Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

On Gessel-Kaneko's identity for Bernoulli numbers

“…In 2009, Chen and Sun [2], obtained two identities by applying Zeilberger's algorithm n+3 k=0 n + 3 k (n + k + 3)(n + k + 2) B n+k = 0,…”

Some results for Carlitz’s q-Bernoulli numbers and polynomials

“…which is a q-analogue of a formula on the classical Bernoulli numbers due to Agoh (see, e.g., [1,25]) which can be derived by applying the extended Zeilberger's algorithm (see, e.g., [7]). In particular, the case r = 1 in (2.28) was first discovered by Kaneko [17].…”

Recurrence for values of the zeta function

“…The previous relations derive from the following identity 8) easily proved by using the binomial expansion…”