Abstract
For positive integers a, b, c and d, let N(a, b, c, d; n) be the number of representations of n as \(ax^2+by^2+cz^2+dw^2\) and T(a, b, c, d; n) be the number of representations of n as \(a\frac{X(X+1)}{2}+b\frac{Y(Y+1)}{2}+c\frac{Z(Z+1)}{2}+d\frac{W(W+1)}{2}\), where x, y, z, w are integers, and n, X, Y, Z, W are nonnegative integers. Recently, Sun (J. Ramanujan Math. Soc., 35 (2020), 373–389) established a number of relations between T(a, b, c, d; n) and N(a, b, c, d; n), and posed some conjectures on such relations. In this paper, we confirm several Sun’s conjectures by employing Ramanujan’s theta function identities.
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1 Introduction
Let \(\mathbb {N}^{+}\), \(\mathbb {N}\) and \(\mathbb {Z}\) denote the set of positive integers, the set of nonnegative integers and the set of integers, respectively. Let \(\mathbb {Z}^4=\mathbb {Z}\times \mathbb {Z}\times \mathbb {Z}\times \mathbb {Z}\) and \(\mathbb {N}^4=\mathbb {N}\times \mathbb {N}\times \mathbb {N}\times \mathbb {N}\). For \(a, b, c, d\in \mathbb {N}^{+}\) and \(n\in \mathbb {N}\), we define
and
We adopt the convention that \(N(a,b,c,d;0)=T(a,b,c,d;0)=1\).
Attributed to Jacobi [1], we now know that
Legendre [2] later discovered that
By Ramanujan’s theta function identities, it is easy to establish the following relation:
from which Legendre’s formula (1.2) follows immediately from Jacobi’s formula (1.1). It should be mentioned that Bateman and Knopp [3, Lemma 2.7] found the following relation between \(r_k(n)\) and \(t_k(n)\),
where \(r_k(n)\) (resp. \(t_k(n)\)) be the number of representations of n as the sum of k squares (resp. triangular numbers). For \(a, b, c, d\in \mathbb {N}^{+}\) with \(1\le a+b+c+d\le 7\), Adiga, Cooper and Han [4] proved that
where
and \(\dot{\imath }_{\dot{\jmath }}\) denotes the number of elements in \(\{a,b,c,d\}\) which are equal to \(\dot{\jmath }\). Moreover, when \(a+b+c+d=8\), Baruah, Cooper and Hirschhorn [5] found the relation between T(a, b, c, d; n) and \(N(a,b,c,d;8n+8)-N(a,b,c,d;2n+2)\).
Seeking the relation between the number of representations as the sum of triangular numbers and the number of representations as the sum of squares is an interesting topic. In [6], Sun conjectured that for several values of a, b, c, d, T(a, b, c, d; n) is a linear combination of N(a, b, c, d; m) and \(N(a,b,c,d;4\,m)\) with \(m=8n+a+b+c+d\). It was proved by the present authors [7]. For more information on this issue, see [7] and references therein. Recently, Sun [8] established seventeen transformation formulae for the number of representing an integer as linear combinations of triangular numbers, and obtained many relations between T(a, b, c, d; n) and N(a, b, c, d; n). For example,
At the end of paper [8], Sun posed some new conjectures about the relations between T(a, b, c, d; n) and N(a, b, c, d; n).
The aim of this paper is to confirm some of Sun’s conjectures in [8]. The main results are listed as follows.
Theorem 1
Let \(n\in \mathbb {N}^{+}\). Then
Theorem 2
Let \(n\in \mathbb {N}^{+}\). Then
Theorem 3
Let \(n\in \mathbb {N}^{+}\). Then
Theorem 4
Let \(n\in \mathbb {N}^{+}\). Then
Remark 1
The proofs of Theorem 3 and Theorem 4 are similar to that of Theorem 1, which are omitted.
The paper is organized as follows. In Sect. 2, we present the necessary preliminaries. In Sects. 3 and 4, we prove Theorems 1 and 2, respectively.
2 Preliminaries
Ramanujan’s general theta function f(a, b) is defined by
By [9, p. 34, Entry 18 (iv)], we see that for \(n\in \mathbb {Z}\),
From the well-known Jacobi triple product identity in [9, p. 35, Entry 19],
where
Setting \((a,b)=(q,q^2)\) and \((a,b)=(q,q^5)\) in (2.2), we readily obtain, respectively
and
where \(f_k:=(q^k;q^k)_\infty \).
Two special cases of f(a, b) are
The generating functions for N(a, b, c, d; n) and T(a, b, c, d; n) are
Next, we require some theta function identities which will be used later.
Lemma 1
Proof
From [9, p. 40, Entry 25 (i), (ii)], we derive (2.9). Taking \((\mu ,\nu )=(2,1)\) in [9, p. 69, eq. (36.8)], we immediately obtain (2.10). Using (25.2.2) and (25.2.4) in [10, p. 219] to conclude (2.11). \(\square \)
The known ordinary methods used to deal with similar identities are not suitable for Sun’s conjectures considered here. To prove Sun’s statement, we devise the following two lemmas, which play an important role in our later proofs.
