Department of Mathematical Sciences; Florida Atlantic University, Boca Raton
SVM HeaderParse 0.2
SVM HeaderParse 0.1
A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n-1])+# 1 (d[m-r 1 ,n-s 1 ]+d[m-s 1 ,n-r 1 ])++# k (d[m-r k ,n-s k ] +d[m-s k ,n-r k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x + u<y<x-l. With a solution we mean a formula that expresses d[m, n] as a sum of di#erences of recursions without the band restriction. Depending on the application, the boundary conditions can take di#erent forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x # 0, and d[x- l, x] = 0 for all x # l (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), d[x + u, x]=0 for all x # 0, and d[x - l +1,x]=d[x-l+1,x-1] for x # l (ordinary lattice paths with weighted left turns), and d[y, y - u +1]=d[y-1,y-u+1]for all y # u and d[x - l +1,x]=d[x-l+1,x-1] for x # l. The first theorem is a gene...