A representation theorem for (q-)holonomic sequences

Chomsky and Schützenberger showed in 1963 that the sequence dL(n), which counts the number of words of a given length n in a regular language L, satisfies a linear recurrence relation with constant coefficients for n, i.e., it is C-finite. It follows that every sequence s(n) which satisfies a linear recurrence relation with constant coefficients can be represented as dL1 (n)− dL2 (n) for two regular languages. We view this as a representation theorem for C-finite sequences. Holonomic or P-recursive sequences are sequences which satisfy a linear recurrence relation with polynomial coefficients. q-holonomic sequences are the q-analog of holonomic sequences. In this paper we prove representation theorems of holonomic and q-holonomic sequences based on position specific weights on words, and for holonomic sequences, without using weights, based on sparse regular languages.