A New Identity for Appell Polynomials

Farid Bencherif LA3C, Faculty of Mathematics USTHB Algiers Algeria

                      fbencherif@usthb.dz

Nouara Mokhtari^{1} and Schehrazade Zerroukhat LATN, Faculty of Mathematics USTHB Algiers Algeria

                      nmokhtari1@usthb.dz
                   mokhtari.nouara@gmail.com
                     szerroukhat@usthb.dz

Abstract In the present paper, we generalize an identity for some Appell polynomials, from which we deduce many explicit formulas for generalized Bernoulli and Euler numbers and polynomials.

1 Introduction and main result

The classical and generalized Bernoulli numbers and polynomials as well as the classical and generalized Euler numbers and polynomials have been extensively studied [4, 8]. In the present paper, we attempt to improve some results related to Bernoulli and Euler polynomials and extend a generalization of the Appell polynomials identity published in [2]. Let ℤ, ℕ and ℂ denote the set of integers, non-negative integers and complex numbers respectively. A polynomial sequence (A_{n}(x))_{n∈ℕ} of ℂ[x] is an Appell polynomial sequence [1] if A_{0}(x) is a non-zero constant polynomial and A′_{n}(x) = nA_{n−1}(x) for n ≥ 1. The exponential generating series of the sequence (A_{n}(x))_{n∈ℕ} can be formulated as follows:

1Corresponding author.

∑_{n=0}^{∞} A_{n}(x)\frac{z^{n}}{n!} = S(z)e^{xz},

where S(z) is a formal power series of ℂ[[z]] with non-zero constant term. We focus now on the following theorem which generalizes [2, Theorem 1.1] and adds other properties.

Theorem 1. Let S(z) = ∑_{n=0}^{∞} a_{n} \frac{z^{n}}{n!} be a formal series of ℂ[[z]] with a_{0} = 1. For every α ∈ ℂ

and (A_{n}^{(α)}(x))_{n∈ℕ} consider the Appell polynomial sequence defined by

∑_{n=0}^{∞} A_{n}^{(α)}(x)\frac{z^{n}}{n!} = S^{α}(z)e^{xz}.  (1)

Then for all λ, α ∈ ℂ, ℓ ∈ ℤ, and non-negative integers n, m, with m ≥ n, we have

A_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}A_{n}^{(−kℓ)}(x + λℓ(α + k))  (2)

and for m ≥ ⌊\frac{n}{2}⌋, we have

A_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}A_{n}^{(−kℓ)}(x + a_{1}ℓ(α + k)).  (3)

By using Relation (2) for λ = 0 and ℓ = 1, we obtain [2, Theorem 1.1].

2 Applications

In this section, we give some applications of Theorem 1. Let us first recall that several sequences of remarkable polynomials in ℂ[x] are Appell polynomial sequences. Indeed, it is clear that for α ∈ ℂ, the sequences of generalized Bernoulli polynomials (B_{n}^{(α)}(x))_{n∈ℕ} and generalized Euler polynomials (E_{n}^{(α)}(x))_{n∈ℕ} defined by (see [9, 14])

∑_{n=0}^{∞} B_{n}^{(α)}(x)\frac{z^{n}}{n!} = (\frac{z}{e^{z} − 1})^{α} e^{xz}  (4)

and

∑_{n=0}^{∞} E_{n}^{(α)}(x)\frac{z^{n}}{n!} = (\frac{2}{e^{z} + 1})^{α} e^{xz}  (5)

are Appell polynomial sequences, associated with the formal series (\frac{z}{e^{z}−1})^{α}, (\frac{2}{e^{z}+1})^{α} respectively. The classical Bernoulli polynomials B_{n}(x) and classical Euler polynomials E_{n}(x) are defined by B_{n}(x) = B_{n}^{(1)}(x) and E_{n}(x) = E_{n}^{(1)}(x).

