IOURNAL OF COMBINATORIAL THEORY, series A 56, 309-3 11 (199 1)

Note

On a Conjecture of Krammer

JOHN GREENE *

Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901-4408

Communicated by George Andrew

Received June 27, 1989

In a letter to David Bressoud passed on to George Andrews [ 11, Daan Krammer made the following conjecture:

Ln/2J 2k- 1 1+2 1 k [ 1 ( _ 1 )k q -k(k ~ 1 J/2 =

k=l Y

where q = e2niin. Here

i

n

0 5 ’ if 5{n

1+4cos$ (1)

if 514

0 5 n = i 1, -1, if if n n = 2 1 or or 4 3 (mod (mod 5) 5),

Lx J denotes the greatest integer function and [;I, is the Gaussian q-binomial coefficient defined by

n n [I [I o 4= n q=l,

n [I =(l-q”)(l-q”-‘)(l-q~Pk+‘)

k, (l-q)(l-qZ)...(l-qk) . (2)

By convention, [z] = 0 if k < 0 or k > n. Krammer was led to his conjecture through manipulations of series

related to the Rogers-Ramanujan identities. These manipulations suggested that the left-hand side of (1) should have a nice evaluation and after com-

* Partially supported by NSF Grant DMS 8801131

309 0097-3165/91 $3.00

Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

310 JOHN GREENE

puting this sum for several values of n, he arrived at (1). In this note we show that, in fact, (1) follows from the finite form of one of the Rogers- Ramanujan identities [2, p. 50, Example lo]

qk2= f (-l)kqkUk+lM2 k= -m [ L)(n !5k)il,’ (3)

First, note that

2k-1 il 1 y-q *k-*)...(l-qk)

k 4 (l-q)...(l-qk)

=(-1)kqk(3k-‘)‘2

(l-q-k)...(l-q’-*k)

(l-q),..(l-qk) ’

Since q” = 1, we may write

2k-1 [ 1 k Y =(-l)kqW--lW n;k . [ 1 4

Let S be the left-hand side of (I). By (4),

BY (3)1

s= -1 +2 f (-l)kqk(5k+‘)/2

k=O [Lib If 5k)Jl,’

(4)

But since q”= 1, [z],=O unless k=O or k=n. Setting n= 5m +r with r = 0, + 1, or +2, we now proceed by cases.

If r= f2, Li(n-5k)J can never be 0 or n, so S= -1 in this case. Ifr=l, Lf(n-5k)J=O if k=m. If r= -1, L$(n-5k)J=n if k= -m.

Thus, in these cases

Since q *mn/2 = ( - 1 )“, S = 1 in these cases. Finally, if r=O, then Li(n-5k)J=O if k=m and L$(n-5k)J=n if

k= -m, so

ONACONJECTUREOFKRAMMER 311

as desired.

REFERENCES

1. G. ANDREWS, Private communication. 2. G. ANDREWS, The theory of partitions, in “Encyclopedia of Mathematics and its Applica-

tions,” Vol. 2, Addison-Wesley, Reading, MA 1976.

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