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Some modular relations for the Göllnitz–Gordon functions by an even–odd method
Abstract
In this paper, we derive some new modular relations which only involve Göllnitz–Gordon functions by using an even–odd method. We also give new proofs of some modular relations of the same nature established earlier by Huang and Chen.
... Identity (13) is from [8]. Replacing q by −q in (5), (10), and (13) and then employing (4), we arrive at (7), (11), and (14), respectively. Identity (9) is from [12], (12) is Lemma 2.6 of [1], and (15) is from [11]. ...
... Replacing q by −q in (5), (10), and (13) and then employing (4), we arrive at (7), (11), and (14), respectively. Identity (9) is from [12], (12) is Lemma 2.6 of [1], and (15) is from [11]. ...
... Utilizing (11) in (29), we obtain ∞ n=0 a 3 (2n + 1)q n = 2 g 7 4 g 3 6 g 9 2 g 12 ...
Naika and Harishkumar (2022) defined the restricted cubic partition function a ℓ (n), which counts the number of ℓ-regular cubic partitions of any positive integer n where the first occurrence of each distinct odd part may be overlined, and proved infinite families of congruences for the particular cases a 3 (n) and a 5 (n). In this paper, we extend and generalize some of the results of Naika and Harishkumar, and also prove new congruences of a ℓ (n) for ℓ = 2k, 2k + 1, 4k, 4k + 1, 3, 4, 5, 8, 9, 16, where ℓ > 2 and k > 0 are integers.
... In this paper, we prove several infinite families of congruences for the partition function M s,t (n) for (s, t) ∈ { (3,9), (4,6), (4,9), (4,12), (6,9), (9,12)} by using theta-function and qseries identities. Congruences for M s,t (n) are proved in Section 3. Section 2 is devoted to list some preliminary results. ...
... In this paper, we prove several infinite families of congruences for the partition function M s,t (n) for (s, t) ∈ { (3,9), (4,6), (4,9), (4,12), (6,9), (9,12)} by using theta-function and qseries identities. Congruences for M s,t (n) are proved in Section 3. Section 2 is devoted to list some preliminary results. ...
... 4,12 (n)q n ≡ ψ(q) (mod 2).(3.117) ...
A partition of a positive integer n is said to be simultaneously s-regular and t-regular partition if none of the parts are divisible by s and t. In this paper, we establish many infinite families of congruences for simultaneously s-regular and t-regular partition function by considering some particular values of s and t.
... Recently, Williams [4] employed his product-to-sum formula [3] in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. His main results can be stated as the following theorem: Theorem 1.1 For n, k ∈ N 0 , a 1 (2 2k+1 (8n + 5)) = a 2 (3 2k+1 (3n + 2)) = a 3 (3 2k (3n + 2)) = a 4 (3 2k+1 (6n + 5)) = a 5 (3 2k+1 (12n + 8)) = a 5 (3 2k+1 (12n + 11)) = a 7 (2 2k (8n + 7)) = a 8 (2 2k+1 (8n + 7)) = a 9 (3 2k+1 (3n + 2)) = a 10 (2 2k (8n + 7)) = 0, where the generating functions of the a i (n) are given by where ( j, a, b, c) ∈ { (1,2,4,4), (2,1,9,9), (3,1,9,9), (4,1,9,9), (5,1,9,9), (7,1,4,16), (8, 2, 4, 16), (9, 1, 9, 81), (10, 4, 4, 16)} and ( j, d, e) ∈ {(1, 16, 10), (2,9,6), (3,3,2), (4,9,6), (5,9,6), (7,8,7), (8,16,14), (9, 9, 6), (10, 4, 3), (10, 32, 28)}. ...
... Recently, Williams [4] employed his product-to-sum formula [3] in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. His main results can be stated as the following theorem: Theorem 1.1 For n, k ∈ N 0 , a 1 (2 2k+1 (8n + 5)) = a 2 (3 2k+1 (3n + 2)) = a 3 (3 2k (3n + 2)) = a 4 (3 2k+1 (6n + 5)) = a 5 (3 2k+1 (12n + 8)) = a 5 (3 2k+1 (12n + 11)) = a 7 (2 2k (8n + 7)) = a 8 (2 2k+1 (8n + 7)) = a 9 (3 2k+1 (3n + 2)) = a 10 (2 2k (8n + 7)) = 0, where the generating functions of the a i (n) are given by where ( j, a, b, c) ∈ { (1,2,4,4), (2,1,9,9), (3,1,9,9), (4,1,9,9), (5,1,9,9), (7,1,4,16), (8, 2, 4, 16), (9, 1, 9, 81), (10, 4, 4, 16)} and ( j, d, e) ∈ {(1, 16, 10), (2,9,6), (3,3,2), (4,9,6), (5,9,6), (7,8,7), (8,16,14), (9, 9, 6), (10, 4, 3), (10, 32, 28)}. ...
... Recently, Williams [4] employed his product-to-sum formula [3] in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. His main results can be stated as the following theorem: Theorem 1.1 For n, k ∈ N 0 , a 1 (2 2k+1 (8n + 5)) = a 2 (3 2k+1 (3n + 2)) = a 3 (3 2k (3n + 2)) = a 4 (3 2k+1 (6n + 5)) = a 5 (3 2k+1 (12n + 8)) = a 5 (3 2k+1 (12n + 11)) = a 7 (2 2k (8n + 7)) = a 8 (2 2k+1 (8n + 7)) = a 9 (3 2k+1 (3n + 2)) = a 10 (2 2k (8n + 7)) = 0, where the generating functions of the a i (n) are given by where ( j, a, b, c) ∈ { (1,2,4,4), (2,1,9,9), (3,1,9,9), (4,1,9,9), (5,1,9,9), (7,1,4,16), (8, 2, 4, 16), (9, 1, 9, 81), (10, 4, 4, 16)} and ( j, d, e) ∈ {(1, 16, 10), (2,9,6), (3,3,2), (4,9,6), (5,9,6), (7,8,7), (8,16,14), (9, 9, 6), (10, 4, 3), (10, 32, 28)}. ...
Recently, combining a product-to-sum formula and conditions for the non-representability of integers by certain ternary quadratic forms, Williams gave ten eta quotients such that their Fourier coefficients vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. In this paper, we generalize Williams’ results by utilizing theta function identities.
... Xia and Yao [18] proved that ...
... In [18], Xia and Yao proved that ...
In 2015, Jennings-Shaffer defined the higher order SPT-functions for overpartitions and gave the combinatorial interpretation of . In recent years, some congruences for have been proved. Garvan and Jennings-Shaffer presented a characterization of the parity on . Motivated by their work, in this paper, we give a characterization of congruences modulo 2 on and prove a congruence modulo 4 for and several parity results for by using the generating functions of which denote the number of overpartitions of n whose first residual crank is congruent to r modulo 8.
... Relations (8) and (9) have been proved in [17]. The following identity given by Ramanujan in [16] and Hirschhorn [12], with a simple proof by means of relation (6). ...
... Xia and Yao proved relations (11) and (12) in [17]. ...
In view of the Rogers-Ramanujan theta function, several researchers have
emphasized the subject of integer partitions and their generating functions for years.
The work has not dealt with in an inclusive and closed-form by now. Thus, these generating functions are extremely desirable to be developed in a generalized way. In this
paper, a novel methodology is proposed to build many new modular relations. By incorporating these new relations, generalized forms of regular partitions in the spirit of
Rogers-Ramanujan and G¨ollnitz-Gordon functions with four, six, and nine dissections
are established. As an application of these generalized generating functions, an infinite
family of new congruences modulo 2 has also been developed.
... The remaining two identities of the lemma were found by Robins [21, Chapter 1, (1.25), (1. 26)]. ...
