On eight colour partitions
B.Hemanthkumar, H.S.Sumanth Bharadwaj

TL;DR
This paper investigates the arithmetic properties of the 8-colour partition function, deriving new Ramanujan-type congruences modulo powers of 2 through explicit generating function formulas.
Contribution
It introduces new Ramanujan-type congruences for the 8-colour partition function and provides explicit formulas for its generating functions.
Findings
Established Ramanujan-type congruences modulo higher powers of 2.
Derived explicit formulas for the generating functions of p_8(n).
Enhanced understanding of the arithmetic properties of 8-colour partitions.
Abstract
In this article, we study the arithmetic properties of the partition function , the number of 8-colour partitions of . We prove several Ramanujan type congruences modulo higher powers of 2 for the function by finding explicit formulas for the generating functions.
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Taxonomy
TopicsAdvanced Mathematical Identities · Advanced Combinatorial Mathematics · Analytic Number Theory Research
On Eight Colour Partitions
B. Hemanthkumar
Department of Mathematics, M. S. Ramaiah University of Applied Sciences, Peenya campus, Bengaluru-560 058, Karnataka, India
and
H. S. Sumanth Bharadwaj
Department of Mathematics, M. S. Ramaiah University of Applied Sciences, Peenya campus, Bengaluru-560 058, Karnataka, India
Abstract.
In this article, we study the arithmetic properties of the partition function , the number of -colour partitions of . We prove several Ramanujan type congruences modulo higher powers of 2 for the function by finding explicit formulas for the generating functions.
Key words and phrases:
colour partitions; congruences
2010 Mathematics Subject Classification:
11P83; 05A17
1. Introduction
For any nonnegative integer , let denote the number of partitions of . The generating function of is given by
[TABLE]
Here and throughout this paper, we set
[TABLE]
for any positive integer . By Euler’s pentagonal number theorem [1, pp. 10-12], we have
[TABLE]
In 1919, Ramanujan [6] obtained the generating function for as
[TABLE]
It follows from (1.2) that . Also, he conjectured that for any integer
[TABLE]
where is the reciprocal modulo of 24, and outlined the proof in [7, pp. 156-177], (also see [3]). In 1938, Watson [8] proved the above conjecture and in 1981, Hirschhorn and Hunt [5] gave a simple proof of the conjecture.
Recently, Hirschhorn [4] studied the number of 3-colour partitions of , denoted by and is given by
[TABLE]
He obtained several congruences for modulo high powers of , namely, for all ,
[TABLE]
which are analogous to Ramanujan’s congruences for the partition function.
In this paper we study the partition function, , the number of 8-colour partitions of , which satisfies
[TABLE]
We obtain appropriate generating formulae for and deduce several Ramanujan-type congruences modulo high powers of 2. The main results of this paper are as follows.
Theorem 1.1**.**
For all nonnegative integers and , we have
[TABLE]
and if is not a generalised pentagonal number, then
[TABLE]
We provide some definitions and preliminary results in Section 2. We establish generating functions for in Section 3 and prove Theorem 1.1 in Section 4.
2. Preliminaries
In this section, we present some preliminary results which are used in proving our main results.
Lemma 2.1**.**
We have
[TABLE]
and
[TABLE]
Proof.
Lemma 2.1 is an immediate consequence of dissection formulas of Ramanujan, collected in Berndt’s book [2, Entry 25, p. 40]. ∎
Lemma 2.2**.**
Let
[TABLE]
Then, for any
[TABLE]
Proof.
Squaring (2.1) on both sides, we deduce that
[TABLE]
Setting for in (2.3) and using the fact that , we obtain
[TABLE]
We can rewrite (2.3) as
[TABLE]
and then (2.4) as
[TABLE]
If follows from the definition of and last equation that
[TABLE]
which yields
[TABLE]
thus for any ,
[TABLE]
This completes the proof of Lemma 2.2. ∎
Let us define an operator as follows:
[TABLE]
and
In view of (2.5), we see that
[TABLE]
Also, from Lemma 2.2
[TABLE]
In particular,
[TABLE]
and
[TABLE]
We define an infinite matrix as follows:
- (1)
for all . 2. (2)
for all . 3. (3)
for all . 4. (4)
for all .
Using the definition of and induction, it is easy to prove that
- (5)
for all . 2. (6)
for all . 3. (7)
for all . 4. (8)
for all .
We omit the details of the proof. The first eight rows of are given by
[TABLE]
Lemma 2.3**.**
For any positive integer , we have
[TABLE]
Proof.
The second equality follows from (6). We use mathematical induction to prove the first equality. From (2.7) and (2.8), we see that the above identity holds for . Suppose that (2.10) holds for some integer . From (2.6), we see that
[TABLE]
Using (3),(4) and (8) in the above equation, we arrive at
[TABLE]
This completes the induction. ∎
3. Generating functions
In this section we obtain generating functions for the sequences in Theorem 1.1.
Theorem 3.1**.**
We have
[TABLE]
Proof.
