Further (p, q)-identities related to divisor functions

Further (p, q)-identities related to divisor functions Victor J. W. Guo1 and Jiang Zeng2

arXiv:1312.6537v1 [math.CO] 23 Dec 2013

1

Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, People’s Republic of China [email protected], http://math.ecnu.edu.cn/~jwguo 2

Universit´e de Lyon; Universit´e Lyon 1; Institut Camille Jordan, UMR 5208 du CNRS; 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France [email protected], http://math.univ-lyon1.fr/~zeng

Abstract. Using basic hypergeometric functions and partial fraction decomposition we

give a new kind of generalization of identities due to Uchimura, Dilcher, Van Hammer, Prodinger, and Chen-Fu related to divisor functions. An identity relating Lambert series to Eulerian polynomials is also proved. Keywords: divisor functions; Lambert series; Prodinger’s identity; Van Hamme’s identity; Dilcher’s identity; l’Hˆopital’s rule; Carlitz’s q-Eulerian polynomials; partitions. 2000 Mathematics Subject Classifications: 11B65, 30E05

1. Introduction The divisor function σm (n) for a natural number P n ism defined as the sum of the mth powers of the (positive) divisors of n, i.e., σm (n) = d|n d . Throughout this paper, we assume that |q| < 1. The generating function of σm (n) has an explicit Lambert series expansion (see [5]) ∞ X

∞ X nm q n . σm (n)q = 1 − qn n=1 n=1 n

(1.1)

In 1981, Uchimura [23] rediscovered an identity due to Kluyver [18] (see also Dilcher [9]): ∞ X k=1

(−1)

k−1

k+1 ∞ X q( 2 ) qk = , (q; q)k (1 − q k ) k=1 1 − q k

(1.2)

where (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ) for n > 0. Since then, many authors have given different generalizations of (1.2) (see [2, 4, 9, 11–13, 15–17, 20, 21, 23–25, 27]). For example, Van Hamme [25] gave a finite form of (1.2) as follows:   (k+1) n n X X qk q 2 k−1 n = , (1.3) (−1) k k 1 − q 1 − q k q k=1 k=1 1

where the q-binomial coefficients

n

k q

are defined by

  (q; q)n n = . k q (q; q)k (q; q)n−k Uchimura [24] proved that n X

(−1)

k−1

k=1

k+1   −1  n X n q( 2 ) qk k+m , = k k q 1 − q k+m m 1 − q q k=1

m > 0.

(1.4)

Dilcher [9] established the following multiple series generalization of (3.15): n X

(−1)

k=1

k−1

  (k)+km X n q 2 = k q (1 − q k )m 16k ≤···6k 1

m

q k1 +···+km . (1 − q k1 ) · · · (1 − q km ) 6n

(1.5)

Prodinger [20] proved that n X

(−1)

k−1

k=0 k6=m

k+1   X   n q k−m n n q( 2 ) m (m+1 ) 2 = (−1) q , m q k=0 1 − q k−m k q 1 − q k−m

0 6 m 6 n.

(1.6)

k6=m

Using partial fraction decomposition the second author [27, (7)] obtained the following common generalization of Dilcher’s identity (1.5) and of some identities due to Fu and Lascoux [11, 12]: n X k=i

(−1)

k−i

    (k−i)+km   n k q 2 q i (q; q)i−1 (q; q)n qn qi , (1.7) = hm−1 ,..., k q i q (1 − zq k )m (q; q)i (zq; q)n 1 − zq i 1 − zq n

where 1 6 i 6 n and hk (x1 , . . . , xn ) is the k-th homogeneous symmetric polynomial in x1 , . . . , xn defined by X X hk (x1 , . . . , xn ) = xi1 · · · xik = xα1 1 · · · xαnn . 16i1 6···6ik 6n

α1 +···+αn =k

We note that Ismail and Stanton [17, Theorem 2.2] have rediscovered the i = 1 case of (1.7) as well as some other results in [27]. In this paper we shall give a different kind of generalizations of (1.4)–(1.6). Our starting point is an identity of Chen and Fu [8, (3.3)], which corresponds to the (r, x) = (0, 1) case of the following result. Theorem 1.1. Let m, n > 0 and 0 6 r 6 m. Then m X k=0

k+1 k+1 n X (−1)k p( 2 )−rk (−1)k q ( 2 )+rk r =x . k (p; p)k (p; p)m−k (xpk ; q)n+1 (q; q) k (q; q)n−k (xq ; p)m+1 k=0

2

(1.8)

We shall give several consequences and variations of Theorem 1.1 in Section 3. Recently Liu [19, Proposition 4.1] has obtained the following formula, the left-hand side of which specializes to the Lambert series in (1.2) when a = 1: ∞ X n=1

∞

2

X (1 − aq 2n )an q n aq n = . 1 − aq n (1 − q n )(1 − aq n ) n=1

(1.9)

In fact, Agarwal [1] (see also [6]) has obtained the following generalization of (1.9): ∞ X n=0

∞

2

X (1 − xtq 2n )xn tn q n tn = . 1 − xq n (1 − xq n )(1 − tq n ) n=0

Motivated by the identity (1.9), we shall generalize the expansion of the Lambert series (1.1) by using Carlitz’s q-Eulerian polynomials An (t; q). Recall (see [10]) that these q-Eulerian polynomials An (t; q) are defined by ∞ X

tj [j + 1]nq =

j=0

An (t; q) , (t; q)n+1

(1.10)

where [n]q = (1 − q n )/(1 − q) for any positive integer n. A well-known combinatorial interpretation for An (t; q) is given by X tdes σ q maj σ , An (t; q) = σ∈Sn

where Sn is the set of permutations of {1, . . . , n} and X des σ = 1, maj σ = σ(i)>σ(i+1)

X

i.

