On a Discrete Version of a Theorem of Clausen and Its Applications

We formulate a result that states that specific products of two independent solutions of a real three-term recurrence relation will form a basis for the solution space of a four-term linear recurrence relation (thereby extending an old result of Clausen [7] in the continuous case to this discrete setting). We then apply the theory of disconjugate linear recurrence relations to the study of irrational quantities. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that there exists a four-term linear recurrence relation whose solutions allow us to show that the number is a quadratic irrational if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher order recurrence relations and a transcendence criterion can also be formulated in terms of these principal solutions. The method also generates new accelerated series expansions of ζ(3), ζ(3), ζ(3) and ζ(3) in terms of Apéry’s now classic sequences.