On Eisenstein series, color partition and divisor function

Ramanujan recorded Eisenstein series identities of level 5 of weight 2. The objective of this article is to prove these identities by classical method and also we find some new Eisenstein series identities for level 7. Finally we make use of these results to obtain convolution sums of divisor functions. We will also be introducing certain restricted color partition functions which helps us to represent the convolution sums in a much elegant form than the existing ones.


Introduction
The smallest and the last chapter of the organized portion of the Ramanujan's notebooks is dedicated to Eisenstein series and deducing identities of the form P 1 − n P n = linear combinations of ratios of theta functions. Ramanujan documented identities of level 3,5,7,9,11,17,19,23,25, 31 and 35. The Chapter 21 of the Ramanujan's notebooks Part III edited by Berndt focuses on proofs of the equalities connecting a certain linear combinations of Eisenstein series with theta functions. Cooper and Toh [9] have proved results for n = 5, 7. Cooper [7] proved certain Eisenstein series identities for level 5 using k-parametrization. Cooper and Ye [11] proved some Eisenstein series identities for level 7 using the theory of modular forms. In Sect. 4 of this article, we establish certain new Eisenstein series identities for level 5 of the above type by making use of Bailey's formula which can be deduced from his very-well known 6 ψ 6 summation formula: For convenience, we set σ 1 (n) = σ (n). We define the convolution sum W k (n) by where a, b, k, n ∈ N. Note that W (1,k) (n) = W (k,1) (n) = W k (n). The sum W k (n) have been evaluated for some k s ≤ 64, for all n ∈ N. For a brief history and development of the this field, one may refer [17]. Most of the convolution sums are evaluated using arithmetic or analytic methods and some use theory of quasi-modular forms. The convolution sums for level 5, 7, 10 and 14 have a constant term. In Sect. 5 of this paper, our object is to deduce convolution sums for level 5, 7, 10 and 14 entirely in terms of divisor functions and partition functions by using the identities obtained in the previous sections and to prove certain fascinating congruence properties of these partition functions.

Preliminary results
As usual for any complex number a and q with |q| < 1, we define Further for convenience, we set The following u (resp. U ), v (resp. V ) and w (resp. W ) relations are the important tools to prove our main results:  [25]. Recently Bhargava et al. [5] have found a very simple proof of (2.2), (2.3) and (2.4) using Ramanujan's 1 ψ 1 summation formula. Elementary proofs of (2.5), (2.6) and (2.7) by using only theta functions results of Ramanujan can be found in [26]. We make use of the famous Ramanujan's 1 ψ 1 Summation formula: We also determine some Eisenstein series identities for level 7 in addition to that of Cooper and Ye [11] by using (2.5)-(2.7). Ramanujan's Eisenstein series P(q) and Q(q) are defined by

Theorem 2.2 We have
Berndt [3, p. 139] proved the above theorem by using the modular equation identities. Cooper [8] too proved using the elliptic funtion identities. From the above theorem, we have and For any complex number a and for each positive integer n, let (a) n = a(a + 1)(a + 2) · · · (a + n − 1) and (a) 0 = 1.
The Gauss ordinary hypergeometric series is defined by A modular equation of degree n is a relation between α and β that is induced by the equation In such relation, we say β has n th degree in α. We also define the multiplier m by where The following theorem of Ramanujan's play a central role in transforming a theta function identity into a modular equation and vice-versa.

Theorem 2.3
For q = exp −π K K and z = K , then 20)

Quintic and septic identities for Q n 's
In this section, we deduce Q n 's for n = 1, 2, 5, 7, 10 and 14 in terms of f n for n = 1, 2, 5, 7, 10 and 14.