σ(i)>σ(i+1

The first values of the polynomials An (t; q) are the following: A0 (t; q) = A1 (t; q) = 1; A2 (t; q) = 1 + tq; A3 (t; q) = 1 + 2tq(q + 1) + t2 q 3 . We generalize Liu’s result (1.9) as follows. Theorem 1.2. For m > 0 and |a| 6 1, we have ∞ n X a[n]m p q n=1

1 − aq n

=

∞ 2n n n2 X [n]m p (1 − aq )a q n=1

(1 − q n )(1 − aq n )

+

2 ∞ m  X X [n]m−k an pkn q n +n Ak (q n ; p) m p

k=1

k

n=1

(q n ; p)k+1

.

(1.11)

When p = 1 the polynomials An (t) := An (t; 1) are the Eulerian polynomials (see [10, 22]). For example, we have A1 (t) = 1, A2 (t) = 1 + t, A3 (t) = 1 + 4t + t2 , A4 (t) = 1 + 11t + 11t2 + t3 . Clearly, the above theorem reduces to the following result for p = 1. 3

Theorem 1.3. For m > 0 and |a| 6 1, we have 2 2 ∞ ∞ m  X ∞ X X X m nm−k an q n +n Ak (q n ) nm (1 − aq 2n )an q n anm q n = + . n n )(1 − aq n ) n )k+1 k 1 − aq (1 − q (1 − q n=1 n=1 n=1 k=1

(1.12)

When m = 0, the above theorem reduces to Liu’s formula (1.9). However, our proof for (1.11) is quite different from Liu’s proof for (1.9). Dividing both sides of (1.12) by a and setting a = 0 we obtain ∞ X

m   X q m q 2 Ak (q) n q = + . 1 − q k=1 k (1 − q)k+1 n=1 m n

This is equivalent to the q = 1 case of (1.10) by using the symmetry of the Eulerian polynomials tn−1 An (t−1 ) = An (t) (n > 1) and the recurrence relation A0 (t) = 1,

n−1   X n (t − 1)n−k−1Ak (t). An (t) = k k=0

In the next section we shall give two proofs of Theorem 1.1: the first one uses the q-series theory and the second one uses the finite divided difference theory in [8], but our proof is self-contained. We then derive some new generalizations of (1.4)–(1.6) in the subsequent sections. We will prove Theorem 1.2 in Section 4 with special cases in Section 5. In Section 6 we give a finite analogue of of Uchimura’s connection between partitions and the number of divisors.

2. Two proofs of Theorem 1.1 2.1. First proof Applying the q-binomial theorem [3, (3.3.6)–(3.3.7)]: (x; q)N = 1 = (x; q)N

N   X N j=0 ∞  X j=0

j

(−1)j q (2) xj , j

 N +j−1 j x, j q

4

(2.1)

q

(2.2)

we have m X k=0

k+1 k+1  ∞  m X (−1)k p( 2 )−rk (−1)k p( 2 )−rk X n + j (pk x)j = k j q (p; p)k (p; p)m−k (p x; q)n+1 (p; p)k (p; p)m−k j=0 k=0  m   ∞  X k n+j xj X m (−1)k p(2)+(j−r+1)k = k p j q (p; p)m j=0 k=0  ∞  1 X n+j xj (pj−r+1 ; p)m = j q (p; p)m j=0  ∞  1 X n+r+j xr+j (pj+1; p)m = r+j q (p; p)m j=0    ∞ X n+r+j r+j m + j . (2.3) = x n m p q j=0

Similarly, we have n X k=0

k+1  ∞  (−1)k q ( 2 )+rk xr xr X m + j xj (q j+r+1; q)n , = k j (q; q)k (q; q)n−k (q x; p)m+1 (q; q)n j=0 p

which is equal to the right-hand side of (2.3). This completes the proof.

