We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (1998a), Cluzeau and van Hoeij (2005) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek...

In this paper, we investigate the relations between the small functions and the solutions, first, second derivatives, and differential polynomial of the solutions to the differential equation f ′′ +A1e P f ′ +A0e f = 0, where P (z) = anz + · · · + a0, Q(z) = bnz + · · · + b0 are polynomials with degree n (n ≥ 1), ai, bi (i = 0, 1, . . . , n), anbn 6= 0 are complex constants, Aj(z) 6≡ 0 (j = 0, ...

Abstract Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes’ multiple Bernoulli polynomials at q = 1, cf. [1, 13, 14]. By using q-Volkenborn integration, we can also investigate the properties of the reflection symmetries of these’ q-Bernoulli polyno...