Arithmetic Properties of Partition k-tuples with Odd Parts Distinct

Transcription

1 Journal of Integer Sequences, Vol. 9 06, Article Arithmetic Properties of Partition k-tuples with Odd Parts Distinct M. S. Mahadeva Naika and D. S. Gireesh Department of Mathematics Bangalore University Central College Campus Bangalore Karnataka India msmnaika@rediffmail.com gireeshdap@gmail.com Abstract Letpod k ndenotethenumberofpartitionk-tuplesofnwhereinoddpartsaredistinct and even parts are unrestricted. We establish some interesting infinite families of congruences and internal congruences modulo, 6, and 5 for pod n, pod n, and pod 6 n, respectively. We also find Ramanujan-type congruences modulo 5 for pod 3 n and densities of pod n, pod 3 n, pod n, and pod 6 n modulo, 5, 6, and 5, respectively. Introduction For q <, Ramanujan s theta functions ϕq and ψq are defined by ϕq := + q n = q n = q;q q ;q n= n= This research was partially supported by DST Govt. of India SR/S/MS:739/.

2 and ψq := q n +n/ = n= q n +n = q ;q q;q, where a;q = a aq aq. Let podn denote the number of partitions of n wherein odd parts are distinct and even parts are unrestricted. The generating function of podn is podnq n = q;q q ;q = ψ q. In 00, Hirschhorn and Sellers [5] proved that, for all α 0 and n 0, pod 3 α+3 n+ 3 3α+ + 0 mod 3. They also found some internal congruences such as podn+7 5pod9n+ mod 7. Recently, Wang [0] established new congruences for podn. For example, for each α and n 0, pod 5 α+ n+ 5α+ + 0 mod 5. Let pod k n denote the number of partition k-tuples of n wherein odd parts are distinct and even parts are unrestricted. The generating function of pod k n is pod k nq n = q;q k q ;q k = ψ q k. 3 Chen and Lin [3] established congruences modulo 3 and 5 for pod n. For example, for α and n 0, pod 5 α+ n+ 5α + 0 mod 5. Wang [, 9] has established congruences modulo 7, 9, and satisfied by pod 3 n and congruencesmodulo5, 9, andsatisfiedbypod nbyemployingthetafunctionidentities. For example, for α and n 0, pod 3 3 α+ n+ 3 3α mod 9 and pod 3 α+ n+ 5 3α + 0 mod 9.

3 He also found some internal congruences such as pod 7n+5 pod 9n+ mod 9. In this paper, we establish congruences modulo powers of and modulo 5 for pod k n for k {,3,,6}. In this vein, in Section 3, we find infinite family of congruences and internal congruences modulo satisfied by pod n and we also find density of pod n modulo. In Section, we prove Ramanujan-type congruences modulo 5 for pod 3 n and that pod 3 n is divisible by 5 at least /30 of the time. In Section 5, we establish infinite family of congruences and internal congruences modulo 6 satisfied by pod n following density of pod n modulo 6. In Section 6, we determine infinite family of congruences and internal congruences modulo 5 satisfied by pod 6 n and we also determine density of pod 6 n modulo 5. Preliminaries The following results are useful in proving our main results. Lemma. [, pp. 0 9] We have ϕq = ϕq +qψq, ϕq = ϕq +qψq, 5 ψq = fq 3,q 6 +qψq 9 6 = fq 0,q 5 +qfq 5,q 0 +q 3 ψq 5, 7 ψq = ϕqψq. Lemma. [, Eq..6.7, p. 6] We have fq,q fq,q 3 = ψq qψq 5. 9 Lemma 3. Let hnq n = qψq. Then h5n+3q n ψq mod