... Again, invoking the 2-dissection of 1/f 4 1 from Lemma 2 in the identity above, and then extracting the terms involving q 2n , we obtain Therefore, U 8n+12 (P 3 ) = 0, which is equivalent to (xix). The remaining results in Theorem 5 can be proved in a similar fashion by using some modular relations between S(q) and T (q) found by Huang [19] and Xia and Yao [26] that are analogous to (38) Since the proofs of the results in Theorem 6 are similar in nature, we choose to prove (xv) and (xvi) only, which are somewhat trickier to prove than the others. We set ...
We show that the series expansions of certain q-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let R(q) denote the Rogers-Ramanujan continued fraction having the well-known q-product repesentation If \begin{align*} \sum_{n=0}^{\infty}\alpha(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}\alpha^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}\beta(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}\beta^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} \alpha(5n+r)&=-\alpha^{\prime}(5n+r-2) \quad r\in\{3,4\},\\ \beta(10n+r)&=-\beta^{\prime}(10n+r-6) \quad r\in\{7,9\}. \end{align*}
... (39) Lemma 2.5 was proved by Xia and Yao [13]. Replacing q by −q in (38) and using ...
... Equating the coefficients of q pn on both sides of (64) and then replacing q p by q, we obtain ∞ n=1 which is the α + 1 case of (13). Extracting the terms involving q pn+j for 1 ≤ j ≤ p − 1 in (64), we arrive at (14). ...
Recently Andrews, Lewis and Lovejoy introduced the partition functions PD(n) defined by the number of partitions of n with designated summands and they found several modulo 3 and 4. In this paper, we find several congruences modulo 3 and 4 for PBD3(n), which represent the number of 3-regular bi-partitions of n with designated summands. For example, for each n ≥ 1 and α ≥ 0 PBD3(4 · 3α+2n + 10 · 3α+1) ≡ 0 (mod 3).
... Then, by substituting q 3 by q in (30) we obtain (27) , and (28) can be obtained from (31) by dividing both sides by q and then replacing q 3 by q . In a similar way, we can obtain (29) from (32). ...
In this paper, we explore a multi-restricted set of partitions that lies at the intersection of the classical -regular and t-distinct (parts appearing at most t times) partition sets. We introduce the t-Schur's partitions and overpartitions, investigate other equinumerous classes of partitions, and examine their combinatorial and arithmetic properties.
... The identities (2.10), ( [5]. For (2.14) see [8] and for (2.16) see [11]. ...
Keith (Integers 23, A9, 2023) introduced the simultaneously s-regular, t-regular and s-distinct partition function which counts the total number of partitions of a positive integer n such that none of the parts are divisible by s and t and each part appears fewer than s times. The simultaneously s-regular, t-regular and s-distinct partition function is denoted by , where are integers. In this paper, we prove some infinite families of congruences for the partition function for (s,t)=(3,4), (4, 9), (5x, 5y) and (7x, 7y), where x and y are two positive integers. We also offer congruences of for prime values of s.
... The proofs of Identities (23) and (24) are similar to Identity (22), and hence, are omitted here. Adding Identities (30) and (31) together and then by using Identity (11) with q = q 2 and Lemma 1, we obtain Identity (25). The proofs of Identities (26) and (27) are similar to Identity (25), and hence, are omitted here. ...
In this paper, we establish several modular relations involving the Rogers–Ramanujan–Slater type functions of order eighteen which are analogues to Ramanujan’s well known forty identities. Furthermore, we give partition theoretic interpretations of two modular relations.
... The proofs of (2.13), (2.14) and (2.15) can be seen in [15], [7] and [13], respectively. ...
Let count the number of , -regular partitions respectively. In this paper we shall derive infinite families of congruences for modulo 2 when , for modulo 8, modulo 9 and modulo 12 when and modulo 2 when .
... From Lemma 3.5 in [12] we have ...
For a positive integer , let denote the number of -regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for and . We prove a specific case of a conjecture of Keith and Zanello on self-similarities of modulo 2. We next prove that the series is lacunary modulo arbitrary powers of 2. We also prove that the series is lacunary modulo 2.
... Therefore, U 8n+12 (P 3 ) = 0, which is equivalent to (xix). The remaining results in Theorem 5 can be proved in a similar fashion by using some modular relations between S(q) and T (q) found by Huang [19] and Xia and Yao [26] that are analogous to (38) and (39). In the following table, we mention the locations of the corresponding relations used to prove the results. ...
We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ra-manujan's continued fractions. For example, let R(q) denote the Rogers-Ramanujan continued fraction having the well-known q-product repesentation R(q) = (q; q 5) ∞ (q 4 ; q 5) ∞ (q 2 ; q 5) ∞ (q 3 ; q 5) ∞. If ∞ n=0 α(n)q n = 1 R 5 (q) = ∞ n=0 α (n)q n −1 , ∞ n=0 β(n)q n = R(q) R (q 16) = ∞ n=0 β (n)q n −1 , then α(5n + r) = −α (5n + r − 2) r ∈ {3, 4}, β(10n + r) = −β (10n + r − 6) r ∈ {7, 9}.
... From Lemma 3.5 in [12] we have ...
For a positive integer , let denote the number of -regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for and . We prove a specific case of a conjecture of Keith and Zanello on self-similarities of modulo 2. We next prove that the series is lacunary modulo arbitrary powers of 2. We also prove that the series is lacunary modulo 2.
... For the proof of (2.9), see [7, p.40]. Equation (2.10) was proved by Xia and Yao [13]. For the proof of (2.11), see [8]. ...
For any relatively prime integers r and s, let denotes the number of (r,s)-regular partitions of a positive integer n into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2 for the particular case . In this paper, we prove several infinite families of congruences modulo 2 and 4 for the following pairs of , and . For example, where .
... (3.11) Xia and Yao [XiYa12] proved that ∞ n=0 b 9 (n)q n = f 3 12 f 18 f 2 2 f 6 f 36 ...
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Let denote the number of l-regular partitions of n and denote the number of l-regular bipartitions of n. In this paper, we establish several infinite families of congruences satisfied by for . We also establish a relation between and .
... Xia and Yao [XiYa12] proved that ...
We study the “shift-Ramanujan expansion” to obtain a formulae for the shifted convolution sum of general
functions f, g satisfying Ramanujan Conjecture; here, the shift-Ramanujan expansion is with respect to the integer shift a > 0.
Assuming Delange Hypothesis for the correlation, we get the “Ramanujan exact explicit formula”, a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = the von Mangoldt function; so the correlation of shift 2k, for natural k, corresponds to
2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.
... Lemma 4 was proved by Xia and Yao [18]. ...
In recent work, Bringmann et al. used q-difference equations to compute a two-variable q-hypergeometric generating function for the number of overpartitions where (i) the difference between two successive parts may be odd only if the larger of the two is overlined, and (ii) if the smallest part is odd then it is overlined, given by . They also established the two-variable generating function for the same overpartitions where (i) consecutive parts differ by a multiple of (k+1) unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k+1, enumerated by . As an application they proved that if n is not a square. In this paper, we extend the study of congruence properties of , and we prove congruences modulo 3 and 6 for , congruences modulo 2 and 4 for and , congruences modulo 4 and 5 for , and congruences modulo 3, 6 and 12 for .
... Identity (2) is nothing but Lemma 3.5 in [16]. ...
... Using the idea of Rogers, Watson [23] and Bressoud [12], Huang [16] and Chen and Huang [13] have established several modular relations for the Göllnitz-Gordan functions and Baruah et al. [10] have given alternative proofs some of them by using Schröter's formulas and some simple theta functions identities of Ramanujan. These functions were studied by Xia and Yao [24]. Septic analogues of Rogers-Ramanujan type functions were studied by Hahn [15,17], Nonic analogues of Rogers-Ramanujan type functions were studied by Baruah and Bora [9] and sextodecic analogues of the Rogers-Ramanujan functions were studied by Gugg [14] and Adiga and Bulkhali [3]. ...
... Lemma 6 was proved by Xia and Yao [13]. ...
Andrews, Lewis and Lovejoy introduced the partition functions PD(n) and PDO(n) defined by the number of partitions of n with designated summands and the number of partitions of n with designated summands in which all parts are odd, respectively. They found several generating function identities and congruences modulo 3, 4, and 24 satisfied by the functions. In this paper, we find generating function identities and congruences modulo 4, 9, 12, 36, 48, and 144 for PD_3(n), which represents the number of partitions of n with designated summands, whose parts are not divisible by 3.
... Gugg [14] found new proofs of modular relations, which involve only S(q) and T (q). E. X. W. Xia and X. M. Yao [19] offered new proofs of some modular relations established by Huang [16] and Chen and Huang [17]. They also established some new relations that involve only Göllnitz-Gordon functions. ...
Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations.
In this article, by employing some theta identities of Ramanujan, we establish many Ramanujan-type congruences modulo 2, 5 and 7 for , where .
A k-colored partition of a positive integer n is a k-tuple of partitions such that . Recently, Fu and Tang defined a generalized crank for k-colored partitions by , where denotes the number of parts in . They also proved some inequalities and equalities for which counts the number of k-colored partitions of n with generalized crank congruent to m modulo j. Very recently, Lin, Peng and Toh established some new Andrews–Beck type congruences on which denotes the total number of parts of in each k-colored partition of n with congruent to m modulo j. In this paper, motivated by the work of Fu–Tang and Lin–Peng–Toh, we establish the generating functions for when j=2,3,4 and deduce some new inequalities and equalities for .
Let Bk,𝓁(n) denote the number of (k, 𝓁)-regular bipartitions. Recently, Dou discovered an infinite family of congruences modulo 11 for B3,11(n). She also presented several conjectures on Bk,𝓁(n). In this article, we establish infinite families of congruences for Bk,𝓁(n) modulo certain positive integers, for specific values of k, 𝓁. We next study the distribution of Bk,𝓁(n) for (k, 𝓁) ∈ {(3, 5),(2, 6),(2, 4),(4, 6)}. Moreover, we prove several infinite families of congruences modulo 2 and 3 for Bk,𝓁(n). We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.
Beck introduced two partition statistics NT(r,m,n) and ,which denote the total number of parts in the partition of n with rank congruent to r modulo m and the total number of ones in the partition of n with crank congruent to r modulo m, respectively. In recent years, a number of congruences and identities on NT(r,m,n) and for some small m have been established.In this paper, we prove an identity on NT(r,8,n) and which confirm a conjecture given by Mao.
Let b9(n) denote the number of 9-regular bipartitions of n with distinct even parts. We prove that b9(n) satisfies several infinite families of congruences modulo 4, along with an internal congruence. We also show that b9(6n+4) and b9(24n+6) are almost always divisible by 4. Our proofs mainly rely on arithmetic properties of modular forms and q-series manipulations.
Let and be the set of integers and the set of positive integers, respectively. For and , let N(a, b, c; n) be the number of representations of n as and t(a, b, c; n) be the number of representations of n as . Recently, Sun established many relations between t(a, b, c; n) and N(a,b,c;8n+a+b+c) and listed 43 relations need to be confirmed. More recently, Xia and Zhang proved 19 relations conjectured by Sun. In this paper, by employing Ramanujan’s theta function identities, we prove the remaining 24 relations.
For a positive integer t, let denote the number of t-regular partitions of a nonnegative integer n. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo 2 for for certain values of t. Further, they proposed some conjectures on self-similarities of modulo 2 for certain values of t. In this paper, we prove their conjectures on and . We also prove a self-similarity result for modulo 2.
