ADDITIONAL ANALYSIS OF BINOMIAL RECURRENCE COEFFICIENTS

This paper involves an investigation of $ig(f(n)ig)_{n=1}^{infty},$where $f(n)$ is defined by$$ f(n+1)=sum_{k=1}^n i{n}{k}f(k),qquad nge 1. leqno(0.1)$$Through successive iterations of (0.1), it is shown that$$ f(n+r)=sum_{k=1}^n f(k)sum_{j=0}^{r-1}A_j^r(n)i{n+j}{k}, qquad rge 1,,nge 1.leqno(0.2)$$The $A_j^r(n)$ of (0.2) are the {it binomial recurrence coefficients.} The main result of this paper is arecurrence formula for the $A_j^r(n),$ namely,$$ sum_{j=k}^{r-1}i{j}{k}A_j^r=A_{k-1}^r,leqno(0.3)$$where $A_j^requiv A_j^r(0).$ This paper then provides two applications involving (0.3). The firstinvolves series inversion while the second involves polynomials whose general term has theform $A_j^rx^j.$