3)
and Proof The identity (2.4) is equivalent to (3.5) From the above identity, one may deduce the following:

7)
and (2.4) and the definition of x, we find that Substituting the identity obtained by subtracting 3 times of (3.7) from 5 times of (3.6) in (2.12) and then using (3.10) in the resulting identity, we obtain (3.1). Substituting the identity obtained by adding 5 times of (3.6) with 3 times of (3.7) in (2.13) and then employing (3.10) in the resulting identity, we obtain (3.2).
Replacing q by q 5 in (2.12) and then employing the identity obtained by subtracting 3 times of (3.9) from 5 times of (3.8) in the resulting identity and then using (3.9), we obtain (3.3).
Replacing q by q 5 in (2.13) and then employing the identity obtained by adding 5 times of (3.8) and 3 times of (3.9) in the resulting identity and then using (3.10), we obtain (3.4).

13)
and (3.14) Proof From (2.6), we find that From (2.5) and the definition of y, we have 1 U 2 + 49U 2 = y 3 − 6y − 2 y 3 − 6y + 9 and From the above two equation, we have and The identity (2.6) is equivalent to From the above identity, one may deduce the following identities:

19)
and Using (3.16) and the identity obtained by subtracting 3 times of (3.20) from 5 times of (3.19) in (2.12) and then using (3.15) to eliminate y 3 − 6y + 5 y 3 − 6y + 9 and then employing the definition of y in the resulting identity, we obtain (3.11).
Using (3.16) and the identity obtained by adding 5 times of (3.19) and 3 times of (3.20) in (2.13) and then using (3.15) and the definition of y in the resulting identity, we obtain (3.12).
Replacing q by q 7 in (2.12) and using (3.17) and the identity obtained by subtracting 3 times of (3.22) from 5 times of (3.21) in the resulting identity and then using (3.15) and the definition of y, we obtain (3.13).
Replacing q by q 7 in (2.13) and using (3.17) and the identity obtained by adding 5 times of (3.21) and 3 times of (3.22) in the resulting identity and then using (3.15) and the definition of y, we obtain (3.14).

Eisenstein series of level 10 and 14
This section acts as the bridge between Sects. 3 and 5. In this section, we determine certain new identities relating Eisenstein series of level 5 and 10. We also find certain Eisenstein series identities for level 7 and 14.
Adding the above two identity, we have .
Using the above in (4.1),we have Using the above in (4.3) and then using the resulting identity in (4.2), we find that Expanding each of the summands into geometric series, interchanging the order of summation, summing the inner geometric series and then using definition of P n , we obtain the required result.

Theorem 4.2
Adding the above two identity, we have Setting a = −1, b = −q 5 in (1.1), we find that Using the above in (4.5), we have From [3, Entry 29, p. 45], we find that Squaring (4.7) and using the resulting indentity along with (4.4) and the above identity in (4.6), we find that Expanding each of the summands into geometric series, interchanging the order of summation, summing the inner geometric series and then using definition of P n , we obtain the required result.

Lemma 4.3 We have
Proof Using ψ(q) = f 2 2 f 1 and the defintion of u and w from (2.1), we have Using (2.2), (2.3) and (2.4), we have the required result.

Theorem 4.4 We have
and (4.10) Proof Let β have degree 5 over α, z 1 , z 5 be as defined in (2.15) and m be the multiplier for degree 5 defined by (2.14 and (4.12) Using the above in (4.11), we find that Similarly tranforming the square of the right hand side of (4.9) by using (2.19), (2.22) and (2.23) into modular equations, we have (4.14) From [3, p. 286, Eqs. (13.12), (13.13) and (13.14)], we have α β Using the above in (4.14) and then taking the positive square root, we have Guided by Lemma 4.3, we have considered the positive root. The identity (4.9) follows from (4.13) and (4.16). The proof of (4.10) is similar to that of (4.9).

Corollary 4.5 We have
and Remark From the above corollary, Entry 4(i) of Chapter 21 of Ramanujan's second notebook [19] follows.