✷

2.2. Second proof For any function f we define the finite difference operator with respect to the alphabet A = {a1 , a2 , . . .} by f (ai ) − f (ai+1 ) f ∂i = , i > 1. ai − ai+1 By induction on n > 0 it is easy to see that 1 1 ∂1 · · · ∂n = . y − a1 (y − a1 ) · · · (y − an+1 )

(2.4)

We give a direct proof of the following formula, which was derived from a more general result in [8, Corollary 2.2]. Lemma 2.1. Let m, n > 0. For any polynomial f with degree r 6 m, and alphabets A = {a1 , a2 , . . .} and B = {b1 , b2 , . . .} we have f (a1 ) f (b1 ) ∂1 · · · ∂m = − ∂1 · · · ∂n , (b1 − a1 ) · · · (b1 − an+1 ) (a1 − b1 ) · · · (a1 − bm+1 )

(2.5)

where ∂i acts on B and A, respectively, on the left-hand side and the right-hand side of (2.5). 5

Proof. For f (x) = xr and r 6 m we have f (a1 ) f (b1 ) ∂1 · · · ∂m = − . b1 − a1 (a1 − b1 ) · · · (a1 − bm+1 )

(2.6)

Indeed, if r = 0 it reduces to (2.4). Suppose it is true for degree r − 1. Writing br−1 a1 (b11−a1 )

br1 (b1 −a1 )

=

+ b1r−1 and using the fact that b1r−1 ∂1 · · · ∂m = 0 we have b1r−1 br1 ∂1 · · · ∂m = a1 · ∂1 · · · ∂m . b1 − a1 (b1 − a1 )

The result follows then from the hypothesis assumption. Now, applying the divided difference operators ∂1 , . . . , ∂n with respect to A to both sides of (2.6) successively and using (2.4) we obtain the desired result. ✷ We start from the partial-fraction decomposition n+1 n−k n X (−1)k q ( 2 )−( 2 ) x−n 1 = . Qn+1 i (q; q)k (q; q)n−k (z − xq k+1 ) i=1 (z − xq ) k=0

(2.7)

If we take f (x) = xr with 0 6 r 6 m, ai = xpi (i = 1, . . . , m + 1) and bk = q −k (k = 1, . . . , n + 1), then, by (2.7) and (2.6), we have m+1 m−k m X (−1)k p( 2 )−( 2 ) x−m f (b1 ) f (b1 ) ∂1 · · · ∂n = ∂1 · · · ∂n (b1 − a1 ) · · · (b1 − am+1 ) (p; p)k (p; p)m−k (b1 − xpk+1 ) k=0

=

n+2 m+1 m−k m X (−1)k+n p( 2 )−( 2 ) q ( 2 ) x−m (xpk+1 )r

(p; p)k (p; p)m−k (xqpk+1 ; q)n+1

k=0

,

and n+1 k+1 n X (−1)k+n+1 q n−k+( 2 )+( 2 ) f (a1 ) f (a1 ) ∂1 · · · ∂m = ∂1 · · · ∂m (a1 − b1 ) · · · (a1 − bn+1 ) (q; q) k (q; q)n−k (a1 − bk+1 ) k=0

=

k+1 n+2 n X (−1)k+n+1 q ( 2 )+( 2 )+(m−r)(k+1)

(q; q)k (q; q)n−k (xq k+1 p; p)m+1

k=0

.

It follows from (2.5) that m X k=0

k+1 k+1 n X (−1)k p( 2 )−r(k+1) (−1)k q ( 2 )+r(k+1) xr = . k+1 p; p) (p; p)k (p; p)m−k (xqpk+1 ; q)n+1 (q; q) k (q; q)n−k (xq m+1 k=0

By substituting r by m − r and xp by x we get (1.8). 6

3. Consequences and variations of Theorem 1.1 3.1. An infinite version Letting r = 0 in (1.8), we obtain the following symmetric bibasic transformation formula. Corollary 3.1. For m, n > 0, there holds m X k=0

k+1 k+1 n X (−1)k q ( 2 ) (−1)k p( 2 ) = . (p; p)k (p; p)m−k (xpk ; q)n+1 (q; q)k (q; q)n−k (xq k ; p)m+1 k=0

(3.1)

In fact, we have the following infinite form of (3.1). Theorem 3.2. Let |a|, |b|, |p|, |q| < 1. Then ∞ ∞ (b; q)∞ X (q/b; q)k (xaq k ; p)∞ k (a; p)∞ X (p/a; p)k (xbpk ; q)∞ k a = b , (p; p)∞ k=0 (p; p)k (xpk ; q)∞ (q; q)∞ k=0 (q; q)k (xq k ; p)∞

(3.2)

where (a; q)∞ = limn→∞ (a; q)n . Proof. Using the q-binomial theorem [3, Theorem 2.1]: ∞ X (a; q)n n=0

(q; q)n

xn =

(ax; q)∞ , (x; q)∞

we can write the left-hand side of (3.2) as ∞ ∞ ∞ ∞ (a; p)∞ X (p/a; p)k k X (b; q)j k j (a; p)∞ X (b; q)j j X (p/a; p)k a (p x) = x (apj )k (p; p)∞ k=0 (p; p)k (q; q) (p; p) (q; q) (p; p) j ∞ j=0 j k j=0 k=0

= =

∞ (a; p)∞ X (b; q)j (pj+1; p)∞ j x (p; p)∞ j=0 (q; q)j (apj ; p)∞

∞ X (b; q)j (a; p)j j=0

(q; q)j (p; p)j

xj .

Therefore, by symmetry, both sides of (3.2) are equal to (3.3).

