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DOI: 10.17516/1997-1397-2022-15-1-75-79 УДК 515
Some New Congruence Identities of General Partition for pr(n)
B.R.Srivatsa Kumar* Shruthi Shruthi^ Halgar J. Gowtham*
Manipal Institute of Technology Manipal Academy of Higher Education Manipal - 576 104, India
Received 07.01.2020., received in revised form 10.08.2021, accepted 20.09.2021 Abstract. In the present work, we deduce some new congruences modulo 3 and 5 for pr(n), where r £ { — (3A + 3), — (5A + 3) | Л is any non-negative integer}. Our emphasis throughout this paper is to exhibit the use of q-identities to generate the congruences for pr (n).
Keywords: q-identity, partition congruence, Ramanujan's general partition function congruences. Citation: B.R. Srivatsa Kumar, S. Shruthi, H.J. Gowtham, Some New Congruence Identities of General Partition for pr(n), J. Sib. Fed. Univ. Math. Phys., 2022, 15(1), 75-79. DOI: 10.17516/1997-1397-2022-15-1-75-79.
1. Introduction
For \ab\ < 1, Ramanujan's general theta function f (a, b) is given by
Ek(k + 1) k(k-1) a 2 b 2 .
By Jacobi's triple product identity [5, p. 35], we have
f (a, b) = (-a; ab)x(-b; ab)x(ab; ab)x, where here and throughout the paper, we utilize the following ^-shifted factorial:
(a; q= П(! " aqk)' q < 1'
k=0
One of the special case of f (a, b) as defined by S. Ramanujan [5, p. 36] is as follows
о X-^ n(3n—1)
f(-q)= f(-q, -q2)= E (-i)nq^^ = (q; qU-
n= — co
*sri_vatsabr<8yahoo.com https://orcid.org/0000-0002-5684-9834 [email protected] https://orcid.org/0000-0002-3305-0085
[email protected] https://orcid.org/0000-0001-5276-2363 Corresponding author © Siberian Federal University. All rights reserved
For convenience, we write f (-qn) = fn. Due to Euler, we have
1
f '
J2p(n)qn = f ,
n=0
where p(n) is the number of partitions of n. S. Ramanujan initiated the general partition function
pr (n) as
Y^Pr (n)qn = -, (1)
n=0 fl
for non-zero integer r. For partition function p(n), Ramanujan's so called "most beautiful identity" is given by
to f 5
]Tp(5n + 4) qn = 5 f6,
n=0 f1
which readily implies
p(5n + 4) = 0 (mod 5).
Further he also recorded two more congruences as follows:
p(7n + 5) = 0 (mod 7), p(11n + 6) = 0 (mod 11).
The generalization of the congruences modulo powers of 5 and 7 for all pr (n) was proved by K. G. Ramanathan [16]. Later A. O.L. Atkin [2] found that Ramanathan's proof is not correct. M. Newmann [13-15], studied the function pr(n) and obtained several interesting congruences and identities involving pr (n). The functions pr (n) have been studied by many mathematicians. For the wonderful work one can see [1-4,6,8-10,12,17-22]. For r = -2, R.Hammond and R. Lewis [11] proved that
p-2(5n + I) = 0 (mod 5),
where I e {2,3,4}. Also in [7], W. Y. C. Chen et. al. proved
p-2(25n + 23) = 0 (mod 25)
by using modular forms. More recently D. Tang [24] for pr (n) proved some new congruences for pr(n), where r e {-2, -6, -7}. For example,
p-2 (52S-1n + 7 X 521'21 + 1) = 0 (mod 55), p-6 (525n + 3 X ^ + ^ = 0 (mod 55)
and
p-7 [52S-1n + 13 X ^ 1 + ^ = 0 (mod 55).
24
Motivated by the above work in this paper, we deduce some new congruences modulo 3 and 5 for pr(n), where r e {-(3A + 3), -(5A + 3) | A is any non-negative integer}. By using the binomial theorem, one can easily deduce the below mentioned congruence, and it will be used again and again in our proof without mentioning precisely. For a prime p, we have
fp = fp (mod p).
Now we state our results which we are proving in this paper. The following congruences holds good for any non-negative integer A:
Theorem 1.1. We have
Theorem 1.2. We have
P—(3A+3)(3n +1) = 0 (mod 3), (2)
P—(3A+3)(3n + 2) = 0 (mod 3). (3)
P-(5A+3)(5n + 2) = 0 (mod 5), (4)
P-(5A+3)(5n + 3) = 0 (mod 5), (5)
P-(5A+3)(5n + 4) = 0 (mod 5). (6)
2. Congruences of modulo 3 and 5
Proof of Theorem 1.1. Set r = — (3A + 3) in (1), we observe that
EP-(3A+3) (n)qn = f3A+3 = ff (7)
n=0
From [5, p. 345 Chapter 20, Entry 1], we have
f3 = f3 f- — 3q + 4qV), (8)
where
(U — 3q + 4q3«2)
f3f38
u =
f6f9
Substituting (8) in (7), we have
E P—(3A+3) (n)qn = fi3Af93 (- — 3q + 4q3uA n=0 Vu /
= fAf27^U — 3q + 4q3u^ (mod 3).
On extracting the powers of q3n+1 and q3n+2 in the above congruence, we deduce (2) and (3). □ Proof of Theorem 1.2. From [20], we have
fi = f25(Rq^ — q — q2R(q5)), (9)
where
w (1 q5n—4)(i q5n —1)
R(q) = TT (1 — q )(1 — q )
R(q) 11 (1 — q5n—3)(1 — q5n—2).
n=1 V ' V '
Set r = —(5A + 3) in (1), we see that
TO
Ep—(5A+3)(n)qn = f5A+3 = fi5Afi3 (10)
n=0
= fAf3 (mod 5).
From (9), we observe that
fi=^ - wm+5q3 - 3q5R2 (q5) -q6R3{q5)) • (n)
Utilizing (11) in (12), one can easily see that
£P-(5X+3) (n)qn = f£+6(^qy + R22qr) + 2q5R2(q5) + 4q6R3(q5^ (mod 5) (12)
On extracting the powers of q5n+2, q5n+3 and q5n+4 in the above congruence, we obtain (4), (5)
and (6). The above result is also due to B.R. Srivatsa Kumar et al. [23]. □
References
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21] N.Saikia, J.Chetry, Infinite families of congruences modulo 7 for Ramanujan's general partition function, Ann. Math. Quebec., 42(2018), no. 1, 127-132.
22] Shruthi, B.R.Srivatsa Kumar, Arithmetic identities of Ramanujan's general partition function for Modulo 17, Proceedings of the Jangjeon Mathematical Society, 22(2019), no. 4, 625—630. DOI: 10.17777/pjms2019.22.4.625
23] B.R.Srivatsa Kumar, Shruthi, D.Ranganatha , Some new Congruences Modulo 5 for General Partition Function, Russian Mathematics, 64(2020), 73-78. DOI: 10.26907/0021-3446-2020-7-83-88
24] D.Tang, Congruences modulo powers of 5 for fc-colored partitions, J. Number Theory, 187(2018), 198-214, .
Некоторые новые конгруэнтные тождества общего разбиения для рг (п)
Б. Р. Шриватса Кумар Шрути Шрути Халгар Дж. Гаутам
Технологический институт Манипала Академия высшего образования Манипала Манипал - 576104, Индия
Аннотация. В настоящей работе мы выводим некоторые новые сравнения по модулю 3 и 5 для рг(п), где г £ { — (ЗА + 3), —(5А + 3) | А любое неотрицательное целое число}. В этой статье мы делаем упор на демонстрацию использования д-тождеств для генерации сравнений для рг (п).
Ключевые слова: д-идентичность, конгруэнтность разбиений, общие конгруэнции статистической суммы Рамануджана.
