2}}x^{m} p^{∗}(q^{m}x),  (14)

where p^{∗}(x) = ψ(p(x)).

Proof. Suppose p(x) = ∑_{k=0}^{n} p_{k}x^{k}. Then, by the linearity of the umbral map, we have

ψ(x^{m}p(x)) = ∑_{k=0}^{n} \binom{n}{k}_{q}p_{k} ψ(x^{m+k}) = ∑_{k=0}^{n} \binom{n}{k}_{q}p_{k} q^{\binom{m+k}{2}}x^{m+k}

= q^{\binom{m}{2}}x^{m} ∑_{k=0}^{n} \binom{n}{i}_{q}q^{\binom{k}{2}}p_{k} (q^{m}x)^{k} = q^{\binom{m}{2}}x^{m}p^{∗}(q^{m}x).

This establishes identity (14).

By this lemma, we can easily find the following analogues of identities (12) and (13) for the q-Appell polynomials of type II:

Theorem 5. Let {a_{n}^{∗}(x)}_{n∈ℕ} be the q-Appell sequence of type II associated with the sequence {g_{n}}_{n∈ℕ}. Then, for all m, n ∈ ℕ, we have the identities

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{k^{2}−k}x^{k}a_{m+n−k}^{∗}(q^{k}x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{\binom{k}{2}+nk}g_{n+m−k} x^{k}  (15)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{k^{2}−k}x^{k}a_{m+n−k}^{∗}(q^{k}x) = ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{k^{2}−k}x^{k}a_{n+m−k}^{[n]∗}(q^{k}x). (16)

Proof. By Lemma 4, we have ψ(x^{m}a_{n}^{[r]}(x)) = q^{\binom{m}{2}}x^{m}a_{n}^{[r]∗}(q^{m}x). Hence, by applying ψ to identities (12) and (13) we obtain at once identities (15) and (16).

4 Examples (of type I)

4.1 q-Hermite polynomials

The q-Hermite polynomials (or Rogers-Szegö polynomials) [8, 3] [32, p. 180] are the q-Appell polynomials associated with the numbers g_{n} = 1; namely,

H_{n}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}x^{k} ,  (17)

and have q-exponential generating series

∑_{n≥0} H_{n}(q; x)\frac{t^{n}}{[n]_{q}!} = E_{q}(t) E_{q}(xt).

Since H_{n}^{[m]}(q; x) = H_{n}(q; x), identities (12) and (13) become

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}H_{m+n−k}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk} x^{k},  (18)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}H_{m+n−k}(q; x) = ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}x^{k}H_{m−k}(q; x). (19)

In particular, for x = 1, we have the Galois numbers G_{n}(q) = H_{n}(q; 1) = ∑_{k=0}^{n} \binom{n}{k}_{q}, [18, 28], and the two previous identities become

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}G_{m+n−k}(q) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk},  (20)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}G_{m+n−k}(q) = ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}G_{m−k}(q).  (21)

4.2 Gaussian polynomials

The Gaussian polynomials [18, 19] are the q-Appell polynomials associated with the qnumbers g_{n} = (−1)^{n}q^{\binom{n}{2}}; namely,

g_{n}(q; x) = (x − 1)(x − q) ⋯ (x − q^{n−1}) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{n−k}q^{\binom{n−k}{2}}x^{k} ,

and have q-exponential generating series

∑_{n≥0} g_{n}(q; x)\frac{t^{n}}{[n]_{q}!} = E_{q}(t)^{−1}E_{q}(xt).

In this case, we have

g_{n}^{[m]}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{m+n−k}q^{\binom{m+n−k}{2}}x^{k}

= (−1)^{m}q^{\binom{m}{2}}q^{mn} ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{n−k}q^{\binom{n−k}{2}}(\frac{x}{q^{m}} )^{k}

= (−1)^{m}q^{\binom{m}{2}}q^{mn}g_{n}(q; x/q^{m}) = (−1)^{m}q^{\binom{m}{2}} \frac{g_{m+n}(q; x)}{g_{n}(q; x)}.

