Solving linear recurrences using functionals

Abstract

Many counting problems (especially those related to lattice path counting) give rise to recurrence relations. Several techniques are available for solving recurrence relations (see, e.g., [3]). A technique not mentioned in [3] is the umbral calculus. The main contribution in this respect is due to Niederhausen [4, 5, 6, 7, 8, 9, 10]. Niederhausen finds polynomial solutions to recurrence relations. In fact, if we have a recurrence relation in two variables, then his approach interprets one variable as the index of a polynomial sequence and the other variable as the argument of a polynomial. Moreover, he assumes that the solutions can be represented as so-called Sheffer sequences (for definitions, see Sect. 2). This assumption is true for a large class of recurrences. The umbral calculus provides several explicit formulas for calculations with these Sheffer sequences. These formulas are expressed in terms of linear operators on the vector space of polynomials. The recurrence relation is then translated into an equation involving the so-called delta operator of the Sheffer sequence. If this equation can be solved explicitly, then the basic sequence can be calculated by the transfer formula from umbral calculus (which, in some sense, is equivalent to Lagrange inversion). The deep part of the work of Niederhausen is that he has extended this basic scheme to allow for certain classes of initial values (see references mentioned above). Related to the work of Niederhausen are two papers by Watanabe [18, 19]. In particular, [18] contains a multivariate extension of the Niederhausen approach in case of the ballot problem.