Lemma 2
Proof
Putting \((\mu ,\nu )=(3,2)\) in (36.9) and (36.7) in [9, p. 69] to deduce (2.12) and (2.14), respectively. By [9, p. 46, Entry 30 (ii) and (iii)], we see that
Employing (2.16) in (2.12) and (2.14), we readily achieve (2.13) and (2.15), respectively.\(\square \)
Lemma 3
Proof
Putting \((\mu ,\nu )=(3,2)\) in [9, p. 68, eq. (36.3)], we obtain
Employing (2.1) and (2.5), the above identity can be rewritten as follows,
Next, applying [9, p. 68, eq. (36.2)] with \(A=B=1\), \(\mu =3\), and \(\nu =2\), we find that
Using (2.1) and (2.6) in (2.20),
Combining (2.19) and (2.21) together to deduce that
From (2.9),
From the above two identities for \(\varphi (q)\varphi (q^5)\), we immediately see that
and the desired identities follow. \(\square \)
Lemma 4
Proof
Identity (2.22) follows from [10, eq. (30.10.4)]. Multiplying both sides of (2.22) by \(\frac{f_1f_2}{f_6}\) and applying (2.3), we immediately deduce (2.23). \(\square \)
To end this section, we present the following fact which will be used frequently and without be explicitly mentioned. Let \(\{a(n)\}_0^\infty \) be the sequence defined by
where \(A_i(q)\) is an arbitrary infinite series in q and \(m\ge 2\). For \(0\le i<m\), collecting those terms on each side of (2.24) where the powers of q are of the form \(mn+i\), we have
3 Proof of Theorem 1
It follows from (2.7) and (2.9) that
Picking out the terms involving \(q^{4n}\) and \(q^{4n+2}\) in (3.1), we deduce that, respectively,
and
Substituting (2.11) into (3.3),
from which we derive
which implies that
Substituting (2.13) into (3.6), and using (2.9) and (2.11) to arrive at
Extracting the terms \(q^{2n}\) for \(n\ge 0\), replacing \(q^2\) by q, and employing (2.3)–(2.6),
Applying (2.23) and (2.18), (3.7) can be rewritten as follows,
Combining (3.4) and above identity, (1.3) follows immediately.
It follows that
Employing (2.9) and (2.10), we deduce that
and
Substituting (2.9) and (2.15) into (3.5), we find that
and
Using first (2.3) and (2.6), then (2.10) and (2.22), it follows that
from which we extract
Applying first (2.5), (2.6), (2.3), (2.4) then (2.17) and (2.18) to achieve that
By (3.10) and (3.12), we get the desired relation (1.4).
Collecting those terms in which the power of q is even in (3.9) and then employing (2.8) and (2.9) yields
and
Appealing to (3.3), we have
Equating the coefficients of \(q^{2n+1}\) to obtain
Hence, the relation (1.5) follows immediately from (1.3) and the above relation.
Employing first (2.6), (2.3) then (2.10) and (2.22) in (3.11),
from which we extract the terms involving \(q^{2n}\), then using (2.5) and (2.3) yields
In light of (2.14), (2.11) and (3.6),
Applying (3.8), we conclude that
Equating the coefficient of \(q^n\), and using (1.5), the final relation (1.6) follows immediately. \(\square \)
4 Proof of Theorem 2
which implies that
and
This allows us to find that
from which we extract
Substituting (2.9), (2.11) and (2.13) into (4.4), we see that
This implies that
Using first (2.3)–(2.6), then (2.23) and (2.18) to deduce that
This, with (4.3), yields the relation (1.7).
Applying (2.11) in (4.1) gives
Hence we have
and
By (2.15) and (2.9), (4.5) can be rewritten as follows,
This implies that
If we utilize first (2.6), (2.3), then (2.10) and (2.22), we achieve that
Extracting those terms in which the power of q is even, then using (2.3)–(2.6), (2.17) and (2.18), we deduce that
Combining (4.9) and (4.11) gives the desired relation (1.8).
Selecting terms whose the power of q is even in (4.7), using (2.8) and (2.9), we find that
and
Substituting (4.2) into (4.14), and comparing the coefficients of \(q^{2n+2}\) to deduce that
Hence, relation (1.9) follows immediately from the above relation and (1.7).
Extracting the terms involving \(q^{2n-1}\) in (4.10), using (2.10) and (2.22), gives
and
Invoking (2.14), (2.11) and (4.4),
Utilizing (4.6) to deduce that
This and (1.9) implies (1.10).
Applying (2.11) and (4.13) to deduce that
and
Combining (4.8) and the above identity yields,
which gives
Hence, combine the above relation and (1.8) to derive (1.11).
From (4.15) and (4.12), we have
Comparing the coefficients of \(q^{4n+1}\) on both sides gives
Employing (1.10) and the above relation, we get the final relation (1.12). \(\square \)
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Acknowledgements
The authors would like to thank the referee for helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11871246).
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Cao, L., Lin, B.L.S. Proofs of some conjectures of Sun on representations by linear combinations of triangular numbers. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 17 (2024). https://doi.org/10.1007/s13398-023-01515-6
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DOI: https://doi.org/10.1007/s13398-023-01515-6