Quantum dilogarithm. z n n 2 as q 1. n=1

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1 Quantum dilogarithm Wadim Zudilin (Moscow & Bonn) 14 March 2006 Abstract The q-exponential function e q (z) 1/ (1 zqn ), defined for q < 1 and z < 1, satisfies the remarkable quantum pentagonal identity e q (X)e q (Y ) e q (Y )e q ( Y X)e q (X), where e q (X), e q (Y ) and e q ( Y X) are elements in the algebra C q [[X, Y ]] of formal power series in two elements X, Y linked by the commutation relation XY qy X. Following R. Kashaev we prove that the pentagonal identity is a consequence of the q-binomial theorem and, on the basis of the asymptotics log e q (z) 1 log q n1 z n n 2 as q 1 (already known to S. Ramanujan), it implies the famous Spence Abel 5-term identity for the ordinary dilogarithm (which statement is also nice but more cumbersome). The above asymptotics and implication allow one to attribute e q (z) as a quantum dilogarithm. Let q C satisfy q < 1. Throughout the talk we will use the standard notation (z; q) n (1 z)(1 zq) (1 zq n 1 ) if n 1, 2,...,, (z; q) 0 1 for the q-pochhammer symbol. Note that the q-polynomials [n] q! (1 q) n n k1 1 q k 1 q A talk at Seminar on Algebra, Geometry and Physics (Max Planck Institute for Mathematics in Bonn, March 14, 2006). 1

2 may be viewed as q-factorials since [n] q! n! as q 1. For z C, z < 1, define the q-exponential function e(z) e q (z) 1 (z; q) 1 (1 zqn ). (1) The similarity with the classical exponential function comes from the expansion z n (1 q) n z n e q (z), (2) [n] q! which will be shown below. In addition, this function satisfies the exponential functional identity [5] e(x + Y ) e(x)e(y ), if e(x) e q (X), e(y ) e q (Y ) and e(x + Y ) e q (X + Y ) are viewed as elements in the algebra C q [[X, Y ]] of formal power series in two elements X, Y linked by the commutation relation XY qy X. On the other hand, from (1) we have the asymptotic behaviour log e(z) ( log(1 q n z) ) 1 1 q m1 m1 q mn z m m m1 z m m(1 q m )/(1 q) 1 log q m1 z m m(1 q m ) z m m 2 as q 1 (already mentioned by S. Ramanujan [2]), since (1 q m )/(1 q) m and log q q 1 as q 1. This allows to think of log e(z) (and, thus, of e(z) itself) as of a q-analogue of the dilogarithm function Li 2 (z) n1 z n n 2. Our aim is to show that this q-analogy is somehow deeper than just the above asymptotics. The description below follows R. Kashaev s preprint [3]. First, we prove Theorem 1 (q-binomial theorem). For complex x and y, where y < 1, the following is valid: (x; q) n y n (xy; q). (4) (y; q) 2 (3)

3 Proof. Denote the sum in the left-hand side of (4) by f(x, y). Then manipulations with the series give and implying f(x, y) f(qx, y) xyf(qx, y) f(x, y) f(x, qy) (1 x)yf(qx, y) f(x, qy) f(x, y) 1 y. Finally, n-fold iteration of the latter relation results in n 1 f(x, q n 1 yq k y) f(x, y) q ; k letting now n tend to we deduce that k0 f(x, 0) f(x, y) (y; q) (xy; q). It remains to use f(x, 0) 1 and write the last relation in the required form (4). Specializing x 0, y z in (4) we obtain the promised expansion (2); another specialization x z/y, y 0 gives the expansion Using the formula 1 e q (z) ( 1) n q n(n 1)/2 z n. (5) (z; q) n (z; q) e(zqn ) (zq n ; q) e(z) for n 0, 1, 2,..., we may state the q-binomial theorem by means of the function (1): e(xq n ) e(q n+1 ) yn e(x)e(y) e(q)e(xy). (6) 3

4 Theorem 2. The q-binomial theorem (6) is equivalent to the quantum pentagonal identity e(x)e(y ) e(y )e( Y X)e(X), (7) where e(x) e q (X), e(y ) e q (Y ) and e( Y X) e q ( Y X) are elements in the algebra C q [[X, Y ]] of formal power series in two elements X, Y linked by the commutation relation XY qy X. Proof. We start by mentioning that XY qy X implies X m Y n q mn Y n X m and (Y X) n q n(n 1)/2 Y n X n for n, m 0, 1, 2,.... Therefore, from (2) and e(x)e(y ) m0 m, e(y )e( Y X)e(X) X m (q; q) m Y n q mn (q; q) m Y n X m n,k,m0 m,n,k0 m, Y n k0 ( Y X) k (q; q) k m, m0 ( 1) k Y n (Y X) k X m (q; q) k (q; q) m X m Y n (q; q) m X m (q; q) m (q; q) m (q; q) k Y n+k X m+k min{m,n} k0 (q; q) m k k (q; q) k Y n X m. Comparing corresponding coefficients in the obtained expansions we conclude that identity (7) is equivalent to the series of equalities q mn min{m,n} (q; q) m k0 (q; q) m k k (q; q) k, m, n 0, 1, 2,.... (8) On the other hand, taking complex variables x, y inside the unit disc and applying (2), (5) we see that e(xq n y n ) m, 4 q mn x m y n (q; q) m

