samples/texts/1085779/page_1.md

The Hyder Series

by Syed Shahabudeen

july 2021

Abstract

The Hyder Series is a generelized version of a special type of multi- ple infinite series.In this paper, we will be looking at some main aspect of this series in detail.

1 Introduction

Hyder Series is basically a generelized form of a special type of an infinite series. It is defined as

$${\mathcal{H}}^q({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_k;p_1,p_2,p_3,\dots ,p_k;\beta )=\sum _{m_1,m_2,m_3,\dots ,m_k=0}^{\mathrm{\infty }}\frac{\prod _{1\le i\le k}{\alpha }_i^{m_i}}{{(\sum _{n=1}^k{p}_n{m}_n+\beta )}^q}\backslash mathcal\{H\}^q(\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash alpha\_3,\; \backslash dots,\; \backslash alpha\_k;\; p\_1,\; p\_2,\; p\_3,\; \backslash dots,\; p\_k;\; \backslash beta)\; =\; \backslash sum\_\{m\_1,\; m\_2,\; m\_3,\; \backslash dots,\; m\_k\; =\; 0\}^\{\backslash infty\}\; \backslash frac\{\backslash prod\_\{1\; \backslash le\; i\; \backslash le\; k\}\; \backslash alpha\_i^\{m\_i\}\}\{\backslash left(\; \backslash sum\_\{n=1\}^\{k\}\; p\_n\; m\_n\; +\; \backslash beta\; \backslash right)^q\}$$

where $\sum_{m_1,m_2,m_3,\dots,m_k=0} = \sum_{m_1=0}^{\infty} \sum_{m_2=0}^{\infty} \sum_{m_3=0}^{\infty} \dots \sum_{m_k=0}^{\infty}$

In this paper we'll be looking at some special values, its respective proofs and relation of hyder series to hypergeometric series.

1.1 Notations

The q in the Hyder Notation Stands for the power order of the series. $p_1, p_2, ..., p_k$ are the coefficients of $m_1, m_2, ..., m_k$. If a number is being repeated for n number of times in the first two slots