Proof of Some Conjectures of Melham Using Ramanujan's ₁𝜓₁ Formula

Abstract

We employ Ramanujan's ₁𝜓₁ formula to prove three conjectures of R. S. Melham on representation of an integer 𝑛 as sums of polygonal numbers.

1. Introduction

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

Theorem 1 (see [1]). Let 𝑟{□+□}(𝑛) denote the number of representations of 𝑛 as a sum of two squares, counting order and sign, and let 𝑑_𝑖,𝑗(𝑛) denote the number of positive divisors of 𝑛 congruent to 𝑖 modulo 𝑗. Then 𝑟{□+□}(𝑛)=4𝑑_1,4(𝑛)−𝑑_3,4(𝑛). (1)

The above theorem can also be recasted in terms of Lambert series as ^∞_𝑛=0𝑟{□+□}(𝑛)𝑞^𝑛=1+4^∞_𝑛=0𝑞^4𝑛+11−𝑞^4𝑛+1−𝑞^4𝑛+31−𝑞^4𝑛+3. (2) 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 𝑟{𝑙□+𝑚□}(𝑛) denote the number of representations of 𝑛 as a sum of 𝑙 times a square and 𝑚 times a square. Then 𝑟{□+3□}(𝑛)=2𝑑_1,3(𝑛)−𝑑_2,3(𝑛)+4𝑑_4,12(𝑛)−𝑑_8,12(𝑛). (3)

Theorem 3 (see [5]). Let 𝑟{𝑙Δ+𝑚Δ}(𝑛) denote the number of representations of 𝑛 as a sum of 𝑙 times a triangular number and 𝑚 times a triangular number. Then 𝑟{Δ+3Δ}(𝑛)=𝑑_1,3(2𝑛+1)−𝑑_2,3(2𝑛+1). (4)

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 𝑞-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 𝑟{𝑙□+𝑚□}(𝑛). In this paper, we show that some of these conjectures can also be proved by using Ramanujan’s famous ₁𝜓₁ formula. We prove three conjectures enlisted in the following theorem which have appeared as (6), (7), and (8), respectively, in [11].

Theorem 4. Consider ^∞_𝑛=0𝑟{Δ+5Δ}(𝑛)𝑞^𝑛=^∞_𝑛=0𝑞^3𝑛+𝑞^7𝑛+11−𝑞^20𝑛+5−𝑞^13𝑛+9+𝑞^17𝑛+121−𝑞^20𝑛+15_∞𝑛=0,𝑟{Δ+6Δ}(𝑛)𝑞^𝑛=^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21_∞𝑛=0,𝑟{2Δ+3Δ}(𝑛)𝑞^𝑛=^∞_𝑛=0𝑞^5𝑛1−𝑞^24𝑛+3+𝑞^7𝑛+21−𝑞^24𝑛+9−𝑞^17𝑛+101−𝑞^24𝑛+15−𝑞^19𝑛+161−𝑞^24𝑛+21. (5)(6)(7)

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

2. Notations and Preliminary Results

Ramanujan’s general theta-function 𝑓(𝑎,𝑏) is defined by [5, page 34, (18.1)] 𝑓(𝑎,𝑏)∶=^∞_𝑘=−∞𝑎^{𝑘(𝑘+1)/2}𝑏^{𝑘(𝑘−1)/2},|𝑎𝑏|<1. (8) Jacobi’s famous triple product identity takes the simple form [5, page 35, Entry 19] 𝑓(𝑎,𝑏)=(−𝑎;𝑎𝑏)_∞(−𝑏;𝑎𝑏)_∞(𝑎𝑏;𝑎𝑏)_∞, (9) where, here and throughout the paper, for |𝑞|<1, 𝑎;𝑞_𝑛∶=^𝑛_𝑘=01−𝑎𝑞^𝑘−1,𝑎;𝑞_∞∶=^∞_𝑘=01−𝑎𝑞^𝑘. (10)

We also use the following notation for the sake of brevity of expressions: 𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛;𝑞_∞=𝑎₁;𝑞_∞𝑎₂;𝑞_∞𝑎₃;𝑞_∞⋯𝑎_𝑛;𝑞_∞. (11) One important special case of 𝑓(𝑎,𝑏) is 𝜓𝑞∶=𝑓𝑞,𝑞³=^∞_𝑘=0𝑞^{𝑘(𝑘+1)/2}=𝑞²;𝑞²_∞𝑞;𝑞²_∞, (12) where the product representations in (12) arise from (9).