Hence, identities (12) and (13) become

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}g_{m+n−k}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}(−1)^{n+m−k}q^{\binom{n+m−k}{2}+nk}x^{k},  (22)

\begin{matrix}g_{n}(q; x) ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}g_{m+n−k}(q; x) = \\ = (−1)^{n}q^{\binom{n}{2}} ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}x^{k}g_{n+m−k}(q; x). \end{matrix}  (23)

These two identities can be rewritten in terms of the Pochhammer symbol. More precisely, since (x; q)_{n} = x^{n}g_{n}(q; x^{−1}), we have the identities

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}(x; q)_{m+n−k} = ∑_{k=0}^{m} \binom{m}{k}_{q}(−1)^{n+m−k}q^{\binom{n+m−k}{2}+nk}x^{n+m−k},  (24)

\begin{matrix}(x; q)_{n} ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{n−k}q^{\binom{k}{2}}(x; q)_{m+n−k} \\ = q^{\binom{n}{2}}x^{n} ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}(x; q)_{n+m−k}. \end{matrix}  (25)

4.3 Al-Salam-Carlitz polynomials

The Al-Salam-Carlitz polynomials (or q-Carlitz polynomials) ([4] [13, p. 195] [20]) are defined by the formula

U_{n}^{(α)}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−α)^{n−k}g_{k}(x)

and consequently they have q-exponential generating series

U_{q}^{(α)}(x; t) = ∑_{n≥0} U_{n}^{(α)}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(xt)}{E_{q}(t)E_{q}(αt)} = E_{q}(t)^{−1}E_{q}(αt)^{−1}E_{q}(xt).

This means that they form the q-Appell sequence associated with the q-numbers

u_{n}^{(α)}(q) = (−1)^{n} ∑_{k=0}^{n} \binom{n}{k}_{q}q^{\binom{k}{2}}q^{\binom{n−k}{2}}α^{k}

defined by the q-exponential generating series

u_{q}^{(α)}(t) = ∑_{n≥0} u_{n}^{(α)}(q)\frac{t^{n}}{[n]_{q}!} = E_{q}(t)^{−1}E_{q}(αt)^{−1}.

In this case, identity (12) become

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}U_{m+n−k}^{(α)}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}u_{n+m−k}^{(α)}(q) x^{k} .  (26)

4.4 q-Factorial polynomials

The q-factorial polynomials are the q-Appell polynomials associated with the q-factorial numbers g_{n} = [n]_{q}!; namely,

F_{n}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}[n − k]_{q}! x^{k}.

Hence, they have q-exponential generating series

F_{q}(x, t) = ∑_{n≥0} P_{n}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(xt)}{1 − t}.

Moreover, the polynomials

F_{n}^{[m]}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}[m + n − k]_{q}! x^{k}

have q-exponential generating series

F_{q}^{[m]}(x, t) = ∑_{n≥0} P_{n}^{[m]}(q; x)\frac{t^{n}}{[n]_{q}!} = F_{q}^{[m]}(t)E_{q}(xt),

where

F_{q}^{[m]}(t) = ∑_{n≥0}[m + n]_{q}!\frac{t^{n}}{[n]_{q}!} = [m]_{q}! ∑_{n≥0} \frac{[m + n]_{q}!}{[m]_{q}![n]_{q}!}t^{n} = [m]_{q}! ∑_{n≥0} \binom{m + n}{m}_{q}t^{n},

that is,

F_{q}^{[m]}(t) = \frac{[m]_{q}!}{(t; q)_{m+1}} = \frac{[m]_{q}!}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)}.

Hence, we have

F_{q}^{[m]}(x, t) = \frac{[m]_{q}! E_{q}(xt)}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)}.