(3.3) ✷

When a = pm+1 and b = q n+1 , the identity (3.2) reduces to (3.1). Letting p = q 2 , r = 0, z = q −1 and m, n → ∞ in (1.8), we obtain Corollary 3.3. There holds ∞ X k=0

k

2

(−1) (q; q )k+1q

k(k+1)

=

∞ X q (2k+1)k k=0

∞ (q 2 ; q 2 )∞ X q (2k+1)(k+1) − . (q; q 2)k (q; q 2 )∞ (q 2 ; q 2 )k

7

k=0

Writing (1.8) as m X k=1

=

k+1 k+1 n X (−1)k p( 2 )−rk (−1)k q ( 2 )+rk − (p; p)k (p; p)m−k (pk x; q)n+1 k=1 (q; q)k (q; q)n−k (q k x; p)m+1

xr (p; p)m (xq; q)n − (q; q)n (xp; p)m , (p; p)m (q; q)n (xp; p)m (xq; q)n (1 − x)

(3.4)

and letting x → 1 in (3.4) and applying l’Hˆopital’s rule, we are led to the following result: Corollary 3.4. For m, n > 0 and 0 6 r 6 m, there holds k+1 k+1 n X (−1)k p( 2 )−rk (−1)k q ( 2 )+rk − k (p; p) (q; q)k (q; q)n−k (q k ; p)m+1 k (p; p)m−k (p ; q)n+1 k=1 k=1 ! n m X X qk pk 1 −r + = . − (p; p)m (q; q)n 1 − q k k=1 1 − pk k=1

m X

(3.5)

Note that, by the symmetry of m and n, the identity (3.5) also holds for −n 6 r 6 0. In particular, when m = n and p = q, we have Corollary 3.5. For n > 0 and |r| 6 n, there holds k+1 n X (−1)k−1 q ( 2 )−rk (1 − q 2rk )

k=1

(q; q)k (q; q)n−k

(q k ; q)

n+1

=

r . (q; q)2n

Taking r = 0 in (3.5), we obtain the following bibasic generalization of [15, Corollary 6.4]. Corollary 3.6. For m, n > 0, there holds k+1 k+1 n X (−1)k p( 2 ) (−1)k q ( 2 ) − (p; p)k (p; p)m−k (pk ; q)n+1 (q; q)k (q; q)n−k (q k ; p)m+1 k=1 k=1 ! n m X X 1 qk pk = − . (p; p)m (q; q)n 1 − qk 1 − pk

m X

k=1

(3.6)

k=1

Letting p = q 2 and m = n in (3.6), we have Corollary 3.7. For n > 0, there holds n X k=1

=

k+1 n X (−1)k q k(k+1) (−1)k q ( 2 ) − (q 2 ; q 2 )k (q 2 ; q 2 )n−k (q 2k ; q)n+1 k=1 (q; q)k (q; q)n−k (q k ; q 2 )n+1

n X qk 1 . (q 2 ; q 2 )n (q; q)n k=1 1 − q 2k

(3.7)

8

Letting n tend to ∞ in (3.7), we obtain ∞ X (−1)k (q; q 2 )k q k(k+1)

1 − q 2k

k=1

=

−

∞ X k=1

∞ X k=1

k

q . 1 − q 2k

∞ q (2k+1)(k+1) q (2k+1)k (q 2 ; q 2 )∞ X + (q; q 2)k (1 − q 2k ) (q; q 2 )∞ k=0 (q 2 ; q 2 )k (1 − q 2k+1 )

Note that (see [12, (3.1)]) the coefficient of q n in ∞ X k=1

∞

X q 2k−1 qk = = q + q 2 + 2q 3 + q 4 + 2q 5 + 2q 6 + 2q 7 + q 8 + 3q 9 + · · · 1 − q 2k 1 − q 2k−1 k=1

counts the number of (positive) odd divisors of n.

3.2. Two generalizations of Uchimura’s identity In this section, we give two generalizations of Uchimura’s identity (1.4). Theorem 3.8. For m, n > 0 and 0 6 r 6 m, there holds m X k=1

k+1 k+1 n X (−1)k q ( 2 )+rk (−1)k q ( 2 )−rk r −x (q; q)k (q; q)m−k (xq k ; q)n+1 (q; q)k (q; q)n−k (xq k ; q)m+1 k=1

n

m

X q k (q; q)k−1 1 − xr X q k (q; q)k−1 − + − xr 1−x (xq; q)k (xq; q)k k=1 k=1

1 = (q; q)m (q; q)n

!

.