5 and e(x)e(y)e(xy) 1 m0 m,n,k0 m, x m (q; q) m y n (xy) k k0 (q; q) k (q; q) m (q; q) k x m+k y n+k min{m,n} x m y n k0 (q; q) m k k (q; q) k. Thus (8) is equivalent to e(xq n ) y n e(x)e(y) e(xy), which is exactly the q-binomial theorem (6). Theorem 3. The limiting case q 1 of the q-binomial theorem (4) (or, equivalently, of the quantum pentagonal identity (7)) is the equality ( ) ( ) ( ) x y xy Li 2 (x) + Li 2 (y) Li 2 + Li 2 Li 2 1 y 1 x (1 x)(1 y) log(1 x) log(1 y), 0 < x < 1, 0 < y < 1. (9) Remark. Formula (9) is due to N. Abel [1] but an equivalent formula was published by W. Spence [6] nearly twenty years earlier. Another equivalent form of (9) (see (21) below) was given by L. Rogers [4]. Proof. Without loss of generality assume that q is sufficiently close to 1, namely, that max{x, y, 1 y(1 x)} < q < 1. (10) The easy part of the theorem is the asymptotics of the right-hand side of (4): log (xy; q) (y; q) log e(y) e(xy) 1 ( Li2 (xy) Li 2 (y) ) as q 1, (11) log q which is obtained on the basis of (3). 5

6 For the left-hand side of (4), write Then the sequence (x; q) n y n c n, where c n (x; q) n y n > 0. (12) d n c n+1 c n 1 xqn y > 0, n 0, 1, 2,..., (13) 1 qn+1 satisfies d n+1 (1 xqn+1 )(1 q n+1 ) d n (1 xq n )(1 q n+2 ) 1 qn (1 q)(q x) (1 xq n )(1 q n+2 ) < 1 q n (1 q)(q x), n 0, 1, 2,... (14) (we use 0 < x < q < 1), i.e. is strictly decreasing. On the other hand, 1 y(1 x) < q implies while d 0 c 1 c 0 1 x 1 q y > 1, lim d 1 xq n n lim y y < 1; n n 1 qn+1 thus, there exists the unique index N 1 such that d N 1 c N c N 1 1 and d N c N+1 c N < 1. (15) Solving the inequality c n+1 /c n < 1 or, equivalently, (1 xq n )y < 1 q n+1 we obtain n > T, where T 1 log q log 1 y q xy, (16) hence N [T ]. From (13) (15) we conclude that c N is the main term contributing the sum in (12), namely, 1 < c n c N < const. 6

7 This result implies log Note now that from (16) ( ) e(q)e(xq N ) c n log c N log e(x)e(q N+1 ) yn ( ) e(q)e(xq T ) log e(x)e(q T +1 ) yt as q 1. (17) q T 1 y q xy, whence the asymptotics in (17) may be continued as follows: log ( c n log e(q) + log e x 1 y ) ( log e(x) log e q 1 y ) q xy q xy + log y log q log 1 y q xy 1 ( ( 1 y Li 2 (x) + Li 2 log q ) + log y log 1 y ) ( Li 2 (1) Li 2 x 1 y ) as q 1, (18) where (3) is used. Comparing asymptotics (11) and (18) of the both sides of (4) we arrive at the identity ( ) ( 1 y Li 2 (x) + Li 2 Li 2 (1) Li 2 x 1 y ) + log y log 1 y Li 2 (xy) Li 2 (y). (19) Take x 0 in (19) to get Li 2 (y) + Li 2 (1 y) Li 2 (1) + log y log(1 y) 0. This identity, in particular, implies Li 2 ( 1 y ) Li 2 (1) Li 2 ( 1 1 y ) log 1 y ( log 1 1 y ). (20) 7

8 Substituting (20) into (19) results in ( ) ( ) x(1 y) y(1 x) Li 2 (xy) + Li 2 + Li 2 + log 1 y log 1 x Li 2 (x) + Li 2 (y). (21) Finally, changing variable x x(1 y)/(), ỹ y(1 x)/(), hence 1 x 1 x, 1 ỹ 1 y, x x 1 ỹ, y ỹ 1 x, reduces identity (21) to the required form (9). I would like to end the exposition by noting that any q-hypergeometric summation or transformation identity produces an identity for the dilogarithm function, e.g., Heine s summation (x; q) n (y; q) n (xyz; q) n z n (xz; q) (xy; q) (z; q) (xyz; q) results in a lengthy 3-variable formula for Li 2 (too cumbersome to be stated here). But finding an appropriate non-commutative analogue (like the pentagonal identity for the q-binomial theorem) seems to be always tricky. References [1] N. H. Abel, Note sur la fonction ψx x + x2 + x xn +, n 2 Œuvres complèts, vol. 2, Imprimerie de Grondahl & Son, Christiania (1881), pp [2] B. C. Berndt, Ramanujan s notebooks. Part IV, Springer-Verlag, New York (1994). [3] R. M. Kashaev, The q-binomial formula and the Rogers dilogarithm identity, E-print math.qa/ (2004). [4] L. J. Rogers, On function sum theorems connected with the series n1 xn /n 2, Proc. London Math. Soc. 4 (1907), [5] M.-P. Schützenberger, Une interprétation de certaines solutions de l equation fonctionelle: F (x + y) F (x)f (y), C. R. Acad. Sci. Paris 236 (1953),

9 [6] W. Spence, An essay on logarithmic transcendents, London and Edinburgh (1809). 9

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