Binomial sums about Bernoulli, Euler and Hermite polynomials

(Received: 11 November 2020. Received in revised form: 14 December 2020. Accepted: 15 December 2020. Published online: 19 December 2020.) c © 2020 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/) Abstract Binomial sums about Bernoulli, Euler and Hermite polynomials are examined by making use of the symmetric summation theorem on polynomial differences, which is due to Chu and Magli [European J. Combin. 28 (2007) 921–930]. Several summation formulae are also obtained, including Barbero’s recent one on Bernoulli polynomials reported in [Comptes Rendus Math. 358 (2020) 41–44].


Introduction and motivation
In classical analysis and combinatorics, the Bernoulli and Euler numbers play an important role, that are defined respectively by τ e τ − 1 = n≥0 B n τ n n! and 2e τ e 2τ + 1 = n≥0 E n τ n n! .
Both Bernoulli and Euler polynomials can be expressed by the corresponding numbers through the binomial relations B n (x) = n k=0 n k B k x n−k and E n (x) = n k=0 n k E k (0)x n−k .
They can be characterized by the following general polynomials associated to an arbitrary sequence {a n } by the binomial sums A n (x) = n k=0 a k n k x n−k for n = 0, 1, 2, · · · .
Chu and Magli [5] found that these polynomials satisfy the following general algebraic identity, which has interesting applications to classical combinatorial numbers and polynomials, such as Bernoulli and Euler polynomials (cf. [9]). Lemma 1.1 (Symmetric Difference). For two variables x, y and three integer parameters m, n, with m, n being nonnegative, the following algebraic identity holds: Here and forth, χ denotes, for brevity, the logical function with χ(true) = 1 and χ(false) = 0, otherwise. For two integers i, j and a natural number m, the notation "i ≡ m j" stands for that "i is congruent to j modulo m".
There exist numerous summation formulae and identities about the Bernoulli and Euler numbers and polynomials (cf. [1,2,4,6,7]). Recently, Barbero [3] discovered a new identity about Bernoulli polynomials. We find that Barbero's identity is an implication of Lemma 1.1 when = 1, m = n and A n (x) is specified to Bernoulli polynomial. This suggests us to examine further applications of Lemma 1.1. In the next section, we shall prove a general theorem about Bernoulli polynomials, which contains Barbero's identity as the special case = 1. Then in Section 3, an analogous theorem for Euler polynomials will be shown, where three interesting formulae corresponding to < 1, = 1 and = 2 will be highlighted. Finally, we illustrate an application to Hermite polynomials in Section 4, where some unusual identities are deduced.

Bernoulli polynomials
In Lemma 1.1, performing first the replacements n → m, y → n − x and then specifying A n (x) to Bernoulli polynomial, we have the equality (cf. [9]) By iterating the recurrence relation we can reformulate the polynomial According to the reciprocal relation we deduce further the expression Substituting this into (2) and then simplifying the resultant equation, we get the identity where Φ (m, n) is a double sum defined by The rightmost fraction can be expressed as a multiple integral with the integration domain x ≤ y −1 ≤ y −2 ≤ · · · ≤ y 2 ≤ y 1 ≤ i and then reformulated by reversing the integral order as According to the binomial theorem, we get the expression Under the change of variable by Expanding the binomial in the braces "{· · · }" and then evaluating the beta integral by we find the following expression for the afore-displayed integral: By substituting this into (5), we get another double sum expression where Ω n (λ, µ) denotes the convolution of arithmetic progressions: Summing up, we have established the following theorem.
Theorem 2.1. For any variable x and three integer parameters m, n, with m, n being nonnegative, the following algebraic identity holds: When < 1, Theorem 2.1 gives a simpler identity.
When = 1, the double sum Φ 1 (m, n) reduces to a single term in view of (6). In this case, we recover from Theorem 2.1 the following identity. .
It is obvious that the formula due to Barbero [3, Theorem 1] is equivalent to Corollary 2.2 under the replacement x → n−y 2 . However, our formula looks more elegant. When = 2, we find from Theorem 2.1, by taking into account that the following unusual double sum evaluation.

Euler polynomials
Analogously, making first the replacements n → m, y → n−x and then specifying A n (x) to Euler polynomial in Lemma 1.1, we have another equality (cf. [9]) By iterating the recurrence relation we can reformulate the polynomial According to the reciprocal relation we deduce further the expression Substituting this into (7) and then simplifying the resultant equation, we get the following counterpart identity of that in Theorem 2.1 for Euler polynomials.
Theorem 3.1. For any variable x and three integer parameters m, n, with m, n being nonnegative, the following algebraic identity holds: where Ψ (m, n) is a double sum defined by By carrying out exactly the same procedure as that from (4) to (6), we can write Ψ (m, n) in terms of a multiple integral and then derive the following alternative expression whereΩ n (λ, µ) stands for the alternating convolution of arithmetic progressions: Theorem 3.1 contains the following three interesting special cases.