\begin{aligned} P_2&=\prod_{\ \ \ \ \ \,n=1\\n\not\equiv0\ (\mathrm{mod}\ 5)}^{\infty}\prod_{k=0}^{4}(1-e^{2\pi ikn/5}q^{n/5})\\ &=\prod_{\ \ \ \ \ \,n=1\\n\not\equiv0\ (\mathrm{mod}\ 5)}^{\infty}\prod_{\omega}^{}(1-\omega q^{n/5})...
A special case of an Entry in Part II of Ramanujan's Notebooks is such that [ 1+frac{1}{5} left( frac{1}{2} right)<^>{2} + frac{1}{9} left( frac{1cdot 3}{2 cdot 4} right)<^>{2} + cdots = frac{Gamma<^>4 left(frac{1}{4}right)}{16pi <^>2}. ]1+15(12)2...
Open in MATLAB Online Hi. This is my first post so please let me know if I violate any kind of rules. Thank you in advance. I intend to approximate pi by summing a specified number of terms (k). The output I got was nowhere near what I wanted. Could someone help me please? Here...
The Rogers-Ramanujan continued fraction and a quintic iteration for $1/\\pi$ Properties of the Rogers-Ramanujan continued fraction are used to obtain a formula for calculating with quintic convergence. HH Chan,S Cooper,WC Liaw - 《Proceedings of the American Mathematical Society》 被引量: 10发表...
Then we have the asymptotic formula e−Λmεf~Q(α+iε2πm)∼m−N2χ(d,α)c(Q)G(Q,α)SQ,ζ(ε) (22) as ε tends to 0 from right within the real line. Here χ(d,α)=∏i=1Nχi is the product of the 12th root of ζ defined by χi=e(s(pi,qi)/2) where ...
In this paper, we study properties of the coefficients appearing in theq-series expansion of, the infinite Borwein product for an arbitrary primep, raised to an arbitrary positive real power. We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients....
ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural ...
TransformationFormulae and Rogers-Ramanuj an Type Identities Abstract Inthis thesis,bymaking useofdifferent methods,many new Rogers-Ramanujantype identitiesandq-seriestransformationformulaehavebeenestablished.Themaincontentsof thisdissertatioUarelistedasfollows: ...
摘要: We present several inequalities for the Ramanujan generalized modular equation function $$\mu _{a}(r)=\pi F(a,1-a;1;1-r^2)/$$μa(r)=πF(a,1-a;1;1-r2)/$$[2\sin (\pi a)F(a,1-a;1;r^2)]$$[2sin(πa)F(a,1-a;1;...
\(\pi _b=\lambda \cup \mu \) and \(\pi _d=\varnothing \) . based on definition 4.3 and the map \(\phi \) , the gordon marking of \(\pi _b\) is described as follows. the marks are labeled by the order \(1<\overline{1}<2<\overline{2}<\cdots \) and are assigned ...