COMBINATORIAL CONVOLUTION SUMS DERIVED FROM DIVISOR FUNCTIONS AND FAULHABER SUMS

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1 GLASNIK MATEMATIČKI Vol. 49(69(014, COMBINATORIAL CONVOLUTION SUMS DERIVED FROM DIVISOR FUNCTIONS AND FAULHABER SUMS Bumkyu Cho, Daeyeoul Kim and Ho Park Dongguk University-Seoul, National Institute for Mathematical Sciences, South Korea Abstract. It is known that certain convolution sums using Liouville identity can be expressed as a combination of divisor functions and Bernoulli numbers. In this article we find seven combinatorial convolution sums derived from divisor functions and Bernoulli numbers. 1. Introduction The symbols N, Z, Q and C denote the set of natural numbers, the ring of integers the field of rational numbers and the field of complex numbers, respectively. The Bernoulli polynomials B k (x, which are usually defined by the exponential generating function te xt e t 1 k0 B k (x tk k!, play an important role in different areas of mathematics including number theory and the theory of finite differences. The Bernoulli polynomials satisfy the following well-known identities : N k B k+1(n +1 B k+1 (0, (k 1 k +1 1 k +1 k ( k +1 ( 1 B N k+1. The Bernoulli numbers B k are defined to be B k : B k ( Mathematics Subect Classification. 33E0, 11A67. Key words and phrases. Divisor functions, convolution sums, Faulhaber s sum. 351

2 35 B. CHO, D. KIM AND H. PARK For n N, k Z 0, and l Q, we define some divisor functions: It is clear that σ k (n : d k, σ k(n : n d odd d k, σ # k (n : ( n d k, ˆσ k,l (n : σ k (n lσ k d N H k (N : k, Ĥ k,l (n : H k (n lh k (n/. ˆσ k,0 (n σ k (n, ˆσ k,1 (n σ k (n σ k(n σ k (n/, ˆσ k, k(n σ k (n, and Ĥ k,l (d 1 k +1 k ( k +1 ( 1 B ˆσ k+1,l (n. The identity n 1 σ(kσ(n k 5 ( 1 1 σ 3(n+ 1 1 n σ(n k1 for the basic convolution sum first appeared in a letter from Besge to Liouville in 186 (see []. For some of the history of the subect, and for a selection of these articles, we mention [3,9,10], and especially [5,11]. The study of convolution sums and their applications is classic and they play an important role in number theory. In this paper we are trying to focus on the combinatorial convolution sums. For positive integers k and n, the combinatorial convolution sum (1.1 k ˆσ k s 1,l (mˆσ,l (n m can be evaluated explicitly in terms of divisor functions and a sum of involving Faulhaber sums. We are motivated by Ramanuan s recursion formula for sums of the product of two Eisenstein series [1, Entry 14, p. 33] and its proof, and also the following propositions.

3 CONVOLUTION SUM 353 Proposition 1.1 ([11]. Let k, n be positive integers. Then k σ k s 1 (mσ (n m k +3 4k + σ k+1(n+ k+1 ( k 6 n σ (n k ( k+1 B σ k+1 (n, where B is the -th Bernoulli number. k r0 where and Proposition 1. ([4]. For any integers k 1 and N 3, we have ( k r [ N 1 ] i1 n 1 σ k r (m;i,nσ r (n m;i,n σ k+1 (n;n N nσ (n;n 1 N 1+( 1N [ N 1 ] (N iσ k (n;i,n i1 ( ( n σk+1 N ; N nσ σ r (n;i,n n d i(n d r ( 1 r ( n N ;, n d i(n σr (n;n d r σ r (n σ r (n/n. n d 0(N The aim of this article is to study seven combinatorial convolution sums of the analogous type (1.1. More precisely, we prove the following theorems and corollaries. Theorem 1.3. Let k,n be positive integers, and let l be a rational number. Then k ˆσ k s 1,l (mˆσ,l (n m (1. 1 ˆσ k+1, l +l(n 1 l ˆσ k,l(n n( l ˆσ,l (n+(1 l Ĥ k,l (d. d r

4 354 B. CHO, D. KIM AND H. PARK Remark 1.4. In Theorem 1.3, l 0 and l 1 recover Proposition 1.1 and Proposition., respectively. For k 1, we have the following corollary. Corollary 1.5. For any n N and l Q, we have n 1 ˆσ 1,l (mˆσ 1,l (n m 1 4ˆσ 3, l +l(n l 6 ˆσ 3,l(n+ 3n(l +1 l ˆσ 1,l (n. 1 Remark 1.6. In the preceding corollary, l recovers [7, (11]. Corollary 1.7. Let k, n be positive integers. Then k σ k s 1 (m/σ (n m 1 σ k+1(n/ 1 4ˆσ k, 1(n n 4ˆσ, (n Ĥ k, 1 (d. Corollary 1.8. Let k, n be positive integers. Then k σk s 1 (mσ (n m 1 σ k+1 (n 1 4 σ k (n 3n 4 ˆσ, (n Ĥ 3 k (d. Theorem 1.9. Let k,n be positive integers and let l be a rational number. Then k ˆσ k s 1,l (mˆσ,l (n m (1.3 1 l 4 ˆσ k+1,l 1(n 1 l ˆσ k,l(n n(1 lˆσ,l/ (n + l(σk+1 (n nσ (n+(1 l k,l (d Ĥ and (1.4 ( k n ˆσ k s 1,l (m 1ˆσ,l (n m+1 ˆσ 1 k+1,l(n+ l(l 1 σ k+1 (n+n(l ˆσ,l (n + l 1 4 ˆσ k+1,l 1(n n(l 1ˆσ,l/ (n l(σk+1(n nσ (n.

5 CONVOLUTION SUM 355 Remark In (1.3, l 0 and l 1 recover Proposition.1 and Proposition., respectively. And in (1.4, l 0 recovers [8, Lemma 6.]. If we insert k 1 in the preceding theorem, one has the following corollary. and n Corollary For n N and l Q we have n 1 ˆσ 1,l (mˆσ 1,l (n m 1 l 8 ˆσ 3,l 1(n l 6 ˆσ 3,l(n n(1 lˆσ 1,l/ (n l 1 ˆσ 1,l(n+l(σ 3 (n nσ 1 (n ˆσ 1,l (m 1ˆσ 1,l (n m+1 1 4ˆσ 3,l(n+ l(l 1 4 σ 3 (n+ n(l ˆσ 1,l (n + l 1 8 ˆσ 3,l 1(n n(l 1ˆσ 1,l/ (n l(σ3(n nσ 1(n. Theorem 1.1. Let k, n be positive integers. Then ( k n 1 k s 1 σ k s 1 (m/4σ (n m 1 4 σ k+1(n/ 1 4 (σ k(n k+1 σ k (n/ k σ k (n/4 n 4 (σ (n+ k σ (n/4 σ# k (n + k k 1 (k +1 1 (k +1 1 (k +1 k where ( k+1 u,v,w (k+1! u!v!w!. ( k +1 B σ k+1 (n/ k ( k +1 ( k +1 u+v+wk+1 B k+1 σ k+1 (n/4 B σ # k+1 (n ( k +1 v 1 B v σ # u,v,w w(n.

6 356 B. CHO, D. KIM AND H. PARK Remark In the preceding theorem, k 1 recovers [11, Theorem 15.].. Proofs of the theorems and corollaries To prove the theorems and corollaries, we need the following propositions. Proposition.1 ([8, Theorem 6.3]. Let k, n be positive integers. Then k σ k s 1 (mσ (n m k +3 4k + σ k+1(n 1 4 σ k+1(n+ k ( k +1 k +1 B σ k+1 (n. ( k 6 n σ (n Proposition. ([6, Identity (10]. Let k, n be positive integers. Then k σ k s 1(mσ (n m 1 (σ k+1(n nσ (n. Proposition.3 ([11, p.17]. Let n be a positive integer. Let f : Z C be an even function. Then (a,b,x,y N 4 4ax+byn (f(a b f(a+b 1 f(0(σ(n/ d(n/ d(n/4 1 df(d / /4 ( 1+ n f(d d ( 1+ n f(d d d l1 l d( f(l d /4 l1 f(l.

7 CONVOLUTION SUM 357 Proof of Theorem 1.3. Let k,n N and l Q. Then the left hand side is (.1 k ˆσ k s 1,l (mˆσ,l (n m ( k n 1 ( ( m σ k s 1 (m lσ k s 1 ( ( n m σ (n m lσ k ( σ k s 1 (mσ (n m ( m ( n m +l σ k σ k ( ( n m l σ k s 1 (mσ ( m +σ k σ (n m k ( ( m l σ k s 1 (m σ k s 1 ( ( n m σ (n m σ k ( +(1 l σ k s 1 (mσ (n m ( m ( n m lσ k s 1 σ. Now the first summation of the right side of (.1 becomes (. k ( ( m σ k s 1 (m σ k s 1 ( ( n m σ (n m σ k ( n m σk s 1 (mσ 1 (σ k+1 (n nσ (n

8 358 B. CHO, D. KIM AND H. PARK by Proposition.. Consider the second summation of the right side of (.1 by the use of Proposition 1.1. It equals (.3 k σ k s 1 (mσ (n m ( k +3 k 4k + σ k+1(n+ 6 n σ (n k +1 k ( k +1 B σ k+1 (n 1 σ k+1(n 1 σ k(n nσ (n k +1 k ( k +1 B σ k+1 (n 1 σ k+1(n 1 σ k(n nσ (n+ H k (d. Using Proposition 1.1, we observe that last summation of the right side of (.1 is (.4 k ( m σ k s 1 ( [ k n ] 1 k +3 n ( k k+1( 4k + σ + 6 n k ( k+1 k+1 1 ( n σ k+1 1 σ k k+1 1 σ k+1 k ( k+1 ( n 1 σ k σ ( n m σ k s 1 (mσ ( n m ( n σ ( n B σ k+1 ( n n ( n σ ( n B σ k+1 ( n n ( n σ + ( d H k.

9 CONVOLUTION SUM 359 Using (., (.3 and (.4, we get k ˆσ k s 1,l (mˆσ,l (n m l (σ k+1 (n nσ (n ( 1 +(1 l σ k+1(n 1 σ k(n nσ (n+ H k (d l(1 l( 1 σ k+1 ( n 1 ( n σ k n σ 1 σ k+1(n l( l ( n σ k+1 1 l + n(l ( ( n σ (n lσ ( n + ( d H k ( ( n σ k (n lσ k (l 1 Ĥ k,l (d 1 ˆσ k+1,l(n l(1 l ( n σ k+1 1 l ˆσ k,l(n n( l ˆσ,l (n+(1 l Ĥ k,l (d. Therefore the proof is completed. Proof of Corollary 1.7. We take l 1 in Theorem 1.3. Then the left hand side of Theorem 1.3 is k L (σ k s 1 (mσ (n m +σ k s 1 (m/σ ((n m/ k + (σ k s 1 (m/σ (n m +σ k s 1 (mσ ((n m/. Observing that replacing m and s by n m and k s respectively in L, k (σ k s 1 (m/σ (n m ( k n 1 m0 σ k s 1 (mσ ((n m/,

10 360 B. CHO, D. KIM AND H. PARK we find k L (σ k s 1 (mσ (n m +σ k s 1 (m/σ ((n m/ k + σ k s 1 (m/σ (n m. Consider the right hand side of Theorem 1.3. It it follows that 1 ˆσ k+1, 1(n+σ k+1 (n/ ˆσ k, 1 (n 3n ˆσ, 1(n+ Ĥ k, 1 (d. By Proposition 1.1 and Theorem 1.3, we deduce that k σ k s 1 (m/σ (n m 1 ˆσ k+1, 1(n+σ k+1 (n/ ˆσ k, 1 (n 3n ˆσ, 1(n + k Ĥ k, 1 (d (σ k s 1 (mσ (n m +σ k s 1 (m/σ ((n m/ 1 ˆσ k+1, 1(n+σ k+1 (n/ ˆσ k, 1 (n 3n ˆσ, 1(n + ( 1 Ĥ k, 1 (d σ k+1(n 1 σ k(n nσ (n+ H k (d σ k+1 ( n 1 ( n σ k n ( n σ + ( d H k σ k+1 (n/ 1 ˆσ k, 1(n n ˆσ, (n+ Ĥ k, 1 (d. This completes the proof. Proof of Corollary 1.8. Let k,n N. Then k σk s 1 (mσ (n m ( k n 1 ˆσ k s 1,1 (mσ (n m

11 CONVOLUTION SUM 361 k σ k s 1 (mσ (n m ( k n 1 σ k s 1 (m/σ (n m. Appealing to Corollary 1.7, we obtain k σk s 1 (mσ (n m 1 σ k+1(n 1 4 σ k(n n σ (n n 4 σ (n Ĥ k (d. Remark.4. Using Corollary 1.7, Corollary 1.8 and Proposition., we deduce k ˆσ k s 1, (mσ (n m for n odd. ( k n 1 σ k s 1 (mσ (n m Proof of Theorem 1.9. Let k,n N and l Q. The left hand side of Theorem 1.9 is equal to k ˆσ k s 1,l (mˆσ,l (n m k (σ k s 1 (mσ (n m +l σ k s 1 (mσ (n m k l (σ k s 1 (mσ (n m +σ k s 1 (mσ (n m k (1 l (σ k s 1 (mσ (n m lσ k s 1 (mσ (n m

12 36 B. CHO, D. KIM AND H. PARK k +l (σ k s 1 (m σ k s 1 (m (σ (n m σ (n m k (1 l (σ k s 1 (mσ (n m lσ k s 1 (mσ (n m k +l σk s 1 (mσ (n m. We can divide it into three parts: (.5 (.6 (.7 k σ k s 1 (mσ (n m, k σ k s 1 (mσ (n m, ( k n 1 σ k s 1 (mσ (n m. By making use of Proposition.1, (.5 can be written as k σ k s 1 (mσ (n m k +3 4k + σ k+1(n 1 ( k 4 σ k+1 (n+ 6 n k +1 k ( k +1 B σ k+1 (n σ (n 1 σ k+1(n 1 4 σ k+1 (n 1 σ k(n nσ (n+ H k (d.

13 CONVOLUTION SUM 363 Also, by Proposition 1.1, (.6 equals k σ k s 1 (mσ (n m ( k +3 k 4k + σ k+1(n+ 6 n σ (n k +1 k ( k +1 B σ k+1 (n 1 σ k+1(n 1 σ k(n nσ (n+ H k (d. By Proposition., (.7 is k σk s 1 (mσ (n m k Therefore, we obtain ( k n 1 (σ k+1 (n nσ (n. σ k s 1 (mσ (n m k ˆσ k s 1,l (mˆσ,l (n m 1 (1 l( σ k+1(n 1 4 σ k+1 (n 1 σ k(n nσ (n+ H k (d ( 1 l(1 l σ k+1(n 1 σ k(n nσ (n+ H k (d + l(σ k+1 (n nσ (n 1 l 4 ˆσ k+1,l 1(n 1 l ˆσ k,l(n n(1 lˆσ,l/ (n ( + l(σk+1 (n nσ (n+(1 l H k (d. H k (d l This proves (1.3 of Theorem 1.9. Using (1. and (1.3, we get (1.4. Therefore, the proof is completed.

14 364 B. CHO, D. KIM AND H. PARK Proof of Theorem 1.1. Following the technique used in [11], we take f(x x k in Proposition.3. Then the left hand side of Proposition.3 is ((a b k (a+b k (a,b,x,y N 4 4ax+byn (a,b,x,y N 4 4ax+byn k ( k ( k ( ( 1 s (a k s b s (a k s b s s k (a,b,x,y N 4 4ax+byn 4a m ((a k s 1 b (a k s 1 b n m b k k s 1 σ k s 1 (m/4σ (n m The right hand side of Proposition.3 is 1 (1+n/dd k 1 d k+1 (1+n/d(d k d x1 x d( / x k d /4x1 x k. /4 Set S 1 1 (1+n/dd k 1 d k+1 (1+n/d(d k S S 3 d x1 x d( d /4x1 x k x k. / /4 By the definition of divisor functions, we see that (.8 S 1 1 σ k+1(n/ (σ k(n+ k σ k (n/4 + n (σ (n+ k σ (n/4.

15 CONVOLUTION SUM 365 Next, we consider (.9 and (.10 (.11 S / d x1 x d( k k +1 k k +1 d d d x k d d/ x1 k x k + d k ( k +1 k +1 k k k +1 d k k 1 k+1 k d ( k +1 k x1(x k + d d (x 1 k d+1 x1 x1x k k d d+1 1 x k x1 B σ k+1 (n/+ k σ k (n/ B σ # k+1 (n+σ# k (n ( k +1 ( d+1 B ( k+1 ( d+1 B 1 k +1 k+1 i0 ( k +1 1 k +1 u+v+wk+1 k+1 B (d+1 k+1 k+1 ( ( k +1 k +1 ( k +1 v 1 B v σ w # u,v,w (n i B d k+1 i Similarly, from (.9 and (.10, we obtain k ( S k k +1 B σ k+1 (n/+ k σ k (n/ k +1 k +1 1 k +1 k ( k +1 u+v+wk+1 B σ # k+1 (n+σ# k (n ( k +1 v 1 B v σ w # u,v,w (n.

16 366 B. CHO, D. KIM AND H. PARK Finally, the equation S 3 is equivalent to (.1 S 3 d 1 /4 x1 k 1 k +1 x k + (d k /4 ( k +1 B k+1 σ k+1 (n/4+ k σ k (n/4. Hence, by (.8, (.11 and (.1, we obtain 1 σ k+1(n/ (σ k(n+ k σ k (n/4 + n (σ (n+ k σ (n/4 k k +1 1 k +1 k +1 1 k +1 k k ( k +1 B σ k+1 (n/ k σ k (n/ ( k +1 u+v+wk+1 k ( k +1 B σ # k+1 (n σ# k (n ( k +1 v 1 B v σ # u,v,w w(n B k+1 σ k+1 (n/4 k σ k (n/4 1 σ k+1(n/ (σ k(n k+1 σ k (n/ k σ k (n/4 + n (σ (n+ k σ (n/4 σ # k (n k k +1 1 k +1 1 k +1 k +1 k k k ( k +1 B σ k+1 (n/ ( k +1 ( k +1 u+v+wk+1 B k+1 σ k+1 (n/4 B σ # k+1 (n ( k +1 v 1 B v σ w # u,v,w (n.

17 CONVOLUTION SUM 367 Acknowledgements. The first author was supported by the Dongguk University Research Fund of 013 and the corresponding author was supported by the National Institute for Mathematical Sciences (NIMS grant funded by the Korean government (B1403. References [1] B. C. Berndt, Ramanuan s notebooks. Part II, Springer, [] Nesge, Extrait d une Lettre de M. Besge à M. Liouville, Journal de mathématiques et appliquées e série 7 (186, 56. [3] N. Cheng and K. S. Williams, Evaluation of some convolution sums involving the sum of divisors functions, Yokohama Math. J. 5 (005, [4] B. Cho, D. Kim, and H. Park, Evaluation of a certain combinatorial convolution sum in higher level cases, J. Math. Anal. Appl. 406 (013, [5] J. G. Huard, Z. M. Ou, B. K. Spearman, and K. S. Williams, Elementary evaluation of certain convolution sums involving divisor functions, in: Number theory for the millennium, II, A K Peters, Natick, 00, [6] D. Kim and A. Bayad, Convolution identities for twisted Eisenstein series and twisted divisor functions, Fixed Point Theory Appl. 013, 013:81, 3 pp. [7] D.Kim,A.Kim,and Y.Li,Convolution sums arising from divisor functions, J.Korean Math. Soc. 50 (013, [8] D. Kim, A. Kim and A. Sankaranarayanan, Bernoulli numbers, convolution sums and congruences of coefficients for certain generating functions, J. Inequal. Appl. 013, 013:5 6 pp. [9] D. B. Lahiri, On Ramanuan s function τ(n and the divisor function σ k (n. I, Bull. Calcutta Math. Soc. 38 (1946, [10] G. Melfi, On some modular identities, in: Number theory, de Gruyter, Berlin, 1998, [11] K. S. Williams, Number theory in the spirit of Liouville, Cambridge University Press, 011. B. Cho Department of Mathematics Dongguk University-Seoul 6 Pil-dong 3-ga Jung-gu Seoul South Korea bam@dongguk.edu D. Kim National Institute for Mathematical Science Yuseong-daero 1689-gil Daeeon South Korea daeyeoul@nims.re.kr H. Park Department of Mathematics Dongguk University-Seoul, 6 Pil-dong 3-ga Jung-gu Seoul South Korea ph140@dongguk.edu Received: