Homological properties of 0-Hecke modules for dual immaculate quasisymmetric functions

Let $n$ be a nonnegative integer. For each composition $\alpha$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_\alpha$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_\alpha$ and a minimal injective presentation of $\mathcal{V}_\alpha$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_\alpha, {\bf F}_\beta)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_\beta, \mathcal{V}_\alpha)$, where ${\bf F}_\beta$ is the simple $H_n(0)$-module attached to a composition $\beta$ of $n$. We also compute ${\rm Ext}_{H_n(0)}^i(\mathcal{V}_\alpha,\mathcal{V}_{\beta})$ when $i=0,1$ and $\beta \le_l \alpha$, where $\le_l$ represents the lexicographic order on compositions.Comment: 44 pages, to be published in Forum of Math: Sigm

Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \\ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \\ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \\ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi}$ containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right)$$ through creative applications of an integration method that we had previously introduced. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}.$$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $\frac{1}{\pi}$ such as the series $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac{32 G-64 \ln (2)}{\pi }-16 \ln (2),$$ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ series. We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest

Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \\ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \\ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \\ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi}$ containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right)$$ through creative applications of an integration method that we had previously introduced. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}.$$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $\frac{1}{\pi}$ such as the series $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac{32 G-64 \ln (2)}{\pi }-16 \ln (2),$$ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ series. We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest