The formula (1.1) may be iterated to produce the quartic transformation formula(1.2)[Math Processing Error] Other examples, analogous to (1.1) and (1.2), are the following quadratic and cubic transformation formulas given by Ramanujan in his second notebook [26, pp. 258, 260]:(1.3)[Math ...
The proof of the main result illustrates Hilbert's Theorem 90. An example of a particular formula generalizing Ramanujan's formulas for degree 5 is given. A necessary condition for nested radical expressions of depth 2 to be contained in the normal closure of a pure cubic extension of the ...
The 163 appearing here is the same one appearing in the fact that (the Ramanujan constant) is very nearly an integer. Similarly, the factor comes from the j-function identity for . The series is given by (86) (87) (Borwein and Borwein 1993; Beck and Trott; Bailey et al. 2007,...
For example, we show that the coefficients af(n),n≥1, of Ramanujan's mock theta function(1)f(q)=1+∑n=1∞qn2(1+q)2(1+q2)2⋯(1+qn)2=1+∑n=1∞af(n)qn are given byaf(n)=−124n−1ℑ(∑Q∈QnF(zQ)ωQ), where(2)F(z)=−140⋅E4(z)+4E4(2z)−9E4(...
In [13] Ramanujan introduced the following trigonometrical sum. Definition 1.1. The Ramanujan sum is defined by c q (n) =(h,q)=1 e 2πinh/q , (1.1) where q and n are in N and the summation is over a reduced residue system mod q. ...
Image source: By Joshua Siktar Well we tried to find a10 and we failed. What gives? We keep moving backwards in the sequence to find a term that we actually KNOW the value of, but we aren't given any. For instance, we are usually told a0 to make the recurrence formula useful. If...
In this thesis, by making use of different methods, many new Rogers-Ramanujan type identities and q-series transformation formulae have been established. 本文用不同的方法得到了很多新的Rogers-Ramanujan型恒等式以及一些新的q-级数变换公式。 更多例句>> 5) Formulas for Summing up Power Series 幂级...
We can regard these formulas as analogues of the famous formulas for certain series given by Cauchy, Ramanujan, Berndt, and so on, as well as those for the Eisenstein series given by Hurwitz.doi:10.1112/blms/bdn014Hirofumi TsumuraDepartment of Mathematics and Information SciencesTokyo Metropolitan...
On the other hand, the generating function of ζ⋆({m}n) [12], [14] is given by∏n=1∞(1−xmnm)−1=∏n=1∞(1+xmnm+x2mn2m+⋯+xkmnkm+⋯), so that ζ⋆({m}n) has another expression∑k=1n∑|α|=nζ(mα1,mα2,…,mαk). Let E(m,n,k) be the sum of...
to appear in Ramanujan Journal. http://arxiv.org/abs/1101.5608.M. Josuat-Verg`es and J. S. Kim. Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity. http://arxiv.org/abs/1101.5608 .M. Josuat-Verges, J.S. Kim. Touchard-Riordan formulas, T-fractions, and ...