Generating Functions for Formal Power Series in Noncommuting Variables1

GENERATING FUNCTIONS FOR FORMAL POWER SERIES IN NONCOMMUTING VARIABLES1

K. GOLDBERG

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

,.. A(xi, • ■ ■ , xn)

oo m °o = a0 4- Y Y ¿3 z_, a'i---'mdu ' ' ' >im)xt¡ ■ ■ ■x„™. Id—1 (8¡)GFn„ i—1 ty-=l Now, for each positive integer m and each ($%, • • • , sm) in V„m we define a correspondence Til...,m between .4(xi, • • • , xn) and a generating function in commuting variables Zi, ■ ■ • , zm as follows:

m oo

\¿) -t si' • -sm • A \Xi, ' ' * , Xn) * / . / . asi. . .3m\1l, ' ' ' , îmjZl ' * ' Zm . y=i ¿y=l This correspondence can be thought of as a linear transformation which takes ax£ • • • x*™into as? • • • s„ for all positive ii, ■ ■ • , im, and takes all the other monomials into 0. We shall prove that this correspondence obeys the following rule. Theorem. Letfiix), ■ ■ ■ ,/n(x) be any formal power series with constant terms equal to 0. If A (xi, • • • , x„) is a formal power series in Received by the editors April 11, 1959 and, in revised form, November 13, 1959. 1 The preparation of this paper was supported (in part) by the Office of Naval Research. 988 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GENERATING FUNCTIONS FOR FORMAL POWER SERIES 989

noncommuting variables xi, • • • , x„ and

* »I" ••••• "\,*1> ' ' ' J •K») * «fi< ■-s„(Zl, - * , Zm), then

Tsl---Sm:Aifiixi), ■ ■ ■ ,fn(xn)) -* ASl...Smifslizi), ■ ■ ■ ,/,n(zm)).

This theorem points out a natural method for handling results about the coefficients in a formal power series in noncommuting variables. There are many ways of choosing a set of generating functions for these coefficients. But the method used in the theorem has the distinct advantage of transforming the set of generating functions into a new set in the same way that the related formal power series are transformed into each other. We have restricted ourselves to transformations with a vanishing constant term for reasons of sim- plicity, and also because we wished to keep the result strictly formal. Transformations with nonzero constant terms would involve us in considerations of infinite sums of the coefficients. Following the proof of the theorem we give an example of its application. The proof of the theorem is a simple exercise in manipulating sums. The only sums in which convergence might be a problem are finite. We assume that .4(xi, • • • , x„) has the expansion (1). Denote the coefficient of xh in the ith power of/„(x) by/^:

!/«W)' = Yf**x • h=i

Then A(fiixi), • • • , fn(xn)) has the expansion

oo oo ao+Y Y ]C ]C an ■■•»m(íi>•••>&.) m=i (soevnm y—1 ¿¿=1

ZT-» .('O Aim) *1 Am 2-1 J'lhi ' ' J*mhmX*l ' ' Xsm■ k=l hk=ik

Therefore the coefficient of x*1 ■ ■ • xhsmin Aifiixi), ■ ■ ■ ,/„(«„)) is

"» hi „ , ,. . 2-1 2-1 asi--Sm\lU ' ' ' ! V)/siÄl " ' ' fsmhm. 3=1 iy=l

This coefficient is well defined because the sums have a finite number of terms. It follows that T,,..., takes .4(/i(xi), • ■ • , f„(x„)) into

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 990 K. GOLDBERG [December

» • • *i ... ,. , . . ■V-v ^T-l ^T> V^ /• • w(ll) Slm> *1 *m 2-1 2-1 2-1 2-1 a'\---sm\ll, ■ ■ • , imUsihi - ' - JsmhmZl • • zm k—1 hk—l y=l ij=l