From [5, page 46, Entry 30] we find that 𝑓(𝑎,𝑏)=𝑓𝑎³𝑏,𝑎𝑏³+𝑎𝑓𝑎⁻¹𝑏,𝑎⁵𝑏³. (13) Also if 𝑎𝑏=𝑐𝑑, then [5, page 45, Entry 29] 𝑓(𝑎,𝑏)(𝑐,𝑑)=𝑓(𝑎𝑐,𝑏𝑑)𝑓(𝑎𝑑,𝑏𝑐)+𝑎𝑓𝑏𝑐⁻¹,𝑎𝑐²𝑑𝑓𝑏𝑑⁻¹,𝑎𝑐𝑑². (14)

We recall Ramanujan’s ₁𝜓₁ formula in the form [5, page 34, (17.6)] ^∞_𝑛=−∞𝑎;𝑞_𝑛𝑏;𝑞_𝑛𝑧^𝑛=𝑎𝑧;𝑞_∞𝑞𝑎⁻¹𝑧⁻¹;𝑞_∞𝑞;𝑞_∞𝑏𝑎⁻¹;𝑞_∞𝑧;𝑞_∞𝑞𝑎⁻¹𝑧⁻¹;𝑞_∞𝑏;𝑞_∞𝑞𝑎⁻¹;𝑞_∞, (15) where 𝑏𝑎⁻¹<|𝑧|<1. (16)

Finally, we note that ^∞_𝑛=0𝑟{𝑎Δ+𝑏Δ}(𝑛)𝑞^𝑛=𝜓𝑞^𝑎𝜓𝑞^𝑏. (17)