We note that from (2.7),
[TABLE]
or
[TABLE]
This proves (3.1). Rewriting (2.9), we see that
[TABLE]
or
[TABLE]
Thus, (3.2) follows from (3.1) and (3.3). ∎
Let us define another infinite matrix by
- (9)
for all . 2. (10)
[TABLE]
Using the properties of the numbers and induction, we can easily see that
- (11)
[TABLE]
and
- (12)
[TABLE]
The first four rows of are given by
[TABLE]
Theorem 3.2**.**
For any positive integer ,
[TABLE]
and
[TABLE]
Proof.
We prove (3.5) and (3.6) by mathematical induction on . From (3.1) and (3.2), it follows that (3.5) and (3.6) hold for .
Suppose that (3.5) holds for some positive integer . Applying the operator to both sides gives
[TABLE]
From Lemma 2.3, we have
[TABLE]
Combining (3.7) and (3.8), we obtain
[TABLE]
Interchanging the order of summantion and using properties (5), (6) and (11) to extend the sums to all positive integers, we have
[TABLE]
which is (3.6). Here last equality follows from properties (10) and (11).
Suppose (3.6) holds for some positive integer . Multiplying the above equation by and applying the operator to the resulting expression, we find that
[TABLE]
From Lemma 2.3, we have
[TABLE]
Combine (3.9) and (3.10), interchange the order of the summation and extend sums to all positive integers, we get
[TABLE]
which is (3.5) with replaced by . This completes the proof of (3.5) and (3.6) by induction. ∎
4. Congruences
For a positive integer , let be the highest power of that divides and by convention.
In this section, we consider the powers of 2 that divide the numbers and . Using this information we prove Theorem 1.1.
Lemma 4.1**.**
For any integers and , we have
[TABLE]
Proof.
The proof easily follows from the definition of and properties (5)-(8). ∎
Lemma 4.2**.**
For all positive integers and , we have
[TABLE]
and
[TABLE]
In both cases, equality holds for .
Proof.
We use mathematical induction to prove these results, passing alternatively between (4.2) and (4.3). From (3.4) it follows that (4.2) and (4.3) hold for and equality hold in each case for .
Assume that (4.2) is true for some positive integer . Consider the case of (4.3).
[TABLE]
By our assumption, and . Hence it follows that . For , using the definition of and properties (5) and (6) of the numbers , we get
[TABLE]
By Lemma 4.1, , whenever . Using this fact and the induction hypothesis,
[TABLE]
Combining (4.5) and (4.6), we obtain
[TABLE]
From (4.7), we conclude that if (4.2) is true for some positive integer then (4.3) is also true for that .
Suppose (4.3) is true for some positive integer . Consider
[TABLE]
By our assumption, and hence . Let , by the definition of and properties (5) and (6), we see that
[TABLE]
which is (4.2) with replaced by . Here second inequality of the last expression follows from the Lemma 4.1. ∎
Proof of Theorem 1.1.
Congruence (1.4) follows from (3.1). By the binomial theorem, we can see that for all positive integers and
[TABLE]
In view of (4.2) and (4.8), we see that
[TABLE]
and
[TABLE]
It follows from (3.5), (4.9) and (4.10) that,
[TABLE]
which implies that
[TABLE]
and
[TABLE]
Congruence (1.5) follows from (4.11). By substituting (2.1) in (4.12) and extracting the terms involving odd and even powers of ,
[TABLE]
and
[TABLE]
Congruences (1.6) and (1.9) follow from (4.13). Congruences (1.8) and (1.12) follow from (4.14).
From (4.3), we have
[TABLE]
[TABLE]
Substituting (2.2) in (4.16) and extracting the terms involving odd powers of , we see that
[TABLE]
which implies that
[TABLE]
Congruences (1.7) and (1.10) follow from (4.17). Congruence (1.11) follows from (4.18). ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Andrews, G. E.: The Theory of Parititions. Cambridge University Press, Cambridge (1998)
- 2[2] Berndt, B. C.: Ramanujan’s Notebooks, Part III. Springer-Verlag, New York (1991)
- 3[3] Berndt, B. C., Ono, K.: Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary. Sém. Lothar. Combin. 42, 1–63 (1999)
- 4[4] Hirschhorn, M. D.:Partitions in 3 colours. Ramanujan J. 45, 399–411 (2018)
- 5[5] Hirschhorn, M. D., Hunt, D. C.: A simple proof of the Ramanujan conjecture for powers of 5. J. Reine Angew. Math. 326, 1–17 (1981)
- 6[6] Ramanujan, S.: Some properties of p ( n ) 𝑝 𝑛 p(n) , the number of partitions of n 𝑛 n . Proc. Cambridge Philos. Soc. 19, 207–210 (1919)
- 7[7] Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa Publishing House, Bombay, (1988)
- 8[8] Watson, G. N.: Ramanujans Vermutung über Zerfällungszahlen. J. Reine Angew. Math. 179, 97–128 (1938)