For k ∈ ℕ, from (4) and (5) we deduce the explicit expressions for B_{n}^{(−k)}(x) and E_{n}^{(−k)}(x):

B_{n}^{(−k)}(x) = \frac{1}{k!}\binom{n + k}{k}^{−1} ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}(x + j)^{n+k}

and

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

The Stirling numbers of second kind S(n, k) (see sequence \underline{A008277} in the On-Line Encyclopedia of Integer Sequences (OEIS) [13]) are defined by

∑_{n=0}^{∞} S(n, k)\frac{z^{n}}{n!} = \frac{1}{k!}(e^{z} − 1)^{k}.

The generalized Stirling numbers of second kind S(n, k, x) [3, Eq. (3.9), p. 152] are defined by

∑_{n=0}^{∞} S(n, k, x)\frac{z^{n}}{n!} = \frac{1}{k!}e^{xz}(e^{z} − 1)^{k}.

We have

∑_{n=0}^{∞} S(n, k, x) = \frac{1}{k!} ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}(x + j)^{n}.

and

S(n, k) = S(n, k, 0) = \frac{1}{k!} ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}j^{n}.

by using the generalized Stirling numbers of second kind, we can express B_{n}^{(−k)}(x) as follows:

B_{n}^{(−k)}(x) = \binom{n + k}{k}^{−1}S(n + k, k, x).  (6)

Corollary 2. Let α, λ ∈ ℂ, ℓ ∈ ℤ and n, m ∈ ℕ. For m ≥ n, we have

B_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}B_{n}^{(−kℓ)}(x + λℓ(α + k)),  (7)

and for m ≥ ⌊\frac{n}{2}⌋, we have

B_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}B_{n}^{(−kℓ)}(x − \frac{ℓ}{2}(α + k)).  (8)

Corollary 3. Let α, λ ∈ ℂ, ℓ ∈ ℤ, n, m ∈ ℕ. For m ≥ n, we have

E_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}E_{n}^{(−kℓ)}(x + λℓ(α + k)),

and for m ≥ ⌊\frac{n}{2}⌋, we have

E_{n}^{(αℓ)}(x) = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}E_{n}^{(−kℓ)}(x − \frac{ℓ}{2}(α + k)).  (9)

Corollaries 2 and 3 are deduced from Theorem 1 by choosing successively S(z) = \frac{z}{e^{z}−1} then S(z) = \frac{2}{e^{z}+1} and using Definitions (4) and (5). The generalized Bernoulli numbers B_{n}^{(α)} and Euler numbers E_{n}^{(α)} are defined by B_{n}^{(α)} = B_{n}^{(α)}(0) and E_{n}^{(α)} = 2^{n}E_{n}^{(α)}(\frac{α}{2}). The classical Bernoulli numbers B_{n} and classical Euler numbers E_{n} \underline{A122045} are defined by B_{n} = B_{n}^{(1)} = B_{n}^{(1)}(0) and E_{n} = E_{n}^{(1)} = 2^{n}E_{n}(\frac{1}{2}). We also define the polynomials B̂_{n}^{(α)} by B̂_{n}^{(α)} = B_{n}^{(α)}(α/2) [15, p. 259].

Corollary 4. Let ℓ, n, m ∈ ℕ. For m ≥ n, we have

B_{n}^{(ℓ)}(x) = ∑_{k=0}^{m}(−1)^{k} \begin{matrix}( & {m + 1 \atop k + 1} & ) \\ ( & {n + kℓ \atop n} & ) \end{matrix} S(n + kℓ, kℓ, x).  (10)

This follows by using Relations (6) and (7) for α = 1.

Replacing x = 0 and m = n in Relation (10) we obtain the following relation which can be found in [6, p. 60]:

B_{n}^{(ℓ)} = ∑_{k=0}^{n}(−1)^{k} \begin{matrix}( & {n + 1 \atop k + 1} & ) \\ ( & {n + kℓ \atop n} & ) \end{matrix}S(n + kℓ, kℓ).