(3.8)

Proof. When p = q, the identity (1.8) may be rewritten as m X k=1

k+1 k+1 n X (−1)k q ( 2 )−rk (−1)k q ( 2 )+rk r −x (q; q)k (q; q)m−k (xq k ; q)n+1 (q; q)k (q; q)n−k (xq k ; q)m+1 k=1

xr 1 = − . (q; q)n (x; q)m+1 (q; q)m (x; q)n+1

(3.9)

On the other hand, since n X q k (q; q)k−1 k=1

(xq; q)k

we have n X q k (q; q)k−1 k=1

(xq; q)k

=

n X k=1

r

−x

1 1−x



(q; q)k−1 (q; q)k − (xq; q)k−1 (xq; q)k

m X q k (q; q)k−1 k=1

(xq; q)k



1 = 1−x

1 − xr 1 = + 1−x 1−x =





(q; q)n 1− (xq; q)n

xr (q; q)m (q; q)n − (xq; q)m (xq; q)n

(q; q)n 1 − xr xr (q; q)m + − . 1−x (x; q)m+1 (x; q)n+1

Combining (3.9) and (3.10), we complete the proof of Theorem 3.8. 9



,

 (3.10) ✷

Theorem 3.9. For m, n > 0, there holds k+1 k+1 n X (−1)k q mk+( 2 ) (−1)k q nk+( 2 ) − (q; q)k (q; q)m−k (xq k ; q)n+1 k=1 (q; q)k (q; q)n−k (xq k ; q)m+1 k=1 ! n m X q k (q; q)k−1 X q k (q; q)k−1 1 = . − (q; q)m (q; q)n k=1 (xq; q)k (xq; q)k k=1

m X

(3.11)

Proof. Letting v → ∞ and z → 0 in the following identity (see [14, 15]) n X (q/z; q)k (vq m ; q)k (z; q)n−k (xz; q)m k=0

(q; q)k (v; q)k (q; q)n−k (xq k ; q)m+1

m X (q/z; q)k (vq n ; q)k (z; q)m−k (xz; q)n

k

z =

k=0

(q; q)k (v; q)k (q; q)m−k (xq k ; q)n+1

zk ,

we have m X k=1

k+1 k+1 n X (−1)k q nk+( 2 ) (−1)k q mk+( 2 ) − (q; q)k (q; q)m−k (xq k ; q)n+1 (q; q)k (q; q)n−k (xq k ; q)m+1

k=1

1 1 = − . (q; q)n (x; q)m+1 (q; q)m (x; q)n+1

The proof of Theorem 3.9 then follows from the r = 0 case of (3.10).

✷

It is easy to see that, when m = 0 and x = q m , both (3.8) (with r = 0) and (3.11) reduce to (1.4).

3.3. A generalization of Prodinger’s identity In this section, we give a generalization of Prodinger’s identity (1.6). Theorem 3.10. For n > 0 and 0 6 m, r 6 n, there holds   k+1     n n −rk ) ( k−m X q X n q 2   −rm n m (m+1 2 ) (−1)k−1 r + = (−1) q  . k−m k−m k m 1 − q 1 − q q q k=0 k=0

(3.12)

k6=m

k6=m

Proof. From the partial fraction decomposition n X k=0

k+1 xr (−1)k q ( 2 )−rk , = (q; q)k (q; q)n−k (1 − xq k ) (x; q)n+1

which can also be obtained by letting n = 0 and then replacing (m, p) by (n, q) in (1.8),

10

we derive that n X

k=0 k6=m

k+1 (−1)k q ( 2 )−rk (q; q)k (q; q)n−k (1 − xq k )

m+1 xr (−1)m q ( 2 )−rm = − (x; q)n+1 (q; q)m (q; q)n−m (1 − xq m ) m+1 xr (q; q)m (q; q)n−m − (−1)m q ( 2 )−rm (x; q)m (xq m+1 ; q)n−m = . (1 − xq m )(x; q)m (xq m+1 ; q)n−m (q; q)m (q; q)n−m

(3.13)

Letting x → q −m in (3.13) and applying l’Hˆopital’s rule, we obtain n X

k=0 k6=m

=

k+1 (−1)k q ( 2 )−rk (q; q)k (q; q)n−k (1 − q k−m )

rq −rm (q; q)m (q; q)n−m + (−1)m q (

Pn q k−m )−rm (q −m ; q) (q; q) m n−m k=0,k6=m 1−q k−m

m+1 2

−q m (q −m ; q)m (q; q)n−m (q; q)m(q; q)n−m

which is equivalent to (3.12).

, (3.14) ✷

Letting m = 0 in (3.12), we get the following generalization of Van Hamme’s identity (1.3). Corollary 3.11. For n > 0 and 0 6 r 6 n, there holds   (k+1)−rk n n X X q 2 qk k−1 n (−1) = r + . k q 1 − qk 1 − qk k=1

k=1

Letting n → ∞ in (3.15), we obtain ∞ −rk (k+1 X 2 ) q qk k−1 (−1) = r + , (q; q)k (1 − q k ) 1 − qk k=1 k=1

∞ X

r > 0,

which is a generalization of Uchimura’s identity (1.2).

