International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 738948, 6 pages

http://dx.doi.org/10.1155/2014/738948

Research Article

Proof of Some Conjectures of Melham Using Ramanujan򢀙s Formula

Bipul Kumar Sarmah

Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam 784028, India

Received 5 February 2014; Accepted 16 June 2014; Published 10 July 2014

Academic Editor: Wolfgang zu Castell

Copyright © 2014 Bipul Kumar Sarmah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We employ Ramanujan򢀙s formula to prove three conjectures of R. S. Melham on representation of an integer as sums of polygonal numbers.

1. Introduction

Jacobi’s classical two-square theorem is as follows.

Theorem 1 (see [1]). Let denote the number of representations of as a sum of two squares, counting order and sign, and let denote the number of positive divisors of congruent to modulo . Then

The above theorem can also be recasted in terms of Lambert series as

Similar representation theorems involving squares and triangular numbers were found by Dirichlet [2], Lorenz [3], Legendre [4], and Berndt [5]. For example, the following two theorems are due to Lorenz and Ramanujan, respectively.

Theorem 2 (see [3]). Let denote the number of representations of as a sum of times a square and times a square. Then

Theorem 3 (see [5]). Let denote the number of representations of as a sum of times a triangular number and times a triangular number. Then

Hirschhorn [6, 7] obtained forty-five similar identities (including those obtained by Legendre and Ramanujan) involving squares, triangular numbers, pentagonal numbers, and octagonal numbers employing dissection of the -series representations of the identities obtained by Jacobi, Dirichlet, and Lorenz. In [8], Baruah and the author obtained twenty-five more such identities involving squares, triangular numbers, pentagonal numbers, heptagonal numbers, octagonal numbers, decagonal numbers, hendecagonal numbers, dodecagonal numbers, and octadecagonal numbers. More works on this topic have been done in [9–11]. In [11], Melham presented 21 conjectured analogues of Jacobi’s two-square theorem which are verified using computer algorithms. In [12], Toh offered a uniform approach to prove these conjectures using known formulae for . In this paper, we show that some of these conjectures can also be proved by using Ramanujan’s famous formula. We prove three conjectures enlisted in the following theorem which have appeared as , , and , respectively, in [11].

Theorem 4. Consider

The next section of this paper is devoted to notations, definitions, and preliminary results.

2. Notations and Preliminary Results

Ramanujan’s general theta-function is defined by [5, page 34, (18.1)]

Jacobi’s famous triple product identity takes the simple form [5, page 35, Entry 19]

where, here and throughout the paper, for ,

We also use the following notation for the sake of brevity of expressions:

One important special case of is

where the product representations in (12) arise from (9).

From [5, page 46, Entry 30] we find that

Also if , then [5, page 45, Entry 29]

We recall Ramanujan’s formula in the form [5, page 34, (17.6)]

where

Finally, we note that

3. Proof of the Conjectures (5)–(7)

Proof of Conjecture (5). In view of (17), we rewrite (5) as

Replacing by and then setting , , and in (15), we have

Expanding the -products on the left side of (19) and noting that , we obtain

Also

Thus, from (20) and (21), we have

Replacing by and then setting , , and in (15), and proceeding as in case of (22), we find that

Multiplying (23) by and adding to (22), we obtain

Setting and in (13), we find that

Thus, (24) reduces to

Now

We have

Similarly

Hence from (27), (28), and (29) we have

Using (30) in (26), we find that

This completes the proof of (18).

Proof of Conjecture (6). In view of (17), we rewrite (6) as

Replacing by and then setting , , and in (15), and proceeding as in case of (22), we find that

Replacing by and then setting , , and in (15), and proceeding in a way similar to obtaining (22), we have

Multiplying (34) by and then adding to (33), we find that

Setting , , , and in (14), we find that

Using (36) in (35), we have

Now

Using (38) in (37), we have

This completes the proof of (32).

Proof of Conjecture (7). In view of (17), we rewrite (7) as

Replacing by and then setting , , and in (15), and proceeding as in case of (22), we find that

Also, replacing by and then setting , , and in (15), and proceeding similar to (22), we obtain

Multiplying (42) by and then adding to (41), we find that

Setting , , , and in (14), we find that

Hence, (43) reduces to

Now

Using (46) in (45), we have

This completes the proof of (40).

4. Concluding Remarks

All conjectures in [11] can easily be reformulated as theta function identity using Ramanujan’s formula. However, these identities might be too complicated to actually have a proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

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P. G. L. Dirichlet, J. Math. 21 (1840) 3, 6; Werke 463, 466

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