Identities (12) and (13) become

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}F_{m+n−k}(x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}[n + m − k]_{q}! x^{k},  (27)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}F_{m+n−k}(x) = ∑_{k=0}^{m∧n} \binom{m}{k}_{q}\binom{n}{k}_{q}[k]!(q − 1)^{k}q^{\binom{k}{2}}x^{k}F_{n+m−k}^{[n]}(x).  (28)

4.5 q-rencontres and q-arrangement polynomials

The q-derangement numbers d_{n}(q) [46, 11, 27] and the q-arrangement numbers a_{n}(q) are respectively defined by the formulas

d_{n}(q) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}[n − k]_{q}! and a_{n}(q) = ∑_{k=0}^{n} \binom{n}{k}_{q}[n − k]_{q}!

and have q-exponential generating series

D_{q}(t) = ∑_{n≥0} d_{n}(q)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(t)^{−1}}{1 − t} and A_{q}(t) = ∑_{n≥0} a_{n}(q)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(t)}{1 − t}.

For q = 1, we recover the derangement numbers d_{n} ([31, p. 65] and \underline{A000166} and the arrangement numbers a_{n} ([31, p. 16] and \underline{A000522}).

The q-rencontres polynomials D_{q}(x; t) and the q-arrangement polynomials A_{q}(x; t) are the q-Appell polynomials associated respectively with the q-numbers d_{n}(q) and a_{n}(q); namely,

D_{n}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}d_{n−k}(q) x^{k} and A_{n}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}a_{n−k}(q) x^{k},

and have q-exponential generating series

D_{q}(x; t) = ∑_{n≥0} D_{n}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(t)^{−1}}{1 − t} E_{q}(xt),

A_{q}(x; t) = ∑_{n≥0} A_{n}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{E_{q}(t)}{1 − t} E_{q}(xt).

Then, by identity (12), we have

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}D_{m+n−k}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}d_{n+m−k}(q) x^{k},  (29)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}A_{m+n−k}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}a_{n+m−k}(q) x^{k}.  (30)

More generally, given m ∈ ℕ, we can consider the generalized q-derangement numbers d_{n}^{(m)}(q) and the generalized q-arrangement numbers a_{n}^{(m)}(q) defined by

d_{n}^{(m)}(q) = ∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}[m + n − k]_{q}! and a_{n}^{(m)}(q) = ∑_{k=0}^{n} \binom{n}{k}_{q}[m + n − k]_{q}!

and having q-exponential generating series

D_{q}^{(m)}(t) = ∑_{n≥0} d_{n}^{(m)}(q)\frac{t^{n}}{[n]_{q}!} = \frac{[m]_{q}! E_{q}(t)^{−1}}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)},

A_{q}^{(m)}(t) = ∑_{n≥0} a_{n}^{(m)}(q)\frac{t^{n}}{[n]_{q}!} = \frac{[m]_{q}! E_{q}(t)}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)}.

Then, the associated q-Appell polynomials are the generalized q-rencontres polynomials D_{q}^{(m)}(x; t) and the generalized q-arrangement polynomials A_{q}^{(m)}(x; t) defined by

D_{n}^{(m)}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}d_{n−k}^{(m)}(q) x^{k} and A_{n}^{(m)}(q; x) = ∑_{k=0}^{n} \binom{n}{k}_{q}a_{n−k}^{(m)}(q) x^{k}

and with q-exponential generating series

D_{q}^{(m)}(x; t) = ∑_{n≥0} D_{n}^{(m)}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{[m]_{q}! E_{q}(t)^{−1}}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)} E_{q}(xt),

A_{q}^{(m)}(x; t) = ∑_{n≥0} A_{n}^{(m)}(q; x)\frac{t^{n}}{[n]_{q}!} = \frac{[m]_{q}! E_{q}(t)}{(1 − t)(1 − qt) ⋯ (1 − q^{m}t)} E_{q}(xt).

In this way, when q = 1, we recover the generalized derangement numbers d_{n}^{(m)} and the generalized arrangement numbers a_{n}^{(m)}, and the associated Appell polynomials D_{n}^{(m)}(x) and A_{n}^{(m)}(x), [7, 15, 16].

For these generalized polynomials, by identity (12), we have

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}D_{m+n−k}^{(μ)}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}d_{n+m−k}^{(μ)}(q) x^{k},  (31)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}A_{m+n−k}^{(μ)}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}a_{n+m−k}^{(μ)}(q) x^{k}.  (32)

In particular, for x = 1, we have D_{n}^{(μ)}(q; 1) = [μ + n]_{q}!. Hence, the first identity becomes

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}[μ + m + n − k]_{q}! = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}d_{n+m−k}^{(μ)}(q).  (33)

4.6 Generalized q-Bernoulli and q-Euler polynomials

The generalized q-Bernoulli numbers B_{n}^{(ν)}(q) and the generalized q-Bernoulli polynomials B_{n}^{(ν)}(q; x), [1], are defined, respectively, by the q-exponential generating series

B_{q}^{(ν)}(t) = ∑_{n≥0} B_{n}^{(ν)}(q) \frac{t^{n}}{[n]_{q}!} = (\frac{t}{E_{q}(t) − 1})^{ν} ,

B_{q}^{(ν)}(x; t) = ∑_{n≥0} B_{n}^{(ν)}(q; x) \frac{t^{n}}{[n]_{q}!} = (\frac{t}{E_{q}(t) − 1})^{ν} E_{q}(xt).

Similarly, the q-numbers Ẽ_{n}^{(ν)}(q) and the generalized q-Euler polynomials E_{n}^{(ν)}(q; x) are defined, respectively, by the q-exponential generating series

Ẽ_{q}^{(ν)}(t) = ∑_{n≥0} Ẽ_{n}^{(ν)}(q) \frac{t^{n}}{[n]_{q}!} = (\frac{2}{E_{q}(t) + 1})^{ν} ,

E_{q}^{(ν)}(x; t) = ∑_{n≥0} E_{n}^{(ν)}(q; x) \frac{t^{n}}{[n]_{q}!} = (\frac{2}{E_{q}(t) + 1})^{ν} E_{q}(xt).

Hence, by identity (12), we have

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}B_{m+n−k}^{(ν)}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}B_{n+m−k}^{(ν)}(q) x^{k},  (34)

∑_{k=0}^{n} \binom{n}{k}_{q}(−1)^{k}q^{\binom{k}{2}}x^{k}E_{m+n−k}^{(ν)}(q; x) = ∑_{k=0}^{m} \binom{m}{k}_{q}q^{nk}Ẽ_{n+m−k}^{(ν)}(q) x^{k}.  (35)

Clearly, when ν = 1 and q = 1, we have the Bernoulli polynomials B_{n}(x) and the Euler polynomials E_{n}(x), [24, 29]. Moreover, we have the Bernoulli numbers B_{n} = B_{n}(0) and Ẽ_{n} = E_{n}(0) = (2−2^{n+2})\frac{B_{n+1}}{n+1} , the Euler numbers E_{n} = 2^{n}E_{n}(1/2), ([14, p. 49] and \underline{A122045}, \underline{A000364}, \underline{A028296}) and the Springer numbers S_{n} = (−1)^{⌈n/2⌉}4^{n}E_{n}(1/4) ([40] and \underline{A001586}). Hence, the above identities can be rewritten as

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}x^{n−k}B_{m+k}(x) = ∑_{k=0}^{m} \binom{m}{k}B_{n+k} x^{m−k},  (36)

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}x^{n−k}E_{m+k}(x) = ∑_{k=0}^{m} \binom{m}{k}Ẽ_{n+k} x^{m−k}.  (37)

In particular, when x = 1/2, we have B_{n}(1/2) = (2^{1−n} − 1)B_{n} and identities (36) and (37) become

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}(2 − 2^{m+k}) B_{m+k} = ∑_{k=0}^{m} \binom{m}{k}2^{n+k} B_{n+k},  (38)

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}E_{m+k} = ∑_{k=0}^{m} \binom{m}{k}2^{n+k}(2 − 2^{n+k+2}) \frac{B_{n+k+1}}{n + k + 1}.  (39)