Let denote the number of -regular partitions of a positive integer n into distinct even parts. In this paper, we prove congruences modulo 2 and 4 for when =3, 5, 7 and 11. We also prove the infinite families of congruences modulo 9, 12, 18 and 24 for For example, for any and , we have \begin{aligned} Ped_9\Big (24 \cdot p^{2\alpha +1}(pn+r)+ 10\cdot p^{2\alpha +2}+1\Big )\equiv 0 \pmod {24}. \end{aligned}
In this paper, we define the partition function the number of [j, k]-partitions of n into even parts distinct, where none of the parts are congruent to (where . We obtain many infinite families of congruences modulo powers of 2 for and congruences modulo powers of 2 and 3 for . For example, for all and where
We study several congruence properties of restricted partition functions such as:
k-color overpartition functions, Andrews' singular overpartitions, Designated summands, -regular cubic partition pairs, -regular bipartition triples and Partition quadruple with t-cores.
In a recent paper, Sun posed six conjectures on the relations between T(a1,a2,⋯,ak; n) and N(a1,a2,⋯,ak; n), where T(a1,a2,⋯,ak; n) denotes the number of representations of n as a1x1(x1+1) 2 + a2x2(x2+1) 2 + ⋯ + akxk(xk+1) 2, where a1,a2,⋯,ak are positive integers, n,x1,x2,⋯,xk are arbitrary nonnegative integers, and N(a1,a2,⋯,ak; n) denotes the number of representations of n as a1x12 + a 2x22 + ⋯ + a kxk2, where this time x1,x2,⋯,xk are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities.
Let , , and denote the number of 9-regular partitions of n wherein odd parts are distinct, even parts are distinct, and the number of 9-regular overpartitions of n, respectively. By considering from an arithmetic point of view, we establish a number of infinite families of congruences modulo 16 and 32, and some internal congruences modulo small powers of 3. A relation connecting above partition functions in arithmetic progressions is obtained as follows. For any ,
6 \text {pod}_{9}(2n + 1) = 2 \text {ped}_{9}(2n + 3) = 3 \overline {A}_{9}(n + 1).
Let (Formula presented.) be the number of overpartitions of (Formula presented.) into parts not divisible by (Formula presented.). In this paper, we find infinite families of congruences modulo 4, 8 and 16 for (Formula presented.) and (Formula presented.) for any (Formula presented.). Along the way, we obtain several Ramanujan type congruences for (Formula presented.) and (Formula presented.). We also find infinite families of congruences modulo (Formula presented.) for (Formula presented.).
Chen and Huang established some elegant modular relations for the Gollnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Gollnitz–Gordon functions. Furthermore, we also provide new proofs of some modular relations for the Gollnitz–Gordon functions due to Gugg.
In this paper, we consider the Rogers-Ramanujan type functions J(q) and K(q) of order ten and establish several modular relations involving these identities, which are analogues to Ramanujan's forty identities for the Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations.
The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.
The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 𝔅k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 𝔅5(n), 𝔅7(n) and 𝔅11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 𝔅5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 𝔅9(n) by establishing the generating functions of 𝔅9(An+B) modulo 4 for some values of A and B.
In view of the modular equation of fifth order, we give a simple proof of Keith’s conjecture which is some infinite families of congruences modulo 3 for the 9-regular partition function. Meanwhile, we derive some new congruences modulo 3 for the 9-regular partition function.
Let t≥2 be an integer. We say that a partition is t-regular if none of its parts is divisible by t, and denote the number of t-regular partitions of n by b
t
(n). In this paper, we establish several infinite families of congruences modulo 2 for b
9(n). For example, we find that for all integers n≥0 and k≥0, b_9 \biggl(2^{6k+7}n+ \frac{2^{6k+6}-1}{3} \biggr)\equiv 0 \quad (\mathrm{mod}\ 2 ).
Recently, C. Adiga and the author have derived general formulas to express the product of two theta functions as linear combinations of other products of theta functions. In this paper, we employ these formulas to establish new identities that are analogous to the famous Schröter's formulas. As applications of these formulas, we establish several new modular relations for Rogers-Ramanujan functions, Göllnitz-Gordon, septic, nonic and dodecic analogues of the Rogers-Ramanujan functions.