Replacing x = 0, m = n, and ℓ = 1 in Relation (10) it follows that

B_{n} = ∑_{k=0}^{n}(−1)^{k} \begin{matrix}( & {n + 1 \atop k + 1} & ) \\ ( & {n + k \atop n} & ) \end{matrix}S(n + k, k).

Note that this formula has been proven by numerous authors: see, for example, [4, Eq. (11), p. 48], [5, Eq. (6), p. 27], [6, p. 59], [7, p. 219 ], [10, Eq. (1.3), p. 91], [12, p. 140].

Corollary 5. Let α ∈ ℂ, ℓ ∈ ℤ and n, m ∈ ℕ. For m ≥ ⌊\frac{n}{2}⌋, the following hold:

B̂_{n}^{(ℓα)} = ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}B̂_{n}^{(−kℓ)},  (11)

and

B̂_{2n}^{(ℓα)} = ∑_{k=0}^{n}(−1)^{k}\binom{α + n}{n − k}\binom{α + k − 1}{k}B̂_{2n}^{(−kℓ)}.

This follows immediately from Relation (8) for x = \frac{1}{2}αℓ. For ℓ = 1 and m = n, Relation (11) can be written as

B̂_{n}^{(α)} = ∑_{k=0}^{n}(−1)^{k}\binom{α + n}{n − k}\binom{α + k − 1}{k}B̂_{n}^{(−k)}.  (12)

Note that (12) is exactly Relation (6.12) of [15, Theorem 6.3].

Corollary 6. For α ∈ ℂ , ℓ ∈ ℕ, n, m ∈ ℕ, with m ≥ ⌊\frac{n}{2}⌋, we have

E_{n}^{(αℓ)} = ∑_{k=0}^{m} \frac{(−1)^{k}}{2^{kℓ}} \binom{α + m}{m − k}\binom{α + k − 1}{k} ∑_{j=0}^{kℓ} \binom{kℓ}{j}(kℓ − 2j)^{n},  (13)

E_{2n}^{(αℓ)} = ∑_{k=0}^{n} \frac{(−1)^{k}}{2^{kℓ}} \binom{α + n}{n − k}\binom{α + k − 1}{k} ∑_{j=0}^{kℓ} \binom{kℓ}{j}(kℓ − 2j)^{2n},

E_{2n}^{(ℓ)} = ∑_{k=0}^{n} \frac{(−1)^{k}}{2^{kℓ}} \binom{n + 1}{k + 1} ∑_{j=0}^{kℓ} \binom{kℓ}{j}(kℓ − 2j)^{2n}.

This follows immediately from Relation (9) for x = \frac{1}{2}αℓ, and by noticing that E_{n}^{(αℓ)} = 0 for n odd. For m = n and ℓ = 1 the relation (13) can be written as follows:

E_{n}^{(α)} = ∑_{k=0}^{n} \frac{(−1)^{k}}{2^{k}} \binom{α + n}{n − k}\binom{α + k − 1}{k} ∑_{j=0}^{k} \binom{k}{j}(k − 2j)^{n}.  (14)

We notice that (14) is nothing else than the relation obtained by Luo [8].

3 Proof of Theorem 1

The following lemma, proved in [2], will be useful for the proof of the main result.

Lemma 7. For m, q ∈ ℕ, we have

1 = ∑_{k=0}^{m}(−1)^{k}\binom{q + m}{m − k}\binom{q + k − 1}{k}x^{q+k} − (−1)^{m}\binom{q + m}{q} ∑_{k=1}^{q} \frac{k}{m + k} \binom{q}{k}(x − 1)^{m+k}.

Let us consider the formal series S(z) = 1+∑_{n=1}^{∞} a_{n} \frac{z^{n}}{n!} . For every α ∈ ℂ, let (A_{n}^{(α)}(x))_{n∈ℕ} be the Appell polynomial sequence defined by (1). For non-negative integers n, m, a given integer ℓ and for every λ ∈ ℂ, we consider the polynomial

P_{λ}(x, α) = A_{n}^{(αℓ)}(x) − ∑_{k=0}^{m}(−1)^{k}\binom{α + m}{m − k}\binom{α + k − 1}{k}A_{n}^{(−kℓ)}(x + λℓ(α + k)).

