Patterns in numbers and infinite sums and products

TextLet aw,B(n)aw,B(n) denote the number of occurrences of the word w in the base B expansion of the non-negative integer n. In this article we generalize the results of Allouche and Shallit [2] by proving the existence of a finite set Lw,BLw,B of pairs (l,cl)(l,cl) where l   is a polynomial with integer coefficients of degree 1 and clcl an integer such that:∑n≥0(−1)aw,B(n)∑(l,cl)∈Lw,Bclf(l(n))={0 if w≠0j,−2⋅(−1)aw,B(0)f(0) if w=0jwhere f is any function that verifies certain convergence conditions.After exponentiating, we recover previous results and obtain new ones such as∏n≥1(3n+13n+2)(−1)n=23,and∏n≥1(9n+79n+8)(−1)a21,3(n)=873.VideoFor a video summary of this paper, please visit https://youtu.be/0wXL7xPkooc.