In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken k-diamond partitions. Recently, Shishuo Fu generalised the notion of broken k-diamond partitions to combinatorial objects which he termed k dots bracelet partitions. Fu denoted the number of k dots bracelet partitions of n by B-k(n) and proved several congruences modulo primes and modulo powers of 2. More recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for B-5(n), B-7(n) and B-11(n). In this note, we prove some congruences modulo powers of 2 for B-5(n). For example, we find that for all integers n >= 0, B-5(16n + 7) equivalent to 0 (mod 2(5)).
Let b9(n)b9(n) denote the number of 9-regular partitions of n . Recently, employing the theory of modular forms, Keith established several congruences modulo 2 and 3 for b9(n)b9(n). He also presented four conjectures on b9(n)b9(n) and two of them have been proved by Lin, and Xia and Yao. The remaining two conjectures are b9(32n+13)≡0(mod12) and b9(64n+13)≡0(mod24) for n⩾0n⩾0. In this paper, employing 2-dissection formulas for certain quotients of theta functions, we prove that b9(32n+13)≡0(mod4) and b9(64n+13)≡0(mod8) for n⩾0n⩾0. Combining these two congruences and the congruence b9(16n+13)≡0(mod3) proved by Keith, we confirm the remaining two conjectures of Keith. We also establish two infinite families of congruences modulo 9 for b9(n)b9(n). For example, we prove that for all integers n⩾0n⩾0 and k⩾1k⩾1, b9(26kn+5×26k−1−13)≡0(mod9).
Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, "Watson, look up at the stars and tell me what you deduce." Watson says, "I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets." Holmes replies, "Watson, you idiot. Somebody stole our tent." When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an "idiot." He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities. Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. In particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought "more natural" proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for the Rogers- Ramanujan functions. In this paper, for 35 of the 40 identities, we oer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. We also establish several new identities for the Rogers-Ramanujan functions. However, we feel that we have still failed to discover most of Ramanujan's thinking about these identities.
We attempt to obtain new modular relations for the Göllnitz–Gordon functions by techniques which have been used by L. J. Rogers, G. N. Watson, and D. Bressoud to prove some of Ramanujan's 40 identities. Also, we give new proofs for some modular relations for the Göllnitz–Gordon functions which have been previously established by using results from L. Rogers and D. Bressoud. Finally, we give applications of those new modular relations to the theory of partitions.
In 1974 B. J. Birch [ 1 ] published a description of some manuscripts of Ramanujan which contained, among other things, a list of forty identities involving the Rogers-Ramanujan functions
At that time nine of these had been proven, and since then twenty-two more of them have been proven, fifteen of them by David Bressoud in his thesis [ 2 ]. Bressoud gives a synopsis of the extant proofs, where he attributes proofs to H. B. C. Darling [ 3 ], L. J. Rogers [ 4 ], L. J. Mordell [ 5 ], and G. N. Watson [ 6 ].
By means of a technique used by Carlitz and Subbarao to prove the quintuple product identity (Proc. Am. Math. Soc. 32(1):42–44,
1972), we recover a general identity (Chu and Yan, Electron. J. Comb. 14:#N7, 2007) for expanding the product of two Jacobi triple products. For applications, we briefly explore identities for certain products
of theta functionsφ(q), ψ(q) and modular relations for the Göllnitz-Gordon functions.
In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang
and S.-L. Chen. We use Schröter’s formulas and some simple theta-function identities of Ramanujan to establish the relations.
We also find some new modular relations of the same nature.
In a manuscript of Ramanujan. published with his Lost Notebook [21, pp. 236-337], there are forty identities involving the Rogers-Ramunujan functions. According to G. N. Watson, the beauty of these identities are comparable to that of the Rogers-Ramanujan identities. In the paper, we establish modular relations involving the Gollnitz-Gordon Functions which are analogous to Ramanujan's forty identities. Furthermore, ae extract interesting partition results from some of the modular relations. (C) 1998 Academic Press.