3.4. A new generalization of Dilcher’s identity Here we shall give a new generalization of Dilcher’s identity (1.5).

11

(3.15)

Theorem 3.12. For m, n > 1, and 0 6 r 6 m + n − 1, there holds   (k)+k(m−r) n X q 2 k−1 n (−1) k q (1 − q k )m k=1 X     X  n q k1 r r q k1 +k2 r + = + m − 1 k =1 1 − q k1 m m − 2 16k 6k 6n (1 − q k1 )(1 − q k2 ) 1 1 2   k1 +···+km X q r +···+ . k 0 16k ≤···6k 6n (1 − q 1 ) · · · (1 − q km ) 1

(3.16)

m

Proof. We proceed by induction on m and on n. For m = 1, the identity (3.16) reduces to (3.15). Suppose that (3.16) is true for some m > 1. We need to show that it is also true for m + 1, namely,   (k)+k(m+1−r) n X q 2 k−1 n (−1) k q (1 − q k )m+1 k=1  X    X  n q k1 +k2 q k1 r r r + + = m − 1 16k 6k 6n (1 − q k1 )(1 − q k2 ) m k =1 1 − q k1 m+1 1 2 1   k1 +···+km+1 X q r +···+ , 0 6 r 6 m + n. (3.17) k 0 16k ≤···6k 6n (1 − q 1 ) · · · (1 − q km+1 ) 1

m+1

We shall prove this induction step (3.17) by induction on n, following the proofs in [16] and [9, Theorem 4]. For n = 1, the left-hand side of (3.17) is equal to q m+1−r , (1 − q)m+1 while the right-hand side of (3.17) is given by  k  m+1−r  r m+1 X q r q q m+1−r q 1+ , = = m+1−k 1−q 1−q 1−q (1 − q)m+1 k=0

since 0 6 r 6 m + 1. This proves that (3.17) holds for n = 1. We now assume that (3.17) holds for n − 1. In order to show that it also holds for n, we need to check that the difference between (3.17) for n and (3.17) for n − 1 is a true identity. This difference is !  n (k2)+k(m+1−r) n X n − 1 q − (−1)k−1 k )m+1 k q k (1 − q q k=1   n  n  X q r q k1 q r = + m 1 − qn m − 1 1 − q n 16k 6n 1 − q k1 1   n X q q k1 +···+km r +···+ . (3.18) 0 1 − q n 16k ≤···6k 6n (1 − q k1 ) · · · (1 − q km ) m

1

12

Applying the relation q (2)+k(m+1−r) 1 − qk k

k !      q (2)+k(m−r)+n n n−1 n = − k q k q k q 1 − qn

one sees that (3.18) is just the induction hypothesis (3.16). This proves that (3.17) holds for all n > 1, and consequently (3.16) holds for all m > 1 and all n > 1. ✷ By the q-binomial theorem (2.1), we have   n X k k−1 n q (2 ) = 1 (−1) k q k=1

(n > 1).

Hence, by Dilcher’s identity (1.5), from (3.16) we deduce the following result. Corollary 3.13. For m, n > 1, and 0 6 r 6 m + n − 1, there holds n X

(−1)

k=1

k−1

  (k)+k(m−r) X   (k)+kj X n m  n q 2 q 2 r k−1 n (−1) = . k m k )j k q (1 − q ) k m − j (1 − q q j=0 k=1

4. Proof of Theorem 1.2 Assume that |aq| < 1. As [n + j]p = [n]p + pn [j]p , we have ∞ n X a[n]m p q n=1

1 − aq n

=

X X ak q nk [n]m p n>1

= = =

k>1

X

X

k nk k nk [n]m + [n]m p a q p a q k>n>1 n>k>1 ∞ ∞ ∞ X ∞ XX X n+k n(n+k) n n(n+j) [n]m a q + [n + j]m p p a q n=1 k=0 n=1 j=1 ∞ ∞ ∞ ∞ X X X X n+k n(n+k) m a q + [n] an q n(n+j) [n]m p p n=1 n=1 j=1 k=0   ∞ ∞ m X X X m [j]kp q n(n+j) . an pnk [n]m−k + p k n=1 j=1 k=1

(4.1)

Note that ∞ X k=0

an+k q n(n+k) +

∞ X

2

an q n(n+j) =

j=1

2

an q n an q n +n + 1 − aq n 1 − qn 2

(1 − aq 2n )an q n = , (1 − q n )(1 − aq n ) 13

(4.2)

and, by (1.10), ∞ X

[j]kp q n(n+j)

=q

n2 +n

j=1

∞ X

[j]kp q n(j−1)

=

qn

j=1

2 +n

Ak (q n ; p) . (q n ; p)k+1

(4.3)

Combining (4.1), (4.2) and (4.3), we complete the proof of (1.11).

5. Some special cases of Theorem 1.2 Letting a = ±1 in (1.11), we get the following result. Corollary 5.1. For m > 0, we have ∞ n X [n]m p q n=1

1 − qn

∞ n X [n]m p q n=1

1 + qn

=

∞ n n2 X [n]m p (1 + q )q n=1

=

1 − qn

+

2 ∞ m  X X [n]m−k pkn q n +n Ak (q n ; p) m p

k=1

∞ n n2 X (−1)n−1 [n]m p (1 + q )q n=1

1 − qn

k

(q n ; p)k+1

n=1

+

,

2 ∞ m  X X (−1)n−1 [n]m−k pkn q n +n Ak (q n ; p) m p

k=1

k

n=1

(q n ; p)k+1

For example, 2 2 ∞ ∞ ∞ X X X [n]p q n [n]p (1 + q n )q n pn q n +n = + , n n n )(1 − pq n ) 1 − q 1 − q (1 − q n=1 n=1 n=1