Similarly, when x = 1/4, we have B_{n}(1/4) = \frac{1}{2^{n}} (\frac{1}{2^{n−1}} − 1) B_{n} − \frac{1}{4^{n}} nE_{n−1} and identities (36) and (37) become

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}((2 − 2^{m+k}) B_{m+k} − (m + k) E_{m+k−1}) = ∑_{k=0}^{m} \binom{m}{k}4^{n+k} B_{n+k},  (40)

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k+⌈ \frac{m+k}{2} ⌉}S_{m+k} = ∑_{k=0}^{m} \binom{m}{k}4^{n+k}(2 − 2^{n+k+2}) \frac{B_{n+k+1}}{n + k + 1}.  (41)

From identities (38) and (40), we also have

∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}(m + k) E_{m+k−1} = ∑_{k=0}^{m} \binom{m}{k}2^{n+k}(1 − 2^{n+k}) B_{n+k}.  (42)

Finally, when x = 1, we have B_{n}(1) = (−1)^{n}B_{n} and E_{n}(1) = (−1)^{n}E_{n}(0) = (−1)^{n}Ẽ_{n} and identities (36) and (37) become

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}B_{m+k} = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}B_{n+k}  (43)

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}(2 − 2^{m+k+2}) \frac{B_{m+k+1}}{m + k + 1}, = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}(2 − 2^{n+k+2}) \frac{B_{n+k+1}}{n + k + 1}. (44)

The first identity (43) is the symmetric Carlitz identity for the Bernoulli numbers [9]. Such an identity can be proved (and generalized) in several ways [12, 17, 30, 42, 44]. In [25], we proved it by using a general method based on umbral calculus. Here, we proved that it is a simple consequence of the general two-parameter identity for the Appell polynomials (specialized to the Bernoulli polynomials). Similarly, the second identity (44) is the analogue symmetric identity for the numbers Ẽ_{n}.

We conclude by noticing that the previous symmetric identities can be extended to the generalized Bernoulli numbers. Indeed, we have B_{n}^{(ν)}(ν) = (−1)^{n}B_{n}^{(ν)} and E_{n}^{(ν)}(ν) = (−1)^{n}Ẽ_{n}^{(ν)}. So, setting q = 1 and x = ν, we obtain

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}ν^{n−k}B_{m+k}^{(ν)} = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}ν^{m−k}B_{n+k}^{(ν)},  (45)

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}ν^{n−k}Ẽ_{m+k}^{(ν)} = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}ν^{m−k}Ẽ_{n+k}^{(ν)}.  (46)

However, Carlitz’s identity (43) cannot be immediately extended to the q-Bernoulli numbers. To obtain such an extension, we need to consider the q-Bernoulli numbers B_{n}^{∗}(q) of type II defined by the q-exponential generating series

B_{q}^{∗}(t) = ∑_{n≥0} B_{n}^{∗}(q)\frac{t^{n}}{[n]_{q}!} = \frac{t}{E_{q}^{∗}(t) − 1},

where

E_{q}^{∗}(t) = E_{q}(−t)^{−1} = ∑_{n≥0} q^{\binom{n}{2}} \frac{t^{n}}{[n]_{q}!}.

Since the q-exponential generating series of the q-Bernoulli numbers can be written as follows

B_{q}(t) = \frac{t}{E_{q}(t) − 1} = \frac{−t E_{q}(t)^{−1}}{E_{q}(t)^{−1} − 1} = B_{q}^{∗}(−t) E_{q}(t)^{−1} ,

we have B_{q}(1; t) = B_{q}^{∗}(−t), that is, B_{n}(q; 1) = (−1)^{n}B_{n}^{∗}(q). Therefore, for ν = 1 and x = 1, identity (34) can be rewritten as

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}_{q}q^{\binom{n−k}{2}}B_{m+k}^{∗}(q) = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}_{q}q^{n(m−k)}B_{n+k}(q).  (47)

