A strange (possible) fact about the Hecke operator T_3 in level 13 and characteristic 2

delta(z) + delta (13z) is a weight 12 modular form of level Gamma_0 (13). Let A in Z/2[[q]] be the mod 2 reduction of the Fourier expansion of this form. (The exponents appearing in A are the odd squares and their products by 13). If n is odd and positive let b_n be A^n and … Read more

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The context of this problem is given by the following link: https://math.stackexchange.com/questions/2270730/whats-the-limit-of-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2 In which it is proposed the problem of finding a closed form for the following nested radical: $$R = … Read more

Rationality of power series whose coefficients are the ranks of a sequence of matrices

Recently, I stumbled several times about the problem to decide whether a certain formal power series f=∞∑n=0dnTn∈Q[[T]] is actually a rational function, where the dn=dimk(ker(An)) are the dimensions of kernels of a sequence of k-linear maps An:kbn→kbn over some field k. More precisely, the An arise as follows: Let R be an infinite dimensional k-algebra … Read more

power series and roots of unity

Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series expansion of $f$, what are possible values of $k$? Any information about what $k$ can or cannot be depending on $p$ and $n$ will be helpful. … Read more

Convergence of Hahn series

Enumerate Q+ with Z+ by a bijiective map f:Z+→Q+. Consider the Hahn series: Pf(x)=+∞∑n=1cnxf(n) where cn∈C, x∈Ω=C∖R− and each xf(n) is chosen as the holomorphic branch mapping R+ to R+. Note these series with finite support form a commutative ring without maximal ideals (and that is where I began to look into them). With f … Read more

Laurent expansion of a principal value integral

Let f(t) be a nice Hölder continuous function. Also, suppose that f is even. I’m interested in evaluating integrals of the form: ∮(1−z)k+1∫10f(t)(1−zt)n+1dtdz, where for the contour integral, one may assume any loop around z=1. Equivalently, I’m interested in calculating Laurent expansions of Fn(z):=∫10f(t)(1−zt)n+1dt about z=1. Note that for z∈(1,∞), Fn(z) is defined as a … Read more

Redundancy in transseries representation of functions?

“Transseries” are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have looked through e.g. Edgar’s Transseries for beginners. The transseries considered there are “expansions around infinity”, so that the powers of the variable $x$ must … Read more

An analogue of rational functions for Hahn series

For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which is also the field of fractions of the formal power series ring $k[[t]]$). The composite embedding $k[t] \hookrightarrow k[[t]] \hookrightarrow k((t))$ induces, … Read more

Smoothness of the radius of convergence

Let (x↦an(x))n be a sequence of smooth functions defined on some fixed interval I. Consider the power series ∑n≥0an(x)tn and denote by R(x) its radius of convergence. Does there exist references in litterature dealing with continuity and smoothness of x↦R(x) ? Examples of questions I want to ask are as follows : Assuming 0<R(x)<∞ for … Read more

Analytic functions in arbitrary rings?

We have developed a rich theory of analytic functions over Rn and Cn. This is pretty reasonable, as analyticity here (local representation by power series) is closely linked to desirable differentiability conditions. However, I see no reason why this idea of power series representation cannot be generalized to any ring R. To create such a … Read more