4 Proof. From 7, it follows that hnq n = qψq = q fq 0,q 5 +qfq 5,q 0 +q 3 ψq 5 = q 5 fq 0,q 5 fq 5,q 0 ψq 5 +q 0 fq 0,q 5 ψq 5 3 +q fq 0,q 5 fq 5,q 0 ψq 5 +6q 7 fq 0,q 5 ψq 5 +q fq 0,q 5 3 fq 5,q 0 +q fq 0,q 5 3 ψq 5 +6q 3 fq 0,q 5 fq 5,q 0 +q 5 fq 5,q 0 +q 3 ψq 5 +q fq 0,q 5 fq 5,q q 9 fq 5,q 0 ψq 5 +q 7 fq 5,q 0 3 ψq 5 +q 6 fq 0,q 5 fq 5,q 0 ψq 5 +q fq 5,q 0 ψq 5 3 +qfq 0,q 5, which yields h5n+3q n qfq,q 3 fq,q ψq 5 +fq,q 3 fq,q +q ψq 5 mod 5. Using 9 in the above equation, we arrive at 0. Lemma. Let gnq n = ψq. Then g5n+q n = ψq qψq 5. Proof. It follows from 7 that gnq n = ψq = fq 0,q 5 +qfq 5,q 0 +q 3 ψq 5 = q 6 ψq 5 +q ψq 5 fq 5,q 0 +q 3 ψq 5 fq 0,q 5 +q fq 5,q 0 +qfq 0,q 5 fq 5,q 0 +fq 0,q 5, which yields g5n+q n = fq,q 3 fq,q +qψq 5. Using 9 in the above equation, we arrive at.

5 Lemma 5. [, Theorem.] For any odd prime, p, ψq = p 3 m=0 Furthermore, m +m p mod p, for 0 m p 3. q m +m f q p +m+p,q p m+p +q p ψq p. 3 Arithmetic properties of pod n In this section, we prove the infinite family of congruences and internal congruences modulo for pod n. 3. Infinite family of congruences modulo Theorem 6. Let p be any odd prime such that p = and α 0. Then pod p α n+ 3pα + q n ψqψq mod 3 and, for all n 0 and ξ p, pod p α+ pn+ξ+ 3pα+ + 0 mod. Proof. We have Invoking and 5, pod nq n = ψ q. 5 pod nq n = ψq ϕ q = ϕ q ψq = + ϕ q+ ϕ q + ψq ϕ q mod from. ψq 5

6 Using in the above equation, we find that which yields pod nq n ϕq +qψq ψq pod n+q n ψq ψq mod, mod. 6 From the binomial theorem, we can see that for any prime p and for each positive integer l, In view of 7, 6 can be expressed as p 3 q;q pl q p ;q p pl mod p l. 7 pod n+q n ψqψq mod, which is the α = 0 case of 3. If we assume that 3 holds for some α 0, then, substituting in 3, pod p α n+ 3pα + q n m=0 p 3 q m +m f q p +m+p,q p m+p +q p ψq p q m +m f m=0 q p +m+p,q p m+p +q p ψq p For any odd prime, p, and 0 m,m p 3/, consider the congruence m +m + m +m 3p 3 mod p, mod. 9 which implies that m + +m + 0 mod p. 0 Since =, the only solution of the congruence 0 is m p = m = p. Therefore, equating the coefficients of q pn+3p 3 from both sides of 9, dividing throughout by q 3p 3 and then replacing q p by q, we obtain pod p α+ n+ 3pα+ + q n ψq p ψq p mod. 6

7 Equating the coefficients of q pn on both sides of and then replacing q p by q, we obtain pod p α+ n+ 3pα+ + q n ψqψq mod, which is the α+ case of 3. Equating the coefficients of q pn+ξ for ξ p from, we arrive at. Corollary 7. Let p be any odd prime such that =. Then pod n is divisible by for at least of all nonnegative integers n. p+ { } Proof. The arithmetic sequences p α+ pn+ξ+ 3pα+ + : α 0 for ξ p, on which pod is 0 modulo, do not intersect. These sequences account for p p + p + p + = 6 p+ of all nonnegative integers. 3. Some internal congruences Theorem. For each n 0, Proof. If p pod 5n+5 pod 6n+3 mod, pod 5n+3 pod 6n+5 mod, 3 pod 6n+7 pod n+ mod, pod 6n+5 pod n+3 mod. 5 anq n = ψqψq, then the authors [6] found that a3nq n = ψqϕq qψq 3 ψq 6. 6 Using 6, we can express as pod 6n+q n ψqϕq qψq 3 ψq 6 mod. 7 Invoking and 7, pod 6n+q n ψq+qψq 3 ψq 6 mod. 7