3. Proof of the Conjectures (5)–(7)

Proof of Conjecture (5). In view of (17), we rewrite (5) as 𝜓𝑞𝜓𝑞⁵=^∞_𝑛=0𝑞^3𝑛+𝑞^7𝑛+11−𝑞^20𝑛+5−𝑞^13𝑛+9+𝑞^17𝑛+121−𝑞^20𝑛+15. (18)
Replacing 𝑞 by 𝑞²⁰ and then setting 𝑧=𝑞³, 𝑎=𝑞⁵, and 𝑏=𝑞²⁵ in (15), we have ^∞_𝑛=−∞𝑞⁵;𝑞²⁰_𝑛𝑞²⁵;𝑞²⁰_𝑛𝑞^3𝑛=𝑞⁸;𝑞²⁰_∞𝑞¹²;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞³;𝑞²⁰_∞𝑞¹⁷;𝑞²⁰_∞𝑞²⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞. (19) Expanding the 𝑞-products on the left side of (19) and noting that (1−𝑞⁵)(𝑞²⁵;𝑞²⁰)_∞=(𝑞⁵;𝑞²⁰)_∞, we obtain ^∞_𝑛=−∞𝑞^3𝑛1−𝑞^20𝑛+5=𝑞⁸;𝑞²⁰_∞𝑞¹²;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞³;𝑞²⁰_∞𝑞¹⁷;𝑞²⁰_∞𝑞⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞. (20) Also ^∞_𝑛=−∞𝑞^3𝑛1−𝑞^20𝑛+5=^∞_𝑛=0𝑞^3𝑛1−𝑞^20𝑛+5+⁻¹_𝑛=−∞𝑞^3𝑛1−𝑞^20𝑛+5=^∞_𝑛=0𝑞^3𝑛1−𝑞^20𝑛+5+^∞_𝑛=1𝑞^−3𝑛1−𝑞^−20𝑛+5=^∞_𝑛=0𝑞^3𝑛1−𝑞^20𝑛+5−^∞_𝑛=1𝑞^17𝑛−51−𝑞^20𝑛−5=^∞_𝑛=0𝑞^3𝑛1−𝑞^20𝑛+5−^∞_𝑛=0𝑞^17𝑛+121−𝑞^20𝑛+15. (21)
Thus, from (20) and (21), we have ^∞_𝑛=0𝑞^3𝑛1−𝑞^20𝑛+5−^∞_𝑛=0𝑞^17𝑛+121−𝑞^20𝑛+15=𝑞⁸;𝑞²⁰_∞𝑞¹²;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞³;𝑞²⁰_∞𝑞¹⁷;𝑞²⁰_∞𝑞⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞. (22) Replacing 𝑞 by 𝑞²⁰ and then setting 𝑧=𝑞³, 𝑎=𝑞⁵, and 𝑏=𝑞²⁵ in (15), and proceeding as in case of (22), we find that ^∞_𝑛=0𝑞^7𝑛1−𝑞^20𝑛+5−^∞_𝑛=0𝑞^13𝑛+81−𝑞^20𝑛+15=𝑞¹²;𝑞²⁰_∞𝑞⁸;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞⁷;𝑞²⁰_∞𝑞¹³;𝑞²⁰_∞𝑞⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞. (23) Multiplying (23) by 𝑞 and adding to (22), we obtain ^∞_𝑛=0𝑞^3𝑛+𝑞^7𝑛+11−𝑞^20𝑛+5−𝑞^13𝑛+9+𝑞^17𝑛+121−𝑞^20𝑛+15=𝑞⁸;𝑞²⁰_∞𝑞¹²;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞³;𝑞²⁰_∞𝑞¹⁷;𝑞²⁰_∞𝑞⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞𝑞+𝑞¹²;𝑞²⁰_∞𝑞⁸;𝑞²⁰_∞𝑞²⁰;𝑞²⁰²_∞𝑞⁷;𝑞²⁰_∞𝑞¹³;𝑞²⁰_∞𝑞⁵;𝑞²⁰_∞𝑞¹⁵;𝑞²⁰_∞=𝑞⁸,𝑞¹²,𝑞²⁰;𝑞²⁰_∞𝑞³,𝑞⁵,𝑞⁷,𝑞¹³,𝑞¹⁵,𝑞¹⁷;𝑞²⁰_∞×𝑞⁷;𝑞²⁰_∞𝑞¹³;𝑞²⁰_∞𝑞²⁰;𝑞²⁰_∞+𝑞𝑞³;𝑞²⁰_∞𝑞¹⁷;𝑞²⁰_∞𝑞²⁰;𝑞²⁰_∞=𝑞⁸,𝑞¹²,𝑞²⁰;𝑞²⁰_∞𝑞³,𝑞⁵,𝑞⁷,𝑞¹³,𝑞¹⁵,𝑞¹⁷;𝑞²⁰_∞×𝑓−𝑞⁷,−𝑞¹³+𝑞𝑓−𝑞³,−𝑞¹⁷. (24) Setting 𝑎=𝑞 and 𝑏=−𝑞⁴ in (13), we find that 𝑓−𝑞⁷,−𝑞¹³+𝑞𝑓−𝑞³,−𝑞¹⁷=𝑓𝑞,−𝑞⁴. (25) Thus, (24) reduces to ^∞_𝑛=0𝑞^3𝑛+𝑞^7𝑛+11−𝑞^20𝑛+5−𝑞^13𝑛+9+𝑞^17𝑛+121−𝑞^20𝑛+15=𝑞⁸,𝑞¹²,𝑞²⁰;𝑞²⁰_∞𝑞³,𝑞⁵,𝑞⁷,𝑞¹³,𝑞¹⁵,𝑞¹⁷;𝑞²⁰_∞𝑓𝑞,−𝑞⁴. (26) Now 𝑓𝑞,−𝑞⁴=−𝑞;−𝑞⁵_∞𝑞⁴;−𝑞⁵_∞−𝑞⁵;−𝑞⁵_∞. (27) We have −𝑞;−𝑞⁵_∞=−𝑞;𝑞¹⁰_∞𝑞⁶;𝑞¹⁰_∞=𝑞²;𝑞²⁰_∞𝑞⁶;𝑞¹⁰_∞𝑞;𝑞¹⁰_∞=𝑞²,𝑞⁶,𝑞¹⁶;𝑞²⁰_∞𝑞,𝑞¹¹;𝑞²⁰_∞. (28) Similarly 𝑞⁴;−𝑞⁵_∞=𝑞⁴,𝑞¹⁴,𝑞¹⁸;𝑞²⁰_∞𝑞⁹,𝑞¹⁹;𝑞²⁰_∞,−𝑞⁵;−𝑞⁵_∞=𝑞¹⁰,𝑞¹⁰,𝑞²⁰;𝑞²⁰_∞𝑞⁵,𝑞¹⁵;𝑞²⁰_∞. (29) Hence from (27), (28), and (29) we have 𝑓𝑞,−𝑞⁴=𝑞²,𝑞⁴,𝑞⁶,𝑞¹⁰,𝑞¹⁰,𝑞¹⁴,𝑞¹⁶,𝑞¹⁸,𝑞²⁰;𝑞²⁰_∞𝑞,𝑞⁵,𝑞⁹,𝑞¹¹,𝑞¹⁵,𝑞¹⁹;𝑞²⁰_∞. (30) Using (30) in (26), we find that ^∞_𝑛=0𝑞^3𝑛+𝑞^7𝑛+11−𝑞^20𝑛+5−𝑞^13𝑛+9+𝑞^17𝑛+121−𝑞^20𝑛+15=𝑞²,𝑞⁴,𝑞⁶,𝑞⁸,𝑞¹⁰,𝑞¹⁰,𝑞¹²,𝑞¹⁴,𝑞¹⁶,𝑞¹⁸,𝑞²⁰,𝑞²⁰;𝑞²⁰_∞𝑞,𝑞³,𝑞⁵,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹¹,𝑞¹³,𝑞¹⁵,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹;𝑞²⁰_∞=𝑞²,𝑞⁴,𝑞⁶,𝑞⁸,𝑞¹⁰,𝑞¹²,𝑞¹⁴,𝑞¹⁶,𝑞¹⁸,𝑞²⁰;𝑞²⁰_∞×𝑞¹⁰,𝑞²⁰;𝑞²⁰_∞×𝑞,𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹¹,𝑞¹³,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹;𝑞²⁰_∞×𝑞⁵,𝑞¹⁵;𝑞²⁰_∞⁻¹=𝑞²;𝑞²_∞𝑞¹⁰;𝑞¹⁰_∞𝑞;𝑞²_∞𝑞⁵;𝑞¹⁰_∞=𝜓𝑞𝜓𝑞⁵. (31) This completes the proof of (18).

