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On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers

N. A. S. Bulkhali, G. Kavya Keerthana, Ranganatha Dasappa

Abstract

In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of two theta functions. Furthermore, as applications, we extract several theorems in the theory of representation of integers as mixed ternary sums involving squares, triangular numbers, generalized pentagonal numbers and generalized octagonal numbers

On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers

Abstract

In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of two theta functions. Furthermore, as applications, we extract several theorems in the theory of representation of integers as mixed ternary sums involving squares, triangular numbers, generalized pentagonal numbers and generalized octagonal numbers

Paper Structure

This paper contains 5 sections, 31 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proof of Theorem \ref{['MainThm3']} and some corollaries
  4. Product of two theta functions
  5. Applications to the theory of representation of integers

Key Result

Theorem 1.1

Let $k$ and $r$ be positive integers with $k>r$, $\gcd (2k,r)=1$ or 2 and $\gcd(2k,k-r)=1$. For any integers $g$, $h$, $u$, $v$, $i$ and $j$ with $g+h=S_1>0$, $g-h=D_1$, $u+v=S_2>0$, $u-v=D_2$, $i+j=S_3>0$, $i-j=D_3$, $S_1=S_2$ and $2\,S_{{1}}=r \left( k-r \right) S_{{3}}$, we have Here $\epsilon_i \in \{-1,1\}$, $\delta_i= \frac{1-\epsilon_i}{2}$ for $i=1,2,3$ and $[x]$ denote the greatest integ

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['MainThm3']}
  • Corollary 3.0.1
  • proof
  • Corollary 3.0.2
  • Corollary 3.0.3
  • Corollary 3.0.4
  • ...and 40 more