On combinatorial functions related to the B\"urman--Lagrange series. Quasi-orthogonality relations

B. I. Selivanov

Diskretnaya Matematika, Tome 10 (1998) no. 1, pp. 127-140.

Résumé

Let $$ g(t)=\sum_{n=m}^\infty g_n\frac{t^n}{n!},\quad g_m\ne 0,\quad m\ge 1, $$ be a formal power series (f.p.s.) over the field $K$ of real or complex numbers. In connection with the Bürman–Lagrange series, it is useful to consider the quantities $$ P^{(m)}(n,k)=\frac{(n-1)!}{(k-1)!}\operatorname{Coef}_{t^{n-k}}[t^n g^{-n/m}(t)],\quad n=1,2,\dots,\quad k=1,\ldots,n, $$ which were introduced by the author and for $m=1$ coincide with the $B$-functions introduced by M. L. Platonov. Using Henrici's method, we show that the set of quantities $$ Q^{(m)}(n,k)=\frac{n!}{k!}\operatorname{Coef}_{t^{n}}[g^{k/m}(t)],\quad n=1,2,\dots,\quad k=1,\ldots,n, $$ forms a quasi-orthogonal to the set $\{P^{(m)}(n,k)\}$, $n=1,2,\dots$, $k=1,\ldots,n$. We describe some properties of the coefficients of the series $x^r(t)$, the $r$th power of a f.p.s. $x(t)$ over the field $K$, where $r\in K$. This research was supported by the Russian Foundation for Basic Research, grant 96–01–00531.