Proof of Conjecture (6). In view of (17), we rewrite (6) as 𝜓𝑞𝜓𝑞⁶=^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21. (32) Replacing 𝑞 by 𝑞²⁴ and then setting 𝑧=𝑞⁷, 𝑎=𝑞³, and 𝑏=𝑞²⁷ in (15), and proceeding as in case of (22), we find that ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞¹⁰;𝑞²⁴_∞𝑞¹⁴;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁷;𝑞²⁴_∞𝑞¹⁷;𝑞²⁴_∞𝑞³;𝑞²⁴_∞𝑞²¹;𝑞²⁴_∞. (33)
Replacing 𝑞 by 𝑞²⁴ and then setting 𝑧=𝑞⁵, 𝑎=𝑞⁹, and 𝑏=𝑞³³ in (15), and proceeding in a way similar to obtaining (22), we have ^∞_𝑛=0𝑞^5𝑛1−𝑞^24𝑛+9−𝑞^19𝑛+101−𝑞^24𝑛+15=𝑞¹⁴;𝑞²⁴_∞𝑞¹⁰;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁵;𝑞²⁴_∞𝑞¹⁹;𝑞²⁴_∞𝑞⁹;𝑞²⁴_∞𝑞¹⁵;𝑞²⁴_∞. (34) Multiplying (34) by 𝑞 and then adding to (33), we find that ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞¹⁰;𝑞²⁴_∞𝑞¹⁴;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁷;𝑞²⁴_∞𝑞¹⁷;𝑞²⁴_∞𝑞³;𝑞²⁴_∞𝑞²¹;𝑞²⁴_∞𝑞+𝑞¹⁴;𝑞²⁴_∞𝑞¹⁰;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁵;𝑞²⁴_∞𝑞¹⁹;𝑞²⁴_∞𝑞⁹;𝑞²⁴_∞𝑞¹⁵;𝑞²⁴_∞=𝑞¹⁰,𝑞¹⁴;𝑞²⁴_∞𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹;𝑞²⁴_∞×𝑓−𝑞⁵,−𝑞¹⁹𝑓−𝑞⁹,−𝑞¹⁵+𝑞𝑓−𝑞⁷,−𝑞¹⁷𝑓−𝑞³,−𝑞²¹. (35) Setting 𝑎=−𝑞, 𝑏=−𝑞¹¹, 𝑐=−𝑞⁴, and 𝑑=−𝑞⁸ in (14), we find that 𝑓−𝑞⁵,−𝑞¹⁹𝑓−𝑞⁹,−𝑞¹⁵+𝑞𝑓−𝑞⁷,−𝑞¹⁷𝑓−𝑞³,−𝑞²¹=𝑓𝑞,𝑞¹¹𝑓−𝑞⁴,−𝑞⁸. (36) Using (36) in (35), we have ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞¹⁰,𝑞¹⁴;𝑞²⁴_∞𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹;𝑞²⁴_∞×𝑓𝑞,𝑞¹¹𝑓−𝑞⁴,−𝑞⁸. (37) Now 𝑓𝑞,𝑞¹¹=−𝑞;𝑞¹²_∞−𝑞¹¹;𝑞¹²_∞𝑞¹²;𝑞¹²_∞=𝑞²;𝑞²⁴_∞𝑞²²;𝑞²⁴_∞𝑞¹²;𝑞¹²_∞𝑞;𝑞¹²_∞𝑞¹¹;𝑞¹²_∞=𝑞²,𝑞¹²,𝑞²²,𝑞²⁴;𝑞²⁴_∞𝑞,𝑞¹¹,𝑞¹³,𝑞²³;𝑞²⁴_∞,𝑓−𝑞⁴,−𝑞⁸=𝑞⁴;𝑞¹²_∞𝑞⁸;𝑞¹²_∞𝑞¹²;𝑞¹²_∞=𝑞⁴,𝑞⁸,𝑞¹²,𝑞¹⁶,𝑞²⁰,𝑞²⁴;𝑞²⁴_∞. (38) Using (38) in (37), we have ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞²,𝑞⁴,𝑞⁸,𝑞¹⁰,𝑞¹²,𝑞¹²,𝑞¹⁴,𝑞¹⁶,𝑞²⁰,𝑞²²,𝑞²⁴,𝑞²⁴;𝑞²⁴_∞𝑞,𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹¹,𝑞¹³,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹,𝑞²³;𝑞²⁴_∞=𝑞²,𝑞⁴,𝑞⁶,𝑞⁸,𝑞¹⁰,𝑞¹²,𝑞¹⁴,𝑞¹⁶,𝑞¹⁸,𝑞²⁰,𝑞²²,𝑞²⁴;𝑞²⁴_∞×𝑞¹²,𝑞²⁴;𝑞²⁴_∞×𝑞,𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹¹,𝑞¹³,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹,𝑞²³;𝑞²⁴_∞×𝑞⁶,𝑞¹²;𝑞²⁴_∞⁻¹=𝑞²;𝑞²_∞𝑞¹²;𝑞¹²_∞𝑞,𝑞²_∞𝑞⁶;𝑞¹²_∞=𝜓𝑞𝜓𝑞⁶. (39) This completes the proof of (32).