2 2 ∞ ∞ ∞ X X X [n]p q n (−1)n−1 [n]p (1 + q 2n )q n (−1)n−1 pn q n +n = + . 1 + qn 1 − q 2n (1 − q n )(1 − pq n ) n=1 n=1 n=1

Letting p = 1 in Corollary 5.1, we get the following result. Corollary 5.2. For m > 0, we have 2 2 ∞ ∞ m  X ∞ X X X m nm−k q n +n Ak (q n ) nm (1 + q n )q n nm q n = + , k n=1 1 − qn 1 − qn (1 − q n )k+1 n=1 n=1 k=1 2 2 ∞ ∞ m  X ∞ X X X m (−1)n−1 nm−k q n +n Ak (q n ) (−1)n−1 nm (1 + q n )q n nm q n = + . n n n )k+1 k 1 + q 1 − q (1 − q n=1 n=1 n=1 k=1

For example, 2 2 ∞ ∞ ∞ X X X nq n n(1 + q n )q n q n +n = + , 1 − qn 1 − qn (1 − q n )2 n=1 n=1 n=1

2 2 ∞ ∞ ∞ X X X nq n (−1)n−1 n(1 + q 2n )q n (−1)n−1 q n +n = + . n 2n n )2 1 + q 1 − q (1 − q n=1 n=1 n=1

Letting m = 1, 2, 3 in (1.12), we obtain 14

.

Corollary 5.3. We have 2 2 ∞ ∞ ∞ X X X anq n n(1 − aq 2n )an q n an q n +n = + , 1 − aq n (1 − q n )(1 − aq n ) n=1 (1 − q n )2 n=1 n=1

2 2 2 ∞ ∞ ∞ ∞ X X X an2 q n n2 (1 − aq 2n )an q n an nq n +n X (1 + q n )an q n +n = + 2 + , n n )(1 − aq n ) n )2 n )3 1 − aq (1 − q (1 − q (1 − q n=1 n=1 n=1 n=1

2 2 2 ∞ ∞ ∞ ∞ X X X X n3 (1 − aq 2n )an q n an n2 q n +n n(1 + q n )an q n +n an3 q n = +3 +3 1 − aq n (1 − q n )(1 − aq n ) (1 − q n )2 (1 − q n )3 n=1 n=1 n=1 n=1

+

2 ∞ X (1 + 4q n + q 2n )an q n +n

(1 − q n )4

n=1

.

Proposition 5.4. We have  n n(n+k) 2 ∞   ∞   m X ∞  X X X aq n n a q n (1 − aq 2n )an q n n = + . n n n m 1 − aq m (1 − q )(1 − aq ) k=1 n=1 m − k (1 − q n )k+1 n=1 n=1

(5.1)

Proof. Similarly to the proof of Theorem 1.2, applying the Chu-Vandermonde convolution formula    X  m  j n n+j , = k m − k m k=0

and the binomial theorem

∞   X j j=k

k

qj =

qk , (1 − q)k+1

we can prove (5.1).

✷

Letting m = 2, 3 in (5.1), we are led to Corollary 5.5. We have

n 2 2 2 ∞ ∞ ∞ ∞ n n X X X (1 − aq 2n )an q n aq nan q n +n X an q n +2n 2 2 = + + , n n )(1 − aq n ) n )2 n )3 1 − aq (1 − q (1 − q (1 − q n=1 n=1 n=1 n=1
n

2 2 ∞ ∞ ∞ ∞ ∞ n n n n n2 +n 2n n n2 X X X X aq (1 − aq )a q a q nan q n +2n X an q n +3n 3 3 2 = + + + . 1 − aq n (1 − q n )(1 − aq n ) (1 − q n )2 (1 − q n )3 (1 − q n )4 n=1 n=1 n=1 n=1 n=1

6. A refinement of Uchimura’s connection between partitions and the number of divisors Uchimura [24] proved the following identity X k>1

+∞ +∞ Y X qk m (1 − q j ). = mq 1 − q k m=1 j=m+1

15

(6.1)

For n ∈ N let d(n) be the number of positive divisors of n. A partition of n is a non decreasing sequence of positive integers π = (n1 , n2 , . . . , nl ) such that n1 > n2 > · · · > nl > 0 and n = n1 + · · · + nl . The number of parts l is called the length of π. The largest part n1 is denoted by g(π) and the smallest part nl is demoted by s(π). A partition into distinct parts is called odd (resp. even) if its length is odd (resp. even). Let t(n) be the sum of smallest parts of odd partitions of n minus the sum of smallest parts of even partitions of n. Bressoud and Subbarao [7] noticed that Uchimura’s identity is equivalent to d(n) = t(n),

(6.2)

and gave a combinatorial proof of (6.1). It is interesting to note that (6.2) was rediscovered by Wang et al. [26]. Let d(n, N) be the number of divisors 6 N of n. Let P(n, N) be the set of partitions π of n into distinct parts such that g(π) − s(π) 6 N − 1. Let t(n, N) be the sum of smallest parts of odd partitions of P(n, N) minus the sum of smallest parts of even partitions of P(n, N). We have the following refinement of (6.2). Theorem 6.1. For all natural numbers n and N, there holds d(n, N) = t(n, N) − t(n − N, N).

(6.3)

For example, when n = 9 and N = 3, we have P(9, 3) = {(9), (5, 4), (4, 3, 2)};

P(6, 3) = {(6), (4, 2), (3, 2, 1)},

and so t(9, 3) − t(6, 3) = (9 + 2 − 4) − (6 + 1 − 2) = 2, which is the number of divisors of 9 less than or equal to 3. Proof. Rewrite (2.2) as follows: ∞ X 1 (q m+1 ; q)N −1 m m = z q . (zq; q)N (q; q) N −1 m=0

Deriving the above identity with respect to z we obtain N ∞ X X 1 qk m(q m+1 ; q)N −1 m−1 m = z q . (zq; q)N k=1 1 − zq k m=0 (q; q)N −1

Multiplying the two sides by (zq; q)n and setting z = 1 we obtain N X k=1

∞ ∞ X X qk m m+1 = mq (q ; q) − mq m+N (q m+1 ; q)N −1. N −1 1 − q k m=1 m=1

Extracting the coefficient of q n we obtain the desired result. 16

✷

Remark 6.2. It is possible to give a combinatorial proof of (6.3) by generalizing Bressoud and Subarao’s combinatorial proof of (6.2). Acknowledgments. This work was partially supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (grant 11371144), and CMIRA COOPERA 2012 de la R´egion Rhˆone-Alpes.

References [1] R.P. Agarwal, Lambert series and Ramanujan, Proc. Indian Acad. Sci. Math. Sci. 103 (1993), 269–293. [2] M. Ando, A Combinatorial proof of an identity for the divisor generating function, Electron. J. Combin. 20 (2) (2013), #P13. [3] G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998. [4] G.E. Andrews, and K. Uchimura, Identities in combinatorics, IV: Differentiation and harmonic numbers, Utilitas Math. 28 (1985), 265–269. [5] T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997, pp. 24-15. [6] J. Arndt, On computing the generalized Lambert series, preprint, arXiv:1202.6525. [7] D. Bressoud and M. Subbarao, On Uchimura’s connection between partitions and the number of divisors, Canad. Math. Bull. 27 (1984), 143–145. [8] S.H.L. Chen and A.M. Fu, A 2n-Point interpolation formula with its applications to qidentities, Discrete Math. 311 (2011), 1793–1802. [9] K. Dilcher, Some q-series identities related to divisor functions, Discrete Math. 145 (1995), 83–93. [10] D. Foata, Eulerian polynomials: from Euler’s time to the present, in: The legacy of Alladi Ramakrishnan in the mathematical sciences, A. Krishnaswami J.R. Klauder and C.R. Rao, Eds., Springer, New York, 2010, pp. 253–273. [11] A.M. Fu and A. Lascoux, q-Identities from Lagrange and Newton interpolation, Adv. Appl. Math. 31 (2003), 527–531. [12] A.M. Fu and A. Lascoux, q-Identities related to overpartitions and divisor functions, Electron. J. Combin. 12 (2005), #R38. [13] N.S.S. Gu and H. Prodinger, Partial fraction decomposition proofs of some q-series identities, preprint, 2011. [14] V.J.W. Guo and J. Zeng, A combinatorial proof of a symmetric q-Pfaff-Saalsch¨ utz identity, Electron. J. Combin. 12 (2005), #N2. [15] V.J.W. Guo and C. Zhang, Some further q-series identities related to divisor functions, Ramanujan J. 25 (2011), 295–306. [16] M. Hoffman, Solution to problem 6407 (proposed by L. Van Hamme), Amer. Math. Monthly 91 (1984), 315–316.

17

[17] M.E.H. Ismail and D. Stanton, Some combinatorial and analytical identities, Ann. Comb. 16 (2012), 755–771. [18] J.C. Kluyver, Vraagstuk XXXVII. (Solution by S.C. van Veen), Wiskundige Opgaven (1919), 92–93. [19] Z.-G. Liu, A q-series expansion formula and the Askey-Wilson polynomials, Ramanujan J. 30 (2013), 193–210. [20] H. Prodinger, Some applications of the q-Rice formula, Random Structures Algorithms 19 (2001), 552–557. [21] H. Prodinger, q-Identities of Fu and Lascoux proved by the q-Rice formula, Quaest. Math. 27 (2004), 391–395. [22] R.P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, Cambridge, 1997. [23] K. Uchimura, An identity for the divisor generating function arising from sorting theory, J. Combin. Theory Ser. A 31 (1981), 131–135. [24] K. Uchimura, A generalization of identities for the divisor generating function, Utilitas Mathematica 25 (1984), 377–379. [25] L. Van Hamme, Advanced problem 6407, Amer. Math. Monthly 40 (1982), 703–704. [26] Z.B. Wang, R. Fokkink, and W. Fokkink, A relation between partitions and the number of divisors, Amer. Math. Monthly 102 (1995), 345–346. [27] J. Zeng, On some q-identities related to divisor functions, Adv. Appl. Math. 34 (2005), 313–315.

18