English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In [l] we stated that knowledge of the coefficients in the formal power series for log e^e" with xyp^yx, could be translated into knowledge of the coefficients in log efMe0(v) for any power series /(x) and g(y) with/(0) =g(0) =0. In the following we prove a generalization of this statement, valid for any formal power series in any number of noncommuting variables. Let ^4(xi, • • • , x„) be an arbitrary formal power series in the noncommuting variables Xi, • • • , x„. It can be thought of as an element of the free associative ring generated by xi, • • • , x„ over an arbitrary ring which contains its coefficients. For each positive integer m let Vnm be the set of ordered ra-tuples (si, ■ ■ ■ , sm) with each Si a positive integer no greater than « and no two consecutive Si equal. Let aSl.. .„m (i\, ■ ■ • , im), with each i¡ a positive integer, and (si, ■ ■ • , sm) in Vnm, be the coefficient of xl\ • • • xÍ£ in A (xi, • • • , x„). Thus, if a0 is the constant term, we can write A (xi, ■ ■ ■ ,xn) uniquely as

Generating functions for formal power series in noncommuting variables.