Proof of Conjecture (7). In view of (17), we rewrite (7) as 𝜓𝑞²𝜓𝑞³=^∞_𝑛=0𝑞^5𝑛1−𝑞^24𝑛+3+𝑞^7𝑛+21−𝑞^24𝑛+9−𝑞^17𝑛+101−𝑞^24𝑛+15−𝑞^19𝑛+161−𝑞^24𝑛+21. (40)
Replacing 𝑞 by 𝑞²⁴ and then setting 𝑧=𝑞⁵, 𝑎=𝑞³, and 𝑏=𝑞²⁷ in (15), and proceeding as in case of (22), we find that ^∞_𝑛=0𝑞^5𝑛1−𝑞^24𝑛+3−𝑞^19𝑛+161−𝑞^24𝑛+21=𝑞⁸;𝑞²⁴_∞𝑞¹⁶;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁵;𝑞²⁴_∞𝑞¹⁹;𝑞²⁴_∞𝑞³;𝑞²⁴_∞𝑞²¹;𝑞²⁴_∞. (41)
Also, replacing 𝑞 by 𝑞²⁴ and then setting 𝑧=𝑞⁷, 𝑎=𝑞⁹, and 𝑏=𝑞³³ in (15), and proceeding similar to (22), we obtain ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+9−𝑞^17𝑛+81−𝑞^24𝑛+15=𝑞¹⁶;𝑞²⁴_∞𝑞⁸;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁷;𝑞²⁴_∞𝑞¹⁷;𝑞²⁴_∞𝑞⁹;𝑞²⁴_∞𝑞¹⁵;𝑞²⁴_∞. (42) Multiplying (42) by 𝑞² and then adding to (41), we find that ^∞_𝑛=0𝑞^5𝑛1−𝑞^24𝑛+3+𝑞^7𝑛+21−𝑞^24𝑛+9−𝑞^17𝑛+101−𝑞^24𝑛+15−𝑞^19𝑛+161−𝑞^24𝑛+21=𝑞⁸;𝑞²⁴_∞𝑞¹⁶;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁵;𝑞²⁴_∞𝑞¹⁹;𝑞²⁴_∞𝑞³;𝑞²⁴_∞𝑞²¹;𝑞²⁴_∞+𝑞²𝑞¹⁶;𝑞²⁴_∞𝑞⁸;𝑞²⁴_∞𝑞²⁴;𝑞²⁴²_∞𝑞⁷;𝑞²⁴_∞𝑞¹⁷;𝑞²⁴_∞𝑞⁹;𝑞²⁴_∞𝑞¹⁵;𝑞²⁴_∞=𝑞⁸,𝑞¹⁶;𝑞²⁴_∞𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹;𝑞²⁴_∞×𝑓−𝑞⁷,−𝑞¹⁷𝑓−𝑞⁹,−𝑞¹⁵+𝑞²𝑓−𝑞⁵,−𝑞¹⁹𝑓−𝑞³,−𝑞²¹. (43) Setting 𝑎=𝑞², 𝑏=−𝑞¹⁰, 𝑐=−𝑞⁵, and 𝑑=−𝑞⁷ in (14), we find that 𝑓−𝑞⁷,−𝑞¹⁷𝑓−𝑞⁹,−𝑞¹⁵+𝑞²𝑓−𝑞⁵,−𝑞¹⁹𝑓−𝑞³,−𝑞²¹=𝑓𝑞²,𝑞¹⁰𝑓−𝑞⁵,−𝑞⁷. (44) Hence, (43) reduces to ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞⁸,𝑞¹⁶;𝑞²⁴_∞𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹;𝑞²⁴_∞×𝑓𝑞²,𝑞¹⁰𝑓−𝑞⁵,−𝑞⁷. (45) Now 𝑓𝑞²,𝑞¹⁰=−𝑞²;𝑞¹²_∞−𝑞¹⁰;𝑞¹²_∞𝑞¹²;𝑞¹²_∞=𝑞⁴;𝑞²⁴_∞𝑞²⁰;𝑞²⁴_∞𝑞¹²;𝑞¹²_∞𝑞²;𝑞¹²_∞𝑞¹⁰;𝑞¹²_∞=𝑞⁴,𝑞¹²,𝑞²⁰,𝑞²⁴;𝑞²⁴_∞𝑞²,𝑞¹⁰,𝑞¹⁴,𝑞²²;𝑞²⁴_∞,𝑓−𝑞⁵,−𝑞⁷=𝑞⁵;𝑞¹²_∞𝑞⁷;𝑞¹²_∞𝑞¹²;𝑞¹²_∞=𝑞⁵,𝑞⁷,𝑞¹²,𝑞¹⁷,𝑞¹⁹,𝑞²⁴;𝑞²⁴_∞. (46) Using (46) in (45), we have ^∞_𝑛=0𝑞^7𝑛1−𝑞^24𝑛+3+𝑞^5𝑛+11−𝑞^24𝑛+9−𝑞^19𝑛+111−𝑞^24𝑛+15−𝑞^17𝑛+141−𝑞^24𝑛+21=𝑞⁴,𝑞⁵,𝑞⁷,𝑞⁸,𝑞¹²,𝑞¹²,𝑞¹⁶,𝑞¹⁷,𝑞¹⁹,𝑞²⁰,𝑞²⁴,𝑞²⁴;𝑞²⁴_∞𝑞²,𝑞³,𝑞⁵,𝑞⁷,𝑞⁹,𝑞¹⁰,𝑞¹⁴,𝑞¹⁵,𝑞¹⁷,𝑞¹⁹,𝑞²¹,𝑞²²;𝑞²⁴_∞=𝑞⁴,𝑞⁸,𝑞¹²,𝑞¹⁶,𝑞²⁰,𝑞²⁴;𝑞²⁴_∞𝑞⁶,𝑞¹²,𝑞¹⁸,𝑞²⁴;𝑞²⁴_∞𝑞²,𝑞⁶,𝑞¹⁰,𝑞¹⁴,𝑞¹⁸,𝑞²²;𝑞²⁴_∞𝑞³,𝑞⁹,𝑞¹⁵,𝑞²¹;𝑞²⁴_∞=𝑞⁴;𝑞⁴_∞𝑞⁶;𝑞⁶_∞𝑞²;𝑞⁴_∞𝑞³;𝑞⁶_∞=𝜓𝑞²𝜓𝑞³. (47) This completes the proof of (40).

4. Concluding Remarks

All conjectures in [11] can easily be reformulated as theta function identity using Ramanujan’s ₁𝜓₁ 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.