OO 00 = SS aSi...Smih,■ ■ •,ù){/»i(si)h • • ■{/.»(«-)}*"• y-i i,=i This last expression is just ASl...Sm(fn(zi), ■ ■ ■ , /„„(z»»)), as one can see by comparing it with the right-hand expression in (2). This proves the theorem. We can now apply this theorem to the formal power series for log Fix)Giy) with £(0) =G(0) = 1. Since we have only two variables, Vnm has only two elements: (1, 2, 1, 2, • • • ) denoting monomials beginning with a power of x, and (2,1,2,1, • • • ) denoting monomials beginning with a power of y. In order to avoid this cumbersome nota- tion, which is necessary only when we have many variables, we will use the subscript x instead of 1212 • • • and the subscript y instead of 2121 • • • . Let&x(i'i, • • • ,im) denote the coefficient of x¡1yi2 • • -inlog£(x)G(y), with byiii, ■ ■ ■ , im) similarly defined. We want to find an expression for m oo BxiZl, ■ • • ,Zm) = Y Y Ox(Íl, ■ • ■ , im)Zl ■ ■ ■ Zm , y-1 ¿y_l

and for £„(zi, ■ ■ ■ , zm), the generating function of bu(ii, ■ • ■ , im). We shall use the following results from [l]. Let ax(ii, ■ ■ • ,im) and ay(ii, ■ ■ ■ ,im) denote the coefficients in log exey, and let Ax(zi, • • • ,zm) and ¿4y(zi, • • • , zm) denote their generating functions. Theorem 2 of [l ] states that

m (3) Ax(zu ■ ■ ■ , zm) = Y Ziem'ziII (e!i - l)(e2i ~ e'0~l »'—1 j^i and

m (4) Ayizi, ■ ■ ■ ,z,n) = Y z¿em"2'Il (e¡i ~ l)(ß2i ~ «'O'1 i=l jj¿i where m' = [m/2] and m"' = [(m—1)/2], with [/] denoting the great- est integer in t. In order to apply the theorem of this paper we define /(x) and g(y) by (5) £(x) = e'<*>, G(y) = ef™ and note that/(0) =g(0) =0. Then we can transform the generating

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use i960] GENERATING FUNCTIONS FOR FORMAL POWER SERIES 991

functions for the coefficients in log e^e" into the generating functions for the coefficients in log e/(x)eaiy) =log Fix)G(y) directly: (6) Bx(zi, • • • , zm) = Axifizi), g(z2), •••,[/, g](zm)) and (7) £„(Zl, • • • , Zm) = ^„(g(zi),/(z2), • • • , [g,f](zm)) where [f, g]izm) denotes/(zm) if m is odd and gizm) if m is even. Applying (5) and (6) to (3) we get Bx(zi, ■ ■ ■ , zm) m—m' m—m' VÍ7n ■ ,\ — 1 = El*««-.)}-'log F(,1W) n ,, , ., : i=i y-i:iv« tyz2i-i) — r(z2j-i) '"' Giz?) — 1 m' (8) • n m ; „ , + y {Giz2i)\-' log c^ y_l t{Z2i-i) — (j(Z2j) i=i "ff ^O^--1)- 1 fr C(hä- i y-i Giz2l) - £(z2y-i) y-i¡y*< Giz2i) - G(s2y) Applying (5) and (7) to (4) we see that £K(zi, ■ ■ ■ , zm) can be ob- tained from (8) by interchanging £ and G throughout and then re- placing the exponent m' in the first term of each sum by m". Since £(0) = 1 the power series for log £(z) is » (-l)*-i log£(z) = Y—T~(Fiz)-l)k. k-i k Thus, an example of (8) is - (-1)*-1 (£(zi) - 1)"- iGizi) - 1)* £*(zi,z2) = p(2i)(g(z2) -i) Y —r— „,\ '—- k~i k F(zi) - G(z2) - log G(z2), as the reader can verify. The generating function (8) shows that bx(ii, ■ ■ ■ , im) is left fixed by any permutation of the odd indexed i¡ and any permutation of the even indexed i¡. For example, bx(ii, i2, H, u, 4) =0^(4, it, ii, *», is). The same is true of by(ii, ■ ■ ■ , im)- If F(z) =Giz) then any permutation of the i,- leaves the coefficients unchanged. If £(z) =Giz) and m is odd (so that m' = m") then bx(ii, • • • , im)=bviii, • • • , im)- Reference 1. K. Goldberg, The formal power series for log exe", Duke Math. J. vol. 23 (1956) pp. 13-22. National Bureau of Standards

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