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We study a monostatic multiple-input multiple-output sensing scenario assisted by a reconfigurable intelligent surface using tensor signal modeling. We propose a method that exploits the intrinsic multidimensional structure of the received echo signal, allowing us to recast the target sensing problem as a nested tensor-based decomposition problem to jointly estimate the delay, Doppler, and angular information of the target. We derive a two-stage approach based on the alternating least squares algorithm followed by the estimation of the signal parameters via rotational invariance techniques to extract the target parameters. Simulation results show that the proposed tensor-based algorithm yields accurate estimates of the sensing parameters with low complexity., Comment: The Asilomar Conference on Signals, Systems, and Computers 2024

This paper addresses the channel estimation problem for beyond diagonal reconfigurable intelligent surface (BD-RIS) from a tensor decomposition perspective. We first show that the received pilot signals can be arranged as a three-way tensor, allowing us to recast the cascaded channel estimation problem as a block Tucker decomposition problem that yields decoupled estimates for the involved channel matrices while offering a substantial performance gain over the conventional (matrix-based) least squares (LS) estimation method. More specifically, we develop two solutions to solve the problem. The first one is a closed-form solution that extracts the channel estimates via a block Tucker Kronecker factorization (BTKF), which boils down to solving a set of parallel rank-one matrix approximation problems. Exploiting such a low-rank property yields a noise rejection gain compared to the standard LS estimation scheme while allowing the two involved channels to be estimated separately. The second solution is based on a block Tucker alternating least squares (BTALS) algorithm that directly estimates the involved channel matrices using an iterative estimation procedure. We discuss the uniqueness and identifiability issues and their implications for training design. We also propose a tensor-based design of the BD-RIS training tensor for each algorithm that ensures unique decoupled channel estimates under trivial scaling ambiguities. Our numerical results shed light on the tradeoffs offered by BTKF and BTALS methods. Specifically, while the first enjoys fast and parallel extraction of the channel estimates in closed form, the second has a more flexible training design, allowing for a significantly reduced training overhead compared to the state-of-the-art LS method.

We give a continued-fraction characterization of Stieltjes moment sequences for which there exists a representing measure with support in $[\xi, \infty)$. The proof is elementary., Comment: LaTeX2e, 10 pages. arXiv admin note: text overlap with arXiv:1903.12455

We introduce a triangular array $\widehat{\sf L}^{(\alpha)}$ of 5-variable homogeneous polynomials that enumerate Laguerre digraphs (digraphs in which each vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weights for peaks, valleys, double ascents, double descents, and loops. These polynomials generalize the classical Laguerre polynomials as well as the rook and Lah polynomials. We show that this triangular array is totally positive and that the sequence of its row-generating polynomials is Hankel-totally positive, under suitable restrictions on the values given to the indeterminates. This implies, in particular, the coefficientwise Hankel-total positivity of the monic unsigned univariate Laguerre polyomials. Our proof uses the method of production matrices as applied to exponential Riordan arrays. Our main technical lemma concerns the total positivity of a large class of quadridiagonal production matrices; it generalizes the tridiagonal comparison theorem. In some cases these polynomials are given by a branched continued fraction. Our constructions are motivated in part by recurrences for the multiple orthogonal polynomials associated to weights based on modified Bessel functions of the first kind $I_\alpha$., Comment: 86 pages, LaTeX2e

We prove the coefficientwise Hankel-total positivity of the even and odd subsequences of Schett polynomials $X_n(x,y,z)$., Comment: LaTeX2e, 12 pages, extended abstract submitted to FPSAC 2024

This paper proposes a tensor-based parametric modeling and estimation framework in multiple-input multiple-output (MIMO) systems assisted by intelligent reflecting surfaces (IRSs). We present two algorithms that exploit the tensor structure of the received pilot signal to estimate the concatenated channel. The first one is an iterative solution based on the alternating least squares algorithm. In contrast, the second method provides closed-form estimates of the involved parameters using the high order single value decomposition. Our numerical results show that our proposed tensor-based methods provide improved performance compared to competing state-of-the-art channel estimation schemes, thanks to the exploitation of the algebraic tensor structure of the combined channel without additional computational complexity., Comment: arXiv admin note: text overlap with arXiv:2305.10499

We show that very simple continued fractions can be obtained for the ordinary generating functions enumerating permutations or D-permutations with a large number of independent statistics, when each cycle is given a weight $-1$. The proof is based on a simple lemma relating the number of cycles modulo 2 to the numbers of fixed points, cycle peaks (or cycle valleys), and crossings., Comment: LaTeX2e, 25 pages, includes 4 figures. Version 2 (34 pages, 6 figures) contains a slightly expanded introduction and a pair of running examples; to appear in the Electronic Journal of Combinatorics. arXiv admin note: text overlap with arXiv:2304.06545, arXiv:2212.07232, arXiv:2003.08192

This letter proposes a model for symbol detection in the uplink of IRS-assisted networks in the presence of channel aging. During the first stage, we model the received pilot signal as a tensor, which serves as a basis for both estimating the channel and configuring the IRS. In the second stage, the proposed tensor approach tracks the aging process to detect and estimate the transmitted data symbols. Our evaluations show that our proposed channel and symbol estimation schemes improve the performance of IRS-assisted systems in terms of the achieved bit error rate and mean squared error of the received data, compared to state of the art schemes.