8 Substituting 6 into and extracting the terms involving q 3n+, pod n+7q n ψq 3 +ψqψq mod, which is equivalent to pod n+7q n ψq 3 + pod n+q n mod. 9 Equatingthecoefficientsofq 3n+ andq 3n+ from9, wearriveatand3, respectively. Equating the coefficients of q 3n and then replacing q 3 by q from 9, pod 5n+7q n ψq+ pod 6n+q n mod. 30 Invoking and 30, pod 5n+7q n pod 6n+q n qψq 3 ψq 6 mod, 3 Equating the coefficients of q 3n and q 3n+ from 3, we arrive at and 5, respectively. Ramanujan-type congruences for pod 3 n In this section, we prove the Ramanujan-type congruences modulo 5 for pod 3 n. Theorem 9. For each α, pod 3 5 α+ n+ µ 5α +3 0 mod 5, 3 where µ = 3,, 9, and 37. Proof. We have pod 3 n n q n = ψq 3. It follows from 7 that pod 3 n n q n ψq mod 5 ψq 5 gnq n mod 5. ψq 5

9 Extracting the terms involving q 5n+, dividing throughout by q and then replacing q 5 by q, pod 3 5n+ n+ q n ψq g5n+q n mod 5 ψq qψq 5 mod 5 from ψq ψq qψq 5 ψq mod 5 using Substituting 7 into 33 and from the Lemma 3, we find that pod 3 5n+ n+ q n fq 0,q 5 +qfq 5,q 0 +q 3 ψq 5 which implies that ψq 5 hnq n mod 5, pod 3 5n+6 n q n ψq 5 ψq h5n+3q n mod 5 Using 7, 3 can be expressed as ψq 5 ψq 5 mod 5 using 0. 3 pod 3 5n+6 n q n ψq 5 mod Extracting the terms involving q 5n from 35, pod 3 5n+6 n q n ψq mod 5 which yields fq 0,q 5 +qfq 5,q 0 +q 3 ψq 5 mod 5, pod 3 65n+39 n+ q n ψq 5 mod From 35, 36, and by induction, we find that for each α, pod 3 5 α n+ 5α+ +3 n+ q n α ψq 5 mod Equating the coefficients of q 5n+ξ for ξ from 37, we arrive at 3. 9

10 Corollary 0. The function pod 3 n is divisible by 5 for at least of all nonnegative 30 integers n. { } Proof. The arithmetic sequences 5 α+ n+ µ 5α +3 : α for µ = 3,, 9, and 37, on which pod 3 is 0 modulo 5, do not intersect. These sequences account for = 7 30 of all nonnegative integers. 5 Arithmetic properties of pod n In this section, we prove the infinite family of congruences and internal congruences modulo 6 for pod n. 5. Infinite family of congruences modulo 6 Theorem. Let p be any prime such that p 3 mod and α 0. Then pod p α n+ pα + q n ψq mod 6 3 and, for all nonnegative integers n and ξ p, pod p α+ pn+ξ+ pα+ + 0 mod Proof. We have Invoking and 0, pod nq n = pod nq n = ψq ϕ q = ϕ q ψq ψ q. 0 = + ϕ q + ϕ q + ψq ϕ q ψq mod 6 using ϕq +qψq ψq mod 6 from 5, 0

11 which implies that pod n+q n ψq mod 6, ψq In view of 7, can be expressed as pod n+q n ψq mod 6, which is the α = 0 case of 3. If we assume that 3 holds for some α 0, then, substituting into 3, pod p α n+ pα + p 3 m=0 q n q m +m f q p +m+p,q p m+p +q p ψq p. 3 For any odd prime, p, and 0 m,m p 3/, consider the congruence m +m + m +m p mod p, which implies that m + +m + 0 mod p. Since = for p 3 mod, the only solution of the congruence is m p = m = p p. Therefore, equating the coefficients of qpn+ from both sides of 3, dividing throughout by q p and then replacing q p by q, we obtain pod p α pn+ p + pα + q n ψq p mod 6. 5 Equating the coefficients of q pn on both sides of 5 and then replacing q p by q, we obtain pod p α+ n+ pα+ + q n ψq mod 6, which is the α+ case of 3. Equating the coefficients of q pn+ξ for ξ p from 5, we arrive at 39.

