A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselve...

For a hypergeometric series ∑ k f(k, a, b, . . . , c) with parameters a, b, . . . , c, Paule has found a variation of Zeilberger’s algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general problem concerning several similar hypergeometric terms f1(k, a, b, . . . , c), f2(k, a, b, . . . , c), . . ., fm(k, a, b, . . . , c). We present an algorithm ...

We formulate a result that states that specific products of two independent solutions of a real three-term recurrence relation will form a basis for the solution space of a four-term linear recurrence relation (thereby extending an old result of Clausen [7] in the continuous case to this discrete setting). We then apply the theory of disconjugate linear recurrence relations to the study of irra...

We obtain new recurrence relations, an explicit formula, and convolution identities for higher-order geometric polynomials. These relations generalize known results polynomials lead to congruences and, in particular, p-Bernoulli numbers.