Theorem 4.6 We have
where T is as defined in the Theorem 3.1.
Proof From (4.17) and (4.18), we have Using the above identity in (4.22), we have From (3.6), we deduce that Employing the above identity in (4.23) and then using (3.10) to eliminate √ x 2 − 20 and √ x 2 − 4 in the resulting identity, we deduce that From (3.8), we deduce that Changing q to q 5 in (4.23) and then employing (4.25) and (3.10) in the resulting identity, we find that Consider −3P 1 + 2P 2 − 5P 5 + 30P 10 = 3 (−P 1 + 2P 2 ) + 4 (−P 2 + 5P 10 ) + 5 (−P 5 + 2P 10 ) . Using and From the above two equations, we have Employing the above equation in (4.29), we have (4.30) The identity (2.5) is equivalent to From the above identity, we have and Changing q to q 2 in (4.30) and then using (4.32) and the definition of y in the resulting identity, we obtain Using the above identity in (4.23) and then using (3.15) and the definition of y in the resulting identity, we obtain Using (3.17) and (3.21), we find that (4.35) Changing q to q 7 in (4.23) and then using the (4.35) and (3.15) and the definition of y in the resulting identity, we obtain Using (4.33), (4.34) and (4.36) in the above equation, we obtain (4.27). Now consider Using (4.33), (4.34) and (4.36) in the above equation, we obtain (4.28).

