  A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if
the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of
characteristic zero. We establish the structure of holonomic recurrence
equation, i.e. linear and homogeneous recurrence equations having polynomial
coefficients, that have $m$-fold hypergeometric term solutions over $K$, for
any positive integer $m$. Consequently, we describe an algorithm, say
$mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis
of the subspace of hypergeometric $(m=1)$ term solutions of holonomic
recurrence equations to the more general case of $m$-fold hypergeometric terms.
  We generalize the concept of hypergeometric type power series introduced by
Koepf (1992), by considering linear combinations of Laurent-Puiseux series
whose coefficients are $m$-fold hypergeometric terms. Thus thanks to
$mfoldHyper$, we deduce a complete procedure to compute these power series;
indeed, it turns out that every linear combination of power series with
$m$-fold hypergeometric term coefficients, for finitely many values of $m$, is
detected.
  On the other hand, we investigate an algorithm to represent power series of
non-holonomic functions. The algorithm follows the same steps of Koepf's
algorithm, but instead of seeking holonomic differential equations, quadratic
differential equations are computed and the Cauchy product rule is used to
deduce recurrence equations for the power series coefficients. This algorithm
defines a normal function that yields together with enough initial values
normal forms for many power series of non-holonomic functions. Therefore,
non-trivial identities are automatically proved using this approach.
  This paper is accompanied by implementations in the Computer Algebra Systems
(CAS) Maxima 5.44.0 and Maple 2019.