12 Corollary. Let p be a prime such that p 3 mod. Then pod n is divisible by 6 for at least of all nonnegative integers n. p+ { } Proof. The arithmetic sequences p α+ pn+ξ+ pα+ + : α 0 for ξ p, on which pod is 0 modulo 6, do not intersect. These sequences account for of all nonnegative integers. p p + p + p + = 6 5. Some internal congruences Theorem 3. For each n 0, Proof. If p+ pod 50n+3 pod 0n+ mod 6, pod 50n+3 pod 0n+5 mod 6, pod 50n+33 pod 0n+7 mod 6, pod 50n+3 pod 0n+9 mod 6. gnq n = ψq, then can be expressed as which yields Invoking and 6, pod n+q n gnq n mod 6, pod 0n+3q n g5n+q n mod 6. 6 pod 0n+3q n ψq qψq 5 mod 6. 7 Substituting into 7, pod 0n+3q n pod n+q n qψq 5 mod 6, equating the coefficients of q 5n+i for i = 0,, 3, and from the above equation, we obtain the desired results.

13 6 Arithmetic properties of pod 6 n In this section, we prove the infinite family of congruences and internal congruences modulo 5 for pod 6 n. 6. Infinite family of congruences modulo 5 Theorem. Let p be any prime such that p 3 mod and α 0. Then pod 6 5p α n+ 5pα +3 q n ψq mod 5 and, for each n 0 and ξ p, pod 6 5p α+ pn+ξ+ 5pα mod 5. Proof. We have pod 6 nq n = In view of 7, can be expressed as Substituting 7 into 9, which yields Invoking 9 and 50, ψ q 6. pod 6 n n q n ψq ψq 5 mod 5. 9 pod 6 n n q n fq0,q 5 +qfq 5,q 0 +q 3 ψq 5 ψq 5 mod 5, pod 6 5n+ n q n fq,q 3 fq,q + qfq,q 3 fq,q ψq 5 ψq ψq pod 6 5n+ n q n ψq +q ψq5 + q ψq 5 ψq mod ψq qψq5 +qψq 5 q ψq5 ψq +qψq5 ψq mod 5, 5 3

14 which implies that pod 6 5n+ n q n ψq mod 5. 5 The remainder of the proof is similar to that of Theorem, but rather than, we use 5. Corollary 5. Let p be a prime such that p 3 mod. Then pod 6 n is divisible by 5 for at least of all nonnegative integers n. 5p+ { } Proof. The arithmetic sequences 5p α+ pn+ξ+ 5pα+ +3 : α 0 for ξ p, on which pod 6 is 0 modulo 5, do not intersect. These sequences account for p 5p + 5p + 5p + = 6 5p+ of all nonnegative integers. 6. Some internal congruences Theorem 6. For each n 0, pod 6 5n+7 3pod 6 5n+ mod 5, pod 6 5n+57 3pod 6 5n+ mod 5, pod 6 5n+ 3pod 6 5n+7 mod 5, pod 6 5n+07 3pod 6 5n+ mod 5. Proof. The proof is similar to that of Theorem 3, but rather than, we use 5. 7 Acknowledgments The authors would like to thank anonymous referee of his/her helpful suggestions and comments. References [] G. E. Andrews and B. C. Berndt, Ramanujans Lost Notebooks, Part I, Springer, 005. [] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, 99. [3] W. Y. C. Chen and B. L. S. Lin, Congruences for bipartitions with odd parts distinct, Ramanujan J. 5 0,

15 [] S. P. Cui and N. S. S. Gu, Arithmetic properties of l-regular partitions, Adv. Appl. Math. 5 03, [5] M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 00, 73. [6] M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic properties arising from Ramanujan s theta functions, Ramanujan J., 06, to appear. [7] S. Radu and J. A. Sellers, Congruences properties modulo 5 and 7 for the pod function, Int. J. Number Theory 7 0, [] L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 05, [9] L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc. 9 05, [0] L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Sequences 05, Article Mathematics Subject Classification: Primary 05A7; Secondary P3. Keywords: congruence, partition k-tuple, partition with odd parts distinct. Concerned with sequences A006950, A735, A736, A73, and A7336. Received November 05; revised version received April 06; May 06. Published in Journal of Integer Sequences, June 06. Return to Journal of Integer Sequences home page. 5