New proof to Somos’s Dedekind eta-function identities of level 10

Michael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities.


Introduction
The Dedekind eta-function η(τ ) is defined by the formula where τ belongs to the upper complex half-plane. Here and all through the paper, we assume |q| < 1 and employ the standard notation (1 − aq n ).
For |x y| < 1, Ramanujan's theta function f(x, y) is defined as x n(n+1)/2 y n(n−1)/2 . Also in Ramanujan's notation, Jacobi's triple-product identity (Berndt 1991, p. 35) is given by The important special cases of f(x, y) (Berndt 1991, p. 36) are Following Ramanujan's notations, for q = e 2πiτ , we set Also after Ramanujan, define For convenience, we write f n = f (−q n ). A theta function identity which relates f 1 , f 2 , f n and f 2n is called the theta function identity of level 2n. Ramanujan documented many modular equations which involve quotients of the function f 1 at different arguments. For example, if (Berndt 1996, p. 206) P := f 1 q 1/6 f 5 and Q := f 2 q 1/3 f 10 , and if (Ramanujan 1988, p. 55) A := f 1 q 1/24 f 2 and B := f 5 q 5/24 f 10 , After the publication of Berndt (1991), many authors including (Adiga et al. 2002(Adiga et al. , 2004(Adiga et al. , 2016Baruah 2002Baruah , 2003Naika 2006;Saikia 2011;Vasuki and Sreeramamurthy 2005;Vasuki 2006;Vasuki and Veeresha 2017;Yi 2004) and many more mathematicians have found additional modular equations of the type (1) and (2) Kumar and Anu Radha (2018). As an application of this, we establish the colored partition identities for the same. Now we list some of the Somos's identities of level 10: and many more. Before concluding this section, we define a modular equation as defined by Ramanujan. The Gauss ordinary hypergeometric series is defined by where (a) n := a(a + 1)(a + 2)...(a + n − 1).
Using the above in (16) Letting q → −q in the above, rewriting x(−q) and y (−q) in terms of f n by employing (17) and then simplifying, we deduce the result. (17) in (3) and then dividing throughout by f 9 1 f 2 5 f 3 10 , we obtain

Second proof of (3) On using
On using P, Q, A and B as defined as in (1) and (2), (18) reduces to 20 Using (19) in (2) But L(P, Q) is nothing but (1) and it verifies (3).
We omit the proof of (4), as the proof is similar to the previous one.
Remark: Using the same method, we can prove the remaining Somos's Dedekind eta-function identities of level 10 which are listed in the previous section.

Applications to colored partition
The identities proved in Section 2 have applications to the theory of partitions. In this section, we demonstrate colored partitions for (3). Similarly for the remaining identities mentioned in section 1, we can establish the same concept. For simplicity, we adopt the standard notation "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." As an example, if 1, 2 and 3 are assigned with two colors, then possible partitions of 3 are 3 i , and v (violet) to differentiate two colors of 1, 2 and 3. Also, the generating function for the number of partitions of n is defined as with k colors, and all the parts are congruent to a (mod b).
Theorem 1 If α(n) represent the number of partitions of n being divided into parts that are congruent to ±1, ±3 modulo 10 with six colors, ±2, ±4 modulo 10 with four colors and +5 with eight colors, respectively. If β(n) is chosen to represent the number of partitions of n into many parts that are congruent to ±1, ±3 modulo 10 with three colors, ±2, ±4 modulo 10 with two colors and +5 with four colors, respectively. If γ (n) indicates the number of partitions of n being split into parts congruent to ±1, ±3 modulo 10 with nine colors, ±2, ±4 modulo 10 with two colors and +5 with four colors, respectively. Then, the following relation holds true: Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.