SCHUR’S OLD DETERMINANT PROVES A BRAND–NEW THEOREM OF GARRETT–ISMAIL–STANTON

HELMUT PRODINGER

Abstract. Garrett, Ismail, and Stanton gave a general formula that contains the Rogers–Ramanujan identities as special cases. We show how easy this is when using a determinant that Schur introduced in 1917.

In a recent difficult paper [2] Garrett, Ismail, and Stanton prove, amongst many other things, the following generalization of the celebrated Rogers–Ramanujan identities:

\begin{aligned}∑_{n≥0} \frac{q^{n^{2}+mn}}{(1 − q)(1 − q^{2}) … (1 − q^{n})} & = (−1)^{m}q^{−\binom{m}{2}}E_{m−2} ∏_{n≥0} \frac{1}{(1 − q^{5n+1})(1 − q^{5n+4})} \\ & − (−1)^{m}q^{−\binom{m}{2}}D_{m−2} ∏_{n≥0} \frac{1}{(1 − q^{5n+2})(1 − q^{5n+3})}, \end{aligned} (1)

with the Schur polynomials, defined by

D_{m} = D_{m−1} + q^{m}D_{m−2},  D_{0} = 1, D_{1} = 1 + q,
E_{m} = E_{m−1} + q^{m}E_{m−2},  E_{0} = 1, E_{1} = 1.

(Another proof, based on generalized Engel expansions, can be found in [1].)

Schur [3] has computed the limits

D_{∞} = ∏_{n≥0} \frac{1}{(1 − q^{5n+1})(1 − q^{5n+4})}, E_{∞} = ∏_{n≥0} \frac{1}{(1 − q^{5n+2})(1 − q^{5n+3})}.

The aim of this note is to give a simple proof of this result, based on the following determinant of Schur:

Schur(x) := |\begin{matrix}1 & xq^{1+m} & & & & … \\ −1 & 1 & xq^{2+m} & & & … \\ & −1 & 1 & xq^{3+m} & & … \\ & & −1 & 1 & xq^{4+m} & … \\ & & & ... & ... & ... \end{matrix}| .

Expanding the determinant with respect to the first column (“top–recursion”) we get

Schur(x) = Schur(xq) + xq^{1+m} Schur(xq^{2}).

Date: April 2, 2000.

1991 Mathematics Subject Classification. 05A30.

Key words and phrases. Rogers–Ramanujan identities, Schur’s determinant.

Setting

Schur(x) = ∑_{n≥0} a_{n}x^{n},

we get, upon comparing coefficients,

a_{n} = q^{n}a_{n} + q^{1+m}q^{2n−2}a_{n−1},

or

a_{n} = \frac{q^{2n−1+m}}{1 − q^{n}} a_{n−1}.

Since a_{0} = 1, iteration leads to

a_{n} = \frac{q^{n^{2}+mn}}{(1 − q)(1 − q^{2}) … (1 − q^{n})},

and thus the left hand side of (1) can be expressed by Schur(1).

On the other hand, Schur(1) is the limit of the finite determinants

Schur_{n} := |\begin{matrix}1 & q^{1+m} & & & & … \\ −1 & 1 & q^{2+m} & & & … \\ & −1 & 1 & q^{3+m} & & … \\ & & −1 & 1 & q^{4+m} & … \\ ⋮ & ⋮ & ⋮ & ... & ... & ... \\ & & & −1 & 1 & q^{n+m} \\ & & & & −1 & 1 \end{matrix}| .

Expanding this determinant with respect to the last row (“bottom–recursion”) we get

Schur_{n} = Schur_{n−1} +q^{n+m} Schur_{n−2} .

We see that the sequences ⟨D_{n+m}⟩_{n} and ⟨E_{n+m}⟩_{n} satisfy this recursion, and thus any linear combination; set

Schur_{n} = λD_{n+m} + μE_{n+m}.

We can determine the parameters λ and μ using the initial conditions Schur_{0} = 1, Schur_{1} = 1 + q^{1+m}, which leads to

λ = \frac{E_{m} − E_{m−1}}{D_{m−1}E_{m} − D_{m}E_{m−1}} = \frac{q^{m}E_{m−2}}{D_{m−1}E_{m} − D_{m}E_{m−1}}

and

μ = \frac{D_{m} − D_{m−1}}{D_{m}E_{m−1} − D_{m−1}E_{m}} = \frac{q^{m}D_{m−2}}{D_{m}E_{m−1} − D_{m−1}E_{m}} .

We can find the nicer form

λ = \frac{q^{m}E_{m−2}}{D_{m−1}E_{m} − D_{m}E_{m−1}} = (−1)^{m}q^{−\binom{m}{2}}E_{m−2},

which amounts to prove that

D_{m−1}E_{m} − D_{m}E_{m−1} = (−1)^{m}q^{\binom{m+1}{2}}.

Andrews, Knopfmacher, and Paule [1] give a nice formula for D_{i}E_{j} − D_{j}E_{i}. We only need a very particular case of that which can be easily proved by induction on m. The beginning m = 0 is trivial; the induction step goes like this:

D_{m}E_{m+1} − D_{m+1}E_{m} = D_{m}(E_{m} + q^{m+1}E_{m−1}) − (D_{m} + q^{m+1}D_{m−1})E_{m}
= q^{m+1}(D_{m}E_{m−1} − D_{m−1}E_{m})
= −q^{m+1}(−1)^{m}q^{\binom{m+1}{2}} = (−1)^{m+1}q^{\binom{m+2}{2}}.

Analogously we obtain

μ = \frac{q^{m}D_{m−2}}{D_{m}E_{m−1} − D_{m−1}E_{m}} = (−1)^{m+1}q^{−\binom{m}{2}}D_{m−2}.

Hence

Schur_{n} = (−1)^{m}q^{−\binom{m}{2}}E_{m−2}D_{n+m} − (−1)^{m}q^{−\binom{m}{2}}D_{m−2}E_{n+m}.

Performing the limit n → ∞ this turns into

Schur(1) = (−1)^{m}q^{−\binom{m}{2}}E_{m−2}D_{∞} − (−1)^{m}q^{−\binom{m}{2}}D_{m−2}E_{∞},

which is (1).

References

[1] G. Andrews, A. Knopfmacher, and P. Paule. An Infinite Family of Engel Expansions of Rogers-Ramanujan Type. Adv. in Appl. Math., to appear 1999.

[2] K. Garrett, M. Ismail, and D. Stanton, Variants of the Rogers–Ramanujan identities, Adv. in Appl. Math. 23 (1999), 274–299.

[3] I. Schur. Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche. S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl., 1917, 302–321, reprinted in I. Schur, Gesammelte Abhandlungen, vol. 2, pp. 117–136, Springer, 1973.

Helmut Prodinger, The John Knopfmacher Centre for Applicable Analysis and Number Theory, Department of Mathematics, University of the Witwatersrand, P. O. Wits, 2050 Johannesburg, South Africa, email: helmut@gauss.cam.wits.ac.za,

homepage: http://www.wits.ac.za/helmut/index.htm