The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analyti...
This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\\sum_{n\\geq 1}\\log n e^{nz}$ and $\\sum_{n\\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\\sum_{m=1}^n \\frac{1}{...
The difference between Γ(β+1−m)Γ(α+β+1−m) and Γ(β+1)Γ(α+β+1)+αm1!⋅Γ(β+n+1)Γ(α+β+n+2)+α(α+1)2!⋅m(m+2n+1)Γ(β+2n+1)Γ(α+β+2n+3)+α(α+1)(α+2)3!⋅m(m+3n+1)(m+3n+2)Γ(β+3n+1)Γ(α+β+3n+4)+⋯ 就段文字本身...
The mathematician Ramanujan introduced a summation PP Vaidyanathan - 《IEEE Transactions on Signal Processing A Publication of the IEEE Signal Processing Society》 被引量: 30发表: 2014年 On sums of Ramanujan sums Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be ...
One among such methods was called ``Ramanujan Summation'' proposed by Indian Mathematician Srinivasa Ramanujan. In this paper, I try to highlight how Ramanujan could have possibly arrived at those values by looking through his notebook jottings and extending further to provide Geometrical meaning ...
The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum cq(n). In a companion paper (Part I), properties of Ramanujan su... Vaidyanathan,P P. - 《Signal Processing IEEE Transactions on》 被引量: 30发表: 2014年 ...
Ramanujan's Summation FormulaQuintuple Product IdentityThis unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details...
s 1ψ1 Summation]]>doi:10.1080/00029890.2004.11920142Additional informationAuthor informationWarren P. JohnsonWARREN P. JOHNSON received his Ph.D from the University of Wisconsin, under the direction of Richard Askey. He has taught at Penn State University, Beloit College, the University of ...
3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . ....
3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . ....