RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7. D. Ranganatha

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1 Indian J. Pure Appl. Math., 83: 9-65, September 07 c Indian National Science Academy DOI: 0.007/s RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 D. Ranganatha Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru , Karnataka, India Siddaganga Institute of Technology, B. H. Road, Tumakuru 57 03, Karnataka, India ddranganatha@gmail.com Received 5 January 07; accepted 8 March 07 Let b l n denote the number of l-regular partitions of n. In 0, using the theory of modular forms, Furcy Penniston presented several infinite families of congruences modulo 3 for some values of l. In particular, they showed that for α, n 0, b 5 3 α+3 n + 3 α+ 0 mod 3. Most recently, congruences modulo powers of 5 for c 5 n was proved by Wang, where c N n counts the number of bipartitions λ, λ of n such that each part of λ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b 5 n, B 5 n, c 5 n modulo powers of 7 for c 9 n. For example, we prove that for j, c 5 5 j n + 5j mod 5 j+, c 9 7 j n + 7j mod 7 j+ b 5 3 α+3 5 j n + 3 α+ 5 j 0 mod 3 5 j. Key words : Congruences; bipartitions; l-regular partitions.. INTRODUCTION Let pn denote the number of unrestricted partitions of the positive integer n. In [7], Ramanujan presented the following generating functions for p5n + p7n + 5. p5n + q n 5 q5 ; q 5 5 q; q 6.

2 50 D. RANGANATHA p7n + 5q n 7 q7 ; q 7 3 q; q + 9q q7 ; q 7 7 q; q 8,. where a; q : It follow from.. that aq k, q <. k0 p5n + 0 mod 5 p7n mod 7. Ramanujan [7] further conjectured that for any integer j, p 5 j n + δ 5 j p 7 j n + δ 7 j 0 mod 5 j,.3 0 mod 7 j,. where 0 < δ l j < l j satisfies the congruence δ l j mod l j. Ramanujan gave a brief sketch of a proof of.3 in an unpublished manuscript [8]. Ramanujan s conjecture. was incorrect a corrected version. was proved by Watson [0] on utilizing modular equation of seventh order. In fact he proved that if j n 0, p 7 j n + δ 7 j 0 mod 7 j.5 p 7 j n + δ 7 j 0 mod 7 j+..6 By utilizing the classical identities of Euler Jacobi, Hirschhorn Hunt [5] Garvan [] gave a simple proof of , respectively. where Very Recently, Ranganatha [9] proved an analogue of.3 for the sequence A 5 n: A 5 5 α n + 5 α 0 mod 5 α, α, A 5 nq n q5 ; q 5 0 q; q.

3 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 5 An l-regular partition is a partition where none of its part is divisible by l. The generating function for b l n, which counts the number of l-regular partitions of n is given by b l nq n ql ; q l q; q..7 Using the theory of modular forms, Furcy Penniston [3] discovered several infinite families of congruences modulo 3 for b l n where l {, 7, 3, 9, 5, 3, 37, 3, 9}. For example, they proved that for all α 0 n 0, b 5 3 α+3 n + 3 α+ 0 mod 3..8 A bipartition of n is an ordered pair of partitions λ, λ such that the sum of all the parts of λ λ equals n. A bipartition λ, λ of n is said to be l-regular bipartition of n if none of the parts of λ λ are divisible by l. The generating function for B l n, the number of l-regular bipartitions of n is given by B l nq n ql ; q l q; q..9 Recently, by following the strategy of Hirschhorn Hunt [5], Wang [, ] established several infinite families of congruences modulo powers of 5 for b 5 n B 5 n. In recent days, a number of mathematicians studied the congruence properties for b l n B l n. For example, see the references listed in [, ]. Let c N n denote the number of bipartitions λ, λ of n such that each part of λ is divisible by N. Then the generating function for c N n is given by c N nq n q; q q N ; q N..0 Congruences for c n modulo powers of 3 was proved by Chan [] modulo powers of 5 was proved by Chan Toh [] Xiong [3]. Soon after, Liu Zhang [6] discovered several infinite families of congruences modulo 3 for c 5 n. Most recently, Wang [] established several Ramanujan-type congruences modulo powers of 5 for c 5 n. In this paper, we prove Ramanujan-type congruences for b 5 n, B 5 n, c 5 n modulo powers of 5 for c 9 n modulo powers of 7. The main results of this paper can be stated as follows:

4 5 D. RANGANATHA Theorem. For j n 0, we have c 5 5 j n + 7 5j mod 5 j. c 5 5 j n + 5j mod 5 j+.. Theorem. For j α, n 0, we have b 5 5 j n + 5 j 0 mod 5 j,.3 B 5 5 j n + 5 j 0 mod 5 j. b 5 3 α+3 5 j n + 3 α+ 5 j 0 mod 3 5 j..5 Theorem.3 For j n 0, we have c 9 7 j n + 5 7j mod 7 j.6 c 9 7 j n + 7j mod 7 j+..7. PRELIMINARIES In this section, we collect number of lemmas which are essential in the proofs of main results of this paper. Let gq n g nq n in the annulus 0 < q < k, where k or +. In [5], Hirschhorn Hunt introduced the huffing operator H modulo 5, that is, Hg n g 5n q 5n

5 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 53 proved that HG j mj, ku j k,. q where G : Gq 5 ;q 5 6 q q;q, u : uq q5 ;q 5 6 q 5 ;q 5 5 the matrix M {mj, k} q 5 q 5 ;q 5 6 j, k is defined as follows: The first five rows of M are for j 6, mj, 0 for k, mj, k 5mj, k + 5mj, k + 5mj 3, k + 5mj, k + mj 5, k. By induction, we can show that mj, k vanishes for all j greater than 5k less than k, that is, mj, k 0 for j 5k + or j k.. Lemma. [5, Lemma.9]. For j, we have HG 6j m6j, j + ku 5j k..3 In view of.., we have the following lemma. Lemma. For j, we have HG 6j+ m6j +, j + ku 5j+ k.. Let π 5 n respectively π 7 n denote the power of 5 respectively 7 in the unique prime factorisation of n. For convention, we let π 5 0 π The following lemma was proved by Hirschhorn Hunt [5].

6 5 D. RANGANATHA where Lemma.3 [5, Lemma.]. For j, k, we have [ ] 5k j π 5 mj, k..5 The huffing operator H 0 modulo 7 is defined as follows. H 0 g n In [], Garvan discovered the following lemma. g 7n q 7n. Lemma. [, Lemma 3.]. For j, we have H 0 ξ j m j, kt k,.6 ξ the m j, k are as defined in [, pp.38-39]. q; q q q 9 ; q 9, T q7 ; q 7 q 7 q 9 ; q 9.7 Lemma.5 We have m j, k 0 for j k or k j +..8 As this can be easily proved by induction, we omit the proof. Lemma.6 [, Lemma 3.]. For j, we have H 0 ξ j m j, j + kt j k..9 The following lemma follows from.6.8. Lemma.7 For j, we have H 0 ξ j m j +, j + kt j k..0 Lemma.8, [Lemma 5.]. For any j, k, we have [ ] π 7 m 7k j j, k..

7 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND CONGRUENCES MODULO POWERS OF 5 FOR c 5 n In this section, we prove the Theorem.. We define the matrix M {aj, k} j, k as follows: a, 5 a, k 0 for k 3. i aj, im6i, i + k, if j is odd, aj +, k 3. i aj, im6i +, i + k, if j is even. Theorem 3. For j, we have c 5 5 j n + 7 5j + 3 q n+ q 5 ; q 5 aj, k u 5k G 6k 3.3 c 5 5 j n + 5j + 3 q n q5 ; q 5 q 5 ; q 5 6 aj, k u 5k G 6k+. 3. PROOF : Setting N 5 in.0, we see that c 5 nq n+ q 5 ; q 5 u G. In view of operator H, we have c 5 5n + q 5n+5 u G H q 5 ; q 5 u q 5 ; q 5 HG 5 q5 q 5 ; q 5 q 5 ; q Replacing q 5 by q in 3.5, we see that 3.3 holds for j. Suppose that 3.3 holds for some j.

8 56 D. RANGANATHA Applying operator H on both sides of 3.3 then using.3, we obtain c 5 5 j n + 5j + 3 q 5n+5 H q 5 ; q 5 aj, k u 5k G 6k q 5 ; q 5 aj, k u 5k HG 6k q 5 ; q 5 aj, ku 5k m6k, i + ku 5k i i q 5 ; q 5 aj, km6k, i + k u i i i q 5 q 5 ; q 5 6 q 5 ; q 5 aj, i q 5 ; q i Replacing q 5 by q in 3.6 then utilizing the definitions of u G, we can rewrite the resulting identity as c 5 5 j n + 5j + 3 q n q5 ; q 5 q 5 ; q 5 6 aj, i u 5i G 6i Applying the operator H on both sides of 3.7 then using., we deduce that c 5 5 j+ n + 7 5j+ + 3 q 5n q 5 ; q 5 H q 5 ; q 5 6 q5 ; q 5 q 5 ; q 5 6 q5 ; q 5 q 5 ; q 5 6 q5 ; q 5 q 5 ; q 5 6 q5 ; q 5 q 5 ; q 5 6 i aj, i u 5i G 6i+ i aj, i u 5i HG 6i+ i aj, iu 5i m6i +, k + iu 5i+ k i i aj, im6i +, k + iu k aj +, k q 5 q 5 ; q 5 6 q 5 ; q 5 6 k. Changing q 5 to q in the above equation then employing the definitions of u G in the resulting identity, we see that 3.3 holds for j +. This completes the proof of 3.3. The proof of 3. follows from

9 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 57 Lemma 3. For j k, we have π 5 aj, k j + [ ] 5k π 5 aj, k j + + [ ] 5k PROOF : Since π 5 a, π 5 a, k + for k, we have 3.8 holds for j. Suppose that 3.8 holds for some j. From.5 3., we have π 5 aj, k π 5 i Again in the view of.5 3., we have π 5 aj +, k π 5 aj, im6i, i + k min {π 5aj, i + π 5 m6i, i + k} i { [ ] [ ]} 5i 5 5k i min j + + i [ ] [ ] 5k 5k 5 j + j i aj, im6i +, i + k min {π 5aj, i + π 5 m6i +, i + k} i { [ ] [ ]} 5i 5 5k i 3 min j i [ ] [ ] 5k 5k 5 j + + j + +, which implies that 3.8 holds for j +. Proof of 3.9 follows from Lemma 3.3 For j, we have aj, j 5 j j mod 5 j+ 3. aj, 3 j 5 j+ j mod 5 j+3. 3.

10 58 D. RANGANATHA PROOF : By 3., we have aj +, aj, m8, + aj, im6i +, i +, 3.3 i aj, aj, m6, + In view of.5, , we find that π 5 aj, im6i +, i + i aj, im6i, i i { } min π 5 aj, i + π 5 m6i +, i + i { [ min i j + + ] [ 5i 5 + ]} i j π 5 aj, im6i, i + i { } min π 5 aj, i + π 5 m6i, i + i { [ min i j + ] [ 5i 5 + ]} i j From 3.3 to 3.6, we deduce aj +, aj, m8, mod 5 j+3, 3.7 aj, aj, m6, mod 5 j From these two congruences, we have aj +, aj, m6, m8, 3860aj, 5 aj, mod 5 j+3 since a, 5, by induction, we obtain 3.. From , arrive at 3.. Theorem. now follows from Theorem 3., CONGRUENCES MODULO POWERS OF 5 FOR b 5 n AND B 5 n To prove our Theorem., we need the following results whose proofs are analogues to that of Theorem 3. thus proofs are omitted.

11 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 59 Theorem. For j, we have b 5 5 j n + 5 j q n+ aj, k u 5k G 6k,. where a, 5, a, k 0 for k aj +, k aj, im6i, i + k.. i Theorem. For j, we have B 5 5 j n + 5 j q n+ dj, k u 5k G 6k,.3 where d, 0, d, 5, d, k 0 for k 3 dj +, k dj, im6i, i + k. i. Lemma.3 For j n 0, we have b 5 5 j n + 5 j 5 j 3 j b 5 n mod 5 j+.5 B 5 5 j n + 5 j 5 j 3 j b 5 n mod 5 j+..6 PROOF : Using.5 by induction on j, we can show that [ ] 5k 5 π 5 aj, k j + π 5 dj, k j + [.7 ] 5k 5, for j, k..8 From.5.7, it follows that π 5 aj, im6i, i + min {π 5aj, k + π 5 m6i, i + } i i { [ ] [ ]} 5i 5 i j + + j

12 60 D. RANGANATHA In view of..9, we have aj +, 35aj, mod 5 j+3..0 Because a, 5, from.0 by induction on j, we find that aj, 5 j 3 j mod 5 j+. Applying the above congruence in., we obtain b 5 5 j n + 5 j q n 5 j 3 j u 5 G 6 5 j 3 j q5 ; q 5 6 q; q 6 mod 5 j+.. Since q; q 5 q 5 ; q 5 mod 5, we have b 5 5 j n + 5 j q n 5 j 3 j q5 ; q 5 5 j 3 j q; q b 5 nq n mod 5 j+,. which implies.5. Proof of.6 follows in a similar way, except that in the place of..7,..8 are used, respectively the numbers dj, satisfies the congruence dj, 5 j 3 j mod 5 j+. Now, congruences.3. follow from.5.6, respectively. Changing j to j n to 3 α+3 n + 3 α+ in.3, we deduce that which is same as b 5 5 j 3 α+3 n + 3 α+ + 5 j 0 mod 5 j, b 5 3 α+3 5 j n + 5j α+ 0 mod 5 j..3 If we replace n by 5 j n + 5j 3 in.8, we obtain b 5 3 α+3 5 j n + 5j α+ 0 mod Finally, congruence.5 follows from.3.. This completes the proof of Theorem

13 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND CONGRUENCES MODULO POWERS OF 7 FOR c 9 n In this section, we prove Theorem.3. We first define the matrix M {gj, k} j, k as follows: g, 7, g, 9 g, k 0 for k 3 5. i gj +, k gj, im i, i + k, i gj, im i +, i + k, if j is odd, if j is even. 5. In view of.8, we note that the summation in 5. is indeed finite. Theorem 5. For j N, we have c 9 7 j n + 5 7j + 5 q n+ q 7 ; q 7 gj, k T k ξ k 5.3 c 9 7 j n + 7j + 5 q n+5 q 9 ; q 9 gj, k T k ξ k+. 5. PROOF : We proceed by induction on j. Setting N 9 in.0, we have c 9 nq n+ q 9 ; q 9 ξ. 5.5 Extracting the terms involving q 7n on both sides of 5.5, we obtain c 9 7n + 5q 7n+7 q 9 ; q 9 7 q7 q 9 ; q 9 q 7 ; q q q 9 ; q 9 8 q 7 ; q Cancelling q 7 on both sides of 5.6 then replacing q 7 by q, we see that 5.3 holds for j. Suppose that 5.3 holds for some j. Applying the operator H 0 on both sides of 5.3 then using

14 6 D. RANGANATHA.9, we obtain c 9 7 j n + 7j + 5 q 7n+7 q 7, q 7 q 7, q 7 q 7, q 7 q 7, q 7 q 7, q 7 gj, k H 0 T k ξ k gj, k H 0 T k ξ k gj, k T k m k, i + kt i k i i gj, km k, i + kt i gj, i i q 7 q 9 ; q 9 q 7 ; q 7 i. 5.7 Replacing q 7 by q in the above equation then employing the definitions of ξ T in the resulting identity, we find that c 9 7 j n + 7j + 5 q n+5 q 9 ; q 9 gj, i T i ξ i Again applying the operator H 0 on both sides of 5.8 using.0, we deduce that c 9 7 j+ n + 5 7j + 5 q 7n+7 q 9 ; q 9 gj, i H 0 T i ξ i+ i q 9 ; q 9 gj, i T i H 0 ξ i+ i q 9 ; q 9 gj, i T i m i +, k + it i k i q 9 ; q 9 gj, i m k +, j + kt k i k q 7 q 9 ; q 9 q 9 ; q 9 gj +, k q 7 ; q Changing q 7 to q in the above equation then applying.7, we obtain c 9 7 j+ n + 5 7j+ + 5 q n+ q 7 ; q 7 gj +, k T k ξ k+. i

15 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 63 This implies that 5.3 holds for j +. This completes the proof of 5.3. The proof of 5. follows from Lemma 5. For all j, k N, we have π 7 gj, k j + [ ] 7k π 7 gj, k j + + [ ] 7k PROOF : Since π 7 g,, π 7 g,, π 7 g, k + for k 3, we have 5.0 holds for j. Suppose that 5.0 holds for some j. From. 5., we have π 7 gj, k π 7 i Again in the view of. 5., we have π 7 gj +, k π 7 gj, im i, i + k min {π 7gj, i + π 7 m i, i + k} i { [ ] [ ]} 7i 7 7k i min j + + i [ ] [ ] 7k 7k 7 j + j i gj, im i +, i + k min {π 7gj, i + π 7 m i +, i + k} i { [ ] [ ]} 7i 7 7k i 5 min j i [ ] [ ] 7k 6 7k 7 j + + j + +, 5.3 which implies that 5.0 holds for j +. This completes the proof of 5.0. Proof of 5. follows from Lemma 5.3 For j, we have gj, j 7 j mod 7 j+ 5.

16 6 D. RANGANATHA gj, j 5 7 j+ mod 7 j PROOF : By 5., we have π 7 gj, im i, i + i π 7 gj, im i +, i + i { } min π 7 gj, i + π 7 m i, i + i { [ min i j + ] [ 7i 7 + ]} 6 i j { } min π 7 gj, i + π 7 m i +, i + i { [ min i j + + ] [ 7i 7 + ]} i j Using in 5., we see that gj, gj, m, mod 7 j+, 5.8 gj +, gj, m 6, mod 7 j Therefore, gj +, m, m 6, gj, 598gj, gj, mod 7 j From 5.0 by induction on j, we deduce 5.. In view of , we have 5.5. Theorem.3 follows from 5., ACKNOWLEDGEMENT The author would like to thank Prof. Chadrashekar Adiga for his advice guidance. The author also thankful to UGC, New Delhi for awarding UGC-BSR Fellowship, under which this work has been done.

17 RAMANUJAN-TYPE CONGRUENCES MODULO POWERS OF 5 AND 7 65 REFERENCES. H. -C. Chan, Ramanujan s cubic continued fraction an analog of his most beautiful identity, Int. J. Number Theory, 6 00, H. H. Chan P. C. Toh, New analogues of Ramanujan s partition idenities, J. Number Theory, 30 00, D. Furcy D. Penniston, Congruences for l-regular partitions modulo 3, Ramanujan J., F. G. Garvan, A simple proof of Watson s partition congruences for powers of 7, J. Austral. Math. Soc. Series A, 36 98, M. D. Hirschhorn D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math., 36 98, J. Liu A. Y. Z. Zhang, Arithmetic properties of a restricted bipartition function, Electron. J. Comb., 05, P3-P8. 7. S. Ramanujan, Some propeties of pn, the number of partitions of n, Proc. Cambridge Philos. Soc., 9 99, S. Ramanujan, The lost notebook other unpublished paper, Narosa, New Delhi, D. Ranganatha, On a Ramanujan-type congruence for bipartitions with 5-cores, J. Integer Sequences, 9 06, Article G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math., , L. Wang, Congruences for 5-regular partitions moudlo powers of 5, Ramanujan J., 06, DOI 0.007/s L. Wang, Congruences modulo powers of 5 for two restricted bipartitions, Ramanujan J., 06, DOI 0.007/s X. H. Xiong, The number of cubic partitions modulo powers of 5, Sci. Sin. Math., 0, -5.