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I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$\xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/... 0answers 102 views ### Szegő curve for partial sum of Taylor series of Riemann \Xi(z) function I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of e^x-1. Entire functions e^z, \cos(z), and \sin(z) can ... 0answers 105 views ### Positive, Uni-modal, Log-concave Combinatorics We define a sequence, \{a_n\}_{n=0}^\infty, to be a uni-modal sequence if for some m,$$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$We define a sequence, \{a_n\}_{... 0answers 129 views ### Generating a series representation for the inverse of the operator f(f) I am considering the following problem: Suppose you are given a function u: C \rightarrow C, find a function g such that g(g) = u (Let's assume that such a function exists). And by "find", I ... 0answers 102 views ### Properties and name of some polynomials I have encountered in a problem some polynomials given by P_k(x) = \prod_{j=0}^{k-2} (kx-j). I need to understand if these polynomials are known, and if they have certain special properties, as ... 0answers 335 views ### Applying the ideas of power series to certain convolutions - which identities transfer? Let's suppose I'm working with some set of functions f_k(n). f_1(n) is essentially the root of my functions, and could be nearly anything, and then f_k(n) = (f_1(n) * f_{k-1}(n)) for some ... 0answers 82 views ### An integral form of sum \sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2} for two real analytic at 0 functions? Fourier|Taylor series parallels Thinking of parallels between Fourier series and Taylor series, you might find out that the integral \frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle ... 0answers 148 views ### Asymptotics to Taylor expansions? I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. http://math.stackexchange.com/questions/1440931/... 0answers 133 views ### Series expansion with remaining log n Hi, I'm studying the asymptotic behavior (n \rightarrow \infty) of the following formula, where k is a given constant.$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2} I'm trying to do a ...
Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps: Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...