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

2021 ◽  
Author(s):  
B. R. Srivatsa Kumar ◽  
Shruthi
Keyword(s):  
Dedekind Eta Function ◽  
Partition Identities

AbstractMichael 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.

scholarly journals New approach to Somos’s Dedekind eta-function identities of level 6

2021 ◽  
Vol 40 (1) ◽  
pp. 187-198
Author(s):  
D. Anu Radha ◽  
B. R. Srivatsa Kumar ◽  
Shruthi
Keyword(s):  
Dedekind Eta Function ◽  
New Approach ◽  
Partition Identities

In the present work, we prove few new Dedekind eta-function identities of level 6 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 6 proved by B. R. Srivatsa Kumar et. al. As an application of this, we establish colored partition identities.

scholarly journals Proofs of some partition identities conjectured by Kanade and Russell

2021 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Lie Algebra ◽  
Partition Identities ◽  
Partition Identity ◽  
Affine Lie Algebra ◽  
Level 2

AbstractKanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

scholarly journals Identities on poly-Dedekind sums

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Lee-Chae Jang
Keyword(s):  
Modular Group ◽  
Reciprocity Relation ◽  
Dedekind Sums ◽  
Transformation Behavior ◽  
Dedekind Eta Function ◽  
Bernoulli Function ◽  
Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

scholarly journals Partition identities II. The results of Bateman and Erdős

2006 ◽  
Vol 117 (1) ◽  
pp. 160-190
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris
Keyword(s):  
Partition Identities

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

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