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 Diskr. Mat., 1998, Volume 10, Issue 1, Pages 127–140 (Mi dm415)

On combinatorial functions related to the Bürman–Lagrange series. Quasi-orthogonality relations

B. I. Selivanov

Abstract: 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
$$P^{(m)}(n,k)=\frac{(n-1)!}{(k-1)!}\operatorname{Coef}_{t^{n-k}}[t^n g^{-n/m}(t)],\quad n=1,2,…,\quad k=1,\ldots,n,$$
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
$$Q^{(m)}(n,k)=\frac{n!}{k!}\operatorname{Coef}_{t^{n}}[g^{k/m}(t)],\quad n=1,2,…,\quad k=1,\ldots,n,$$
forms a quasi-orthogonal to the set $\{P^{(m)}(n,k)\}$, $n=1,2,…$, $k=1,\ldots,n$. 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.

DOI: https://doi.org/10.4213/dm415

Full text: PDF file (927 kB)

English version:
Discrete Mathematics and Applications, 1998, 8:1, 127–140

Bibliographic databases:

UDC: 519.1

Citation: B. I. Selivanov, “On combinatorial functions related to the Bürman–Lagrange series. Quasi-orthogonality relations”, Diskr. Mat., 10:1 (1998), 127–140; Discrete Math. Appl., 8:1 (1998), 127–140

Citation in format AMSBIB
\Bibitem{Sel98} \by B.~I.~Selivanov \paper On combinatorial functions related to the B\"urman--Lagrange series. Quasi-orthogonality relations \jour Diskr. Mat. \yr 1998 \vol 10 \issue 1 \pages 127--140 \mathnet{http://mi.mathnet.ru/dm415} \crossref{https://doi.org/10.4213/dm415} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1669047} \zmath{https://zbmath.org/?q=an:1001.05006} \transl \jour Discrete Math. Appl. \yr 1998 \vol 8 \issue 1 \pages 127--140