Applications to convolution sums of divisor function
In this section, we use some of the results of preceeding sections to evaluate convolution sums. We recall the definition of color partition and also we introduce certain restricted color partition functions which will be used in the evaluation of the convolution sums to avoid the coefficient functions of some infinite series existing in the convolution sums of [9,10,15,16,18,24]. For convenience, we adopt the standard notation (a 1 , a 2 , . . . , a n ; q) ∞ = n i=1 (a i ; q) ∞ and define q m± ; q n = q m , q n−m ; q n , (m < n) ; n, m ∈ N.
Definition 5.1 [13] A positive integer n has l colors if there are l copies of n available colors and all of them are viewed as distinct objects. Partitions of a positive integer into parts with colors are called "colored partitions".
For example, if 1 is allowed to have 2 colors, then all the colored partitions of 2 are 2, 1 r + 1 r , 1 g + 1 g and 1 r + 1 g . Where we use the indices r (red) and g (grey) to distinguish the two colors of 1. Also is the generating function for the number of partitions of n where all the parts are congruent to u (mod v) and have k colors.
We now introduce the following seven restricted color partition funtions for lateral use in this section: Definition 5.2 Let p (d,e) (n) (resp. p (d,o) (n)) denote the number of distinct partitions of n with even (resp. odd) number of parts and parts congruent to 0 (mod 5) with 8 colors, parts not congruent to 0 (mod 5) with 4 colors each.
From the above definition, it is clear that  i. parts which are congruent to ±1, ±3 (mod 10) have 1 color. ii. parts which are not congruent to 0, ±1, ±3 (mod 10) are distinct and have 4 colors each and parts congruent to 0 (mod 10) are distinct and have 8 colors. iii. number of parts which are not congruent to ±1, ±3 (mod 10) being even (resp. odd).
Similarly for n = 5, one can easily find that m ( One can also find the values of coefficients involved in following definitions by using the same technique used in the above two tables.  t (d,o) (n)) denote the number of distinct partitions of n with even (resp. odd) number of parts and parts congruent to 0 (mod 10) with 8 colors, parts congruent to ±2, ±4, 5 (mod 10) with 4 colors each, parts congruent to ±1, ±3 (mod 10) with 5 colors each.
The above definition yields
From the above definition, one can easily see that
From the above definition, we see that From the above definition, we have where A(n) = a (d,e) (n) − a (d,o) (n), with A(0) = 1.
From the above definition, one may find that where As stated in the beginning of this section, the identity (5.8) and (5.17) are same as that deduced by M. Lemire and K. S. Williams [15] and also Cooper and Toh [9] have deduced (5.8) and (5.17), apart from the constant term, which we have represented using partition function. Lemire and Williams have deduced these identities by finding the direct relations between Eisenstein series of weight 2 and 4 from existing identities found in [21]. Cooper and Toh have deduced by employing modular equations and differentiation. Royer [24] too have deduced (5.8), (5.10), (5.17) and (5.19) by the method based on quasimodular forms. Similarly the identity (5.9) is same as that deduced by Cooper and Ye [10] by quasi-modular. Ntienjem [16] have deduced (5.18) and (5.19) by employing modular groups and modular forms. Ramakrishnan, Sahu and Singh [18] have also deduced (5.17), (5.18) and (5.19). Our method is very elementary as we used certain Eisenstein series identities of weight 2 (also deduced by elementary approach) to find the required convolution sums. Theorem 5. 9 We have Proof Adding 4 times of (4.17) to (4.18), we have Squaring the above identity on both sides and then using (2.2), (2.3) and (2.4), we obtain where Adding 8 times of (3.1) and 5 times of (3.3) and then using (2.3), we have Replacing q by q 5 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.11) and using (2.9) and (2.11), we obtain Using (5.12) in the above to eliminate X 1 and then using (2.10) and (5.1) and then equating the coefficients of q n on both sides of the resulting identity, we obtain (5.8). Subtracting 3 times of (4.20) from (4.19), we have Squaring the above on both sides and then using (2.2) and (2.4), we obtain where From Theorem 3.1, (2.2) and (2.4), we have Replacing q by q 2 in (2.9) and in (2.11) and q by q 5 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.13), we find that Using (5.14) in the above to eliminate X 2 and then using (2.10), (5.2) and (5.3) and then equating the coefficients of q n on both sides of the resulting identity, we obtain (5.9). Subtracting (4.20) from 3 times of (4.19), we have Squaring the above on both sides and then using (2.2) and (2.4), we obtain Replacing q by q 10 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.4) and using (2.9) and (2.11), we obtain Using (5.16) in the above to eliminate X 3 and then using (2.10), (5.2) and (5.3) and then equating the coefficients of q n on both sides of the resulting identity, we obtain (5.10).
Squaring the above identity on both sides, we obtain where Adding (3.11) and 49 times of (3.13), we have Replacing q by q 7 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.20) and using (2.9) and (2.11), we obtain Using (5.21) in the above to eliminate X 4 and then using (2.10), (5.4) and (5.5) and then equating the coefficients of q n on both sides of the resulting identity, we obtain (5.17). Subtracting 3 times of (4.28) from (4.27), we have Squaring the above on both sides and then using (2.6), we obtain where From Theorem 3.2 and (2.6), we have Replacing q by q 2 in (2.9) and in (2.11) and q by q 7 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.22), we find that Using (5.23) in the above to eliminate X 5 and then using (2.10), (5.5), (5.6) and (5.7) and then equating the coefficients of q n on both sides of the resulting identity, we obtain (5.18). Subtracting (4.28) from 3 times of (4.27), we have Squaring the above on both sides and then using (2.6), we obtain Replacing q by q 14 in (2.9) and in (2.11) and substituting the resulting identities in the left hand side of (5.24) and using (2.9) and (2.11), we obtain Ramanujan observed certain curious congruence properties of p(n)(number of unrestricted partitions of n) as soon as he noticed the table calculated by Major MacMahon, of the values of p(n), for all values of n from 1 to 200. He [23] found quite simple proofs of the following congruences of p(n): p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7) and p(35n + 19) ≡ 0 (mod 35).
Extracting the terms involving q 5n+1 from both sides of the above and then dividing by q and replacing q 5 by q in the resulting identity, we obtain (5.30). Extracting the terms involving q 5n+2 from both sides of the above and then dividing by q 2 and replacing q 5 by q in the resulting identity, we obtain (5.31). Extracting the terms involving q 5n+4 from both sides of the above and then dividing by q 4 and replacing q 5 by q in the resulting identity, we obtain (5.33).