Recursive initial value problems for Sheffer sequences
The combinatorial problem addressed in this paper is the enumeration of certain lattice paths. The paths of interest are those which take unit steps in the North and East directions and have weighted left turns (East followed by North). Furthermore, such paths are restricted to lie either entirely above or entirely below a line of positive slope. Each of these latter cases leads to a different recursively defined initial value problem which is solved using tools from finite operator calculus. Although the article is self-contained, the non-expert may wish to consult one or both of the following: \textit{H. Niederhausen} [``Lattice path enumeration and umbral calculus'', in N. Balakrishnan (ed.), Advances in combinatorial methods and applications to probability and statistics, Birkhäuser, Boston, 15-27 (1997; Zbl 0880.05003)] and \textit{G.-C. Rota}, \textit{D. Kahaner} and \textit{A. Odlyzko} [On the foundations of combinatorial theory. VIII: Finite operator calculus, J. Math. Anal. Appl. 42, 684-760 (1973; Zbl 0267.05004)].
scientific article; zbMATH DE number 1331478