Let n_{λ} = n if λ ≠ a_{1} and n_{λ} = ⌊\frac{n}{2}⌋ if λ = a_{1}. To prove our theorem it is equivalent to show that P_{λ}(x, α) = 0 for m ≥ n_{λ}. For this, since P_{λ}(x, α) is a polynomial in x and α, this amounts to proving that P_{λ}(x, q) = 0 for every non-negative integer q, provided that m ≥ n_{λ}.

Let D denote the derivation operator of ℂ[x]. In the commutative ℂ-algebra ℂ[[D]] of operators of composition of ℂ[x], consider the operator of translation T_{β} = exp(βD) = ∑_{n=0}^{∞} β^{n} \frac{D^{n}}{n!} for β ∈ ℂ and the automorphism Ω = S(D) = 1 + ∑_{n=1}^{∞} a_{n} \frac{D^{n}}{n!} [11, p. 200], we can write T_{β}(x^{n}) = (x + β)^{n} and Ω^{q}(x^{n}) = A_{n}^{(q)}(x). We remark that for all non-negative integer n, P_{λ}(x, q) = Λ_{λ}(x^{n}) with

Λ_{λ} = Ω^{qℓ} − ∑_{k=0}^{m}(−1)^{k}\binom{q + m}{m − k}\binom{q + k − 1}{k}Ω^{−kℓ} ∘ T_{λℓ(q+k)}.

Then, to get the desired result we must show that Λ_{λ}(x^{n}) = 0 for m ≥ n_{λ}. We can write

Λ_{λ} = Ω^{qℓ} ∘ Ψ_{λ}  (15)

with

Ψ_{λ} = 1 − ∑_{k=0}^{m}(−1)^{k}\binom{q + m}{m − k}\binom{q + k − 1}{k}(Ω^{−ℓ} ∘ T_{λ}^{ℓ})^{q+k}.

Lemma 7 allows us to express Ψ_{λ} as follows:

Ψ_{λ} = (−1)^{m+1}\binom{q + m}{q} ∑_{k=1}^{q} \frac{k}{m + k} \binom{q}{k}(Ω^{−ℓ} ∘ T_{λ}^{ℓ} − 1)^{m+k}.

Let T be a nonzero composition operator. We can write T = ∑_{j≥i} a_{j}D^{j} with a first nonzero coefficient a_{i} ≠ 0. In this case, we say that the composition operator T has order i. Noting then that for every λ, ord(Ω^{−ℓ} ∘ T_{λ}^{ℓ} − 1) ≥ 1, and for the particular case λ = a_{1}, we have ord(Ω^{−ℓ} ∘ T_{λ}^{ℓ} − 1) ≥ 2, we deduce (for k ≥ 1) that ord(Ω^{−ℓ} ∘ T_{λ}^{ℓ} − 1)^{m+k} > m and ord(Ω^{−ℓ} ∘ T_{λ}^{ℓ} − 1)^{m+k} > 2m in the special case where λ = a_{1}. Thus for every integer m ≥ n_{λ}, we have Ψ_{λ}(x^{n}) = 0. Using (15), we obtain the equality Λ_{λ}(x^{n}) = 0 for m ≥ n_{λ}, which completes the proof.

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2010 Mathematics Subject Classification: Primary 11B68; Secondary 05A10. Keywords: Appell polynomial, Bernoulli number, Bernoulli polynomial, Euler number, Euler polynomial.

(Concerned with sequences \underline{A008277} and \underline{A122045}.)

Received June 21 2022; revised versions received June 30 2022; September 16 2022; September 23 2022. Published in Journal of Integer Sequences, October 8 2022.

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