This is the q-analogue of Carlitz’s identity (43), even though it is not symmetric and involves the q-Bernoulli numbers of type I and II. The q-Bernoulli numbers of type II, however, can be expressed in terms of the q-Bernoulli numbers of type I. Indeed, since

B_{q}^{∗}(t) = \frac{t}{E_{q}(−t)^{−1} − 1} = \frac{−t E_{q}(−t)}{E_{q}(−t) − 1} = \frac{−t (1 + E_{q}(−t) − 1)}{E_{q}(−t) − 1} = B_{q}(−t) − t ,

we have B_{n}^{∗}(q) = (−1)^{n}B_{n}(q) − δ_{n,1} = (−1)^{n}(B_{n}(q) + δ_{n,1}). Moreover, by the identity [43]

E_{q}(−t) E_{q^{−1}}(t) = 1 , or E_{q}(−t)^{−1} = E_{q^{−1}}(t) ,

we also have

B_{q}^{∗}(t) = \frac{t}{E_{q}(−t)^{−1} − 1} = \frac{t}{E_{q^{−1}}(t) − 1} = B_{q^{−1}}(t).

Since [n]_{q^{−1}}! = q^{−\binom{n}{k}}[n]_{q}! , we have

B_{q}^{∗}(t) = ∑_{n≥0} B_{n}(q^{−1})\frac{t^{n}}{[n]_{q^{−1}}!} = ∑_{n≥0} q^{\binom{n}{k}}B_{n}(q^{−1})\frac{t^{n}}{[n]_{q}!},

that is,

B_{n}^{∗}(q) = q^{\binom{n}{k}}B_{n}(q^{−1}).

So, replacing this expression in (47), we obtain the identity

(−1)^{n}q^{\binom{n}{2}} ∑_{k=0}^{n} \binom{n}{k}_{q}q^{k(m−n+k)}B_{m+k}(q^{−1}) = (−1)^{m}q^{−\binom{m}{k}} ∑_{k=0}^{m} \binom{m}{k}_{q}q^{n(m−k)}B_{n+k}(q). (48)

In a completely similar way, defining the q-numbers Ẽ_{n}^{∗}(q) of type II by the q-exponential generating series

Ẽ_{q}^{∗}(t) = ∑_{n≥0} Ẽ_{n}^{∗}(q)\frac{t^{n}}{[n]_{q}!} = \frac{2}{E_{q}^{∗}(t) + 1} = \frac{2}{E_{q^{−1}}(t) + 1} = Ẽ_{q^{−1}}(t) ,

we can derive the following identity from identity (35):

(−1)^{n} ∑_{k=0}^{n} \binom{n}{k}_{q}q^{\binom{n−k}{2}}Ẽ_{m+k}^{∗}(q) = (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}_{q}q^{n(m−k)}Ẽ_{n+k}(q).  (49)

Finally, since Ẽ_{n}^{∗}(q) = q^{\binom{n}{k}}Ẽ_{n}(q^{−1}), we also have

(−1)^{n}q^{\binom{n}{2}} ∑_{k=0}^{n} \binom{n}{k}_{q}q^{k(m−n+k)}Ẽ_{m+k}(q^{−1}) = (−1)^{m}q^{−\binom{m}{k}} ∑_{k=0}^{m} \binom{m}{k}_{q}q^{n(m−k)}Ẽ_{n+k}(q). (50)

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2010 Mathematics Subject Classification: Primary 05A19; Secondary 05A30, 05A40, 11B65, 11B68, 33C45, 33D45. Keywords: combinatorial sum, q-binomial sum, q-Appell polynomial, Bernoulli and q-Bernoulli number, Euler number, Springer number, Carlitz’s identity, q-exponential generating series.

(Concerned with sequences \underline{A000166}, \underline{A000364}, \underline{A000522}, \underline{A001586}, \underline{A028296}, and \underline{A122045}.)

Received December 3 2021; revised version received March 7 2023. Published in Journal of Integer Sequences, March 8 2023.

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