Some results on sums of products of Bernoulli polynomials and Euler polynomials

Closed form expressions for Appell polynomials

“…thanks to (22) and Lemma 2. Therefore, the first statement in Theorem 5 follows from Theorem 1, (45), and (46).…”

“…For degenerate Bernoulli polynomials and for Apostol-Euler polynomials, we refer the reader to Wu and Pan [23] and He and Araci [12], respectively. Wang [22], Agoh and Dilcher [5], He [11], and Dilcher and Vignat [9], have obtained convolution identities with the multinomial coefficients replaced by other coefficients. The distinctive feature of Theorem 6 with respect to the aforementioned results is that the right-hand sides in identities (48) and (49) do not contain the Bernoulli polynomials themselves.…”

“…Since the pioneering work by Dilcher [8], many authors have provided different expressions for the sums in (4). See, for instance, Wang [22], Agoh and Dilcher [5], He and Araci [12], Wu and Pan [23], He [11], Dilcher and Vignat [9], and the references therein. As it happens in (5), a common feature of the identities for the sums in (4) usually found in the literature is that the right-hand side of such identities contains the polynomials A(x) themselves.…”

Binomial convolution and transformations of Appell polynomials

“…In his classical result, Dilcher [2] considered the case w 1 = · · · = w m = 1 and obtained an identity involving the products s(m, k)B j (x), where s(m.k) are the Stirling numbers of the first kind. Wang [4] (see also Chu and Zhou [16]) provided identities when at most two of the numbers w j , j = 1, . .…”

Convolutions of the bi-periodic Fibonacci numbers

“…Convolution identities for various types of numbers (or polynomials) have been studied, with or without binomial (or multinomial) coefficients, including Bernoulli, Euler, Genocchi, Cauchy, Stirling and balancing numbers (cf. [1][2][3]6,9,10,15,16,19]). A typical formula is due to Euler, given by…”