5 research outputs found

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

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    Let nn be a nonnegative integer. For each composition Ξ±\alpha of nn, Berg etΒ al.\textit{et al.} introduced a cyclic indecomposable Hn(0)H_n(0)-module VΞ±\mathcal{V}_\alpha with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study VΞ±\mathcal{V}_\alpha's from the homological viewpoint. To be precise, we construct a minimal projective presentation of VΞ±\mathcal{V}_\alpha and a minimal injective presentation of VΞ±\mathcal{V}_\alpha as well. Using them, we compute ExtHn(0)1(VΞ±,FΞ²){\rm Ext}^1_{H_n(0)}(\mathcal{V}_\alpha, {\bf F}_\beta) and ExtHn(0)1(FΞ²,VΞ±){\rm Ext}^1_{H_n(0)}( {\bf F}_\beta, \mathcal{V}_\alpha), where FΞ²{\bf F}_\beta is the simple Hn(0)H_n(0)-module attached to a composition Ξ²\beta of nn. We also compute ExtHn(0)i(VΞ±,VΞ²){\rm Ext}_{H_n(0)}^i(\mathcal{V}_\alpha,\mathcal{V}_{\beta}) when i=0,1i=0,1 and β≀lΞ±\beta \le_l \alpha, where ≀l\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

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    We introduce classes of Ramanujan-like series for 1Ο€\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 Hn=1+12+β‹―+1nH_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} denote the nthn^{\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 1Ο€\frac{1}{\pi} containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula βˆ‘n=1∞(2nn)2Hn32n=Ξ“2(14)4Ο€(1βˆ’4ln⁑(2)Ο€)\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 βˆ‘n=1∞(2nn)2Hn32n(n+1)=8βˆ’2Ξ“2(14)Ο€3/2βˆ’4Ο€3/2+16Ο€ln⁑(2)Ξ“2(14). \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 GG and 1Ο€\frac{1}{\pi} such as the series βˆ‘n=1∞(2nn)2Hn16n(n+1)2=16+32Gβˆ’64ln⁑(2)Ο€βˆ’16ln⁑(2), \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 1Ο€\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 1Ο€2\frac{1}{\pi^2}, especially the constant ΞΆ(3)Ο€2\frac{\zeta(3)}{\pi^2}, which is of number-theoretic interest

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

    Get PDF
    We introduce classes of Ramanujan-like series for 1Ο€\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 Hn=1+12+β‹―+1nH_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} denote the nthn^{\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 1Ο€\frac{1}{\pi} containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula βˆ‘n=1∞(2nn)2Hn32n=Ξ“2(14)4Ο€(1βˆ’4ln⁑(2)Ο€)\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 βˆ‘n=1∞(2nn)2Hn32n(n+1)=8βˆ’2Ξ“2(14)Ο€3/2βˆ’4Ο€3/2+16Ο€ln⁑(2)Ξ“2(14). \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 GG and 1Ο€\frac{1}{\pi} such as the series βˆ‘n=1∞(2nn)2Hn16n(n+1)2=16+32Gβˆ’64ln⁑(2)Ο€βˆ’16ln⁑(2), \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 1Ο€\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 1Ο€2\frac{1}{\pi^2}, especially the constant ΞΆ(3)Ο€2\frac{\zeta(3)}{\pi^2}, which is of number-theoretic interest
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