Bounds for the Length of Recurrence Relations for Convolutions of P-recursive Sequences

We derive an upper bound for the minimal length of a linear recurrence satisfied by a sequence with generating function of the formfe111 · · · fer1g,wheref1, . . . , fr,gare D-finite power series,e1, . . . , erare positive integers, and thefjsatisfy differential equations of a certain type. of introducing our ideas, we give a simple proof of Franel’s Conjecture: The sequence (Sr(n))n∈ N ;given bySr(n)=∑nk=0 (n k)rsatisfies a linear recurrence relation of length ⌊ 1/2(r+1)⌋.