Given the formula (5.1), we first check whether the terms t(c, n, k) are rational functions in k, in which case the sum is of type I. Next, if this is not the case, we extract factors of the form of harmonic numbers H(m)c,k or the form (where we assume 0 = y = ±d)
[x + yk, d]k,
to see if the types III-V can apply. Then we take the quotients of the rest of the terms, and if they are rational functions in k, we find the roots of the numerator and denominator. Now we know the possible type, II-V.