This paper proposes a pilot decoupling-based two-dimensional channel parameter estimation method for intelligent reflecting surface (IRS)-assisted networks. We exploit the combined effect of Terahertz sparse propagation and the geometrical structure of arrays deployed at the base station, the IRS, and the user equipment to develop a low-complexity channel parameter estimation method. By means of a new pilot design along the horizontal and vertical domains, the overall channel parameter estimation problem is decoupled into different domains. Furthermore, with this decoupling, it is possible to simultaneously sense/estimate the channel parameters and to communicate with the sensed node. Specifically, we derive two estimators by decoupling the global problem into sub-problems and exploiting the built-in tensor structure of the sensing/estimation problem by means of multiple rank-one approximations. The Cram\'er-Rao lower bound is derived to assess the performance of the proposed estimators. We show that the two proposed methods yield accurate parameter estimates and outperform state-of-the-art methods in terms of complexity. The tradeoffs between performance and complexity offered by the proposed methods are discussed and numerically assessed.

A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index $i$ is either a cycle valley ($\sigma^{-1}(i)>i<\sigma(i)$) or a cycle peak ($\sigma^{-1}(i)\sigma(i)$). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes--Rogers and Jacobi--Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs., Comment: LaTeX2e, 47 pages, includes 3 figures. Version 2 corrects some minor errors. To appear in the Ramanujan Journal

This paper proposes tensor-based channel estimation for reconfigurable intelligent surface (RIS)-assisted communication networks. We exploit the inherent geometrical structure of the Terahertz propagation channel, including the antenna array geometries at the base station, the RIS, and the user equipment to design a tensor-based channel estimator, referred to as the higher-dimensional rank-one approximations (HDR) method. By exploiting the geometrical structure of the combined base station-RIS-user equipment channel, the proposed HDR estimator recasts parametric channel estimation as a single sixth-order rank-one tensor approximation problem, which can be efficiently solved using higher-order singular value decomposition to deliver parallel estimates of each channel component vector. Numerical results show that the proposed method provides significantly more accurate parameter estimates than competing state-of-the-art tensor-based RIS channel estimation, Khatri-Rao factorization, and least squares methods. For higher-rank channels, the HDR method shows similar spectral efficiency compared to its competitors while having similar computational complexity to the classical least squares estimator.

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $[n]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\mathbf{\phi})$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, \phi_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbf{\phi}$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays., Comment: LaTeX2e, 75 pages, includes 16 figures. Version 2 (37 pages, 2 figures) is the abridged version published in Advances in Applied Mathematics. Version 3 (the default) is identical to Version 1

A D-permutation is a permutation of $[2n]$ satisfying $2k-1 \le \sigma(2k-1)$ and $2k \ge \sigma(2k)$ for all $k$; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings., Comment: LaTeX2e, 92 pages including 11 figures

Reconfigurable intelligent surface is a potential technology component of future wireless networks due to its capability of shaping the wireless environment. The promising MIMO systems in terms of extended coverage and enhanced capacity are, however, critically dependent on the accuracy of the channel state information. However, traditional channel estimation schemes are not applicable in RIS-assisted MIMO networks, since passive RISs typically lack the signal processing capabilities that are assumed by channel estimation algorithms. This becomes most problematic when physical imperfections or electronic impairments affect the RIS due to its exposition to different environmental effects or caused by hardware limitations from the circuitry. While these real-world effects are typically ignored in the literature, in this paper we propose efficient channel estimation schemes for RIS-assisted MIMO systems taking different imperfections into account. Specifically, we propose two sets of tensor-based algorithms, based on the parallel factor analysis decomposition schemes. First, by assuming a long-term model in which the RIS imperfections, modeled as unknown phase shifts, are static within the channel coherence time we formulate an iterative alternating least squares (ALS)-based algorithm for the joint estimation of the communication channels and the unknown phase deviations. Next, we develop the short-term imperfection model, which allows both amplitude and phase RIS imperfections to be non-static with respect to the channel coherence time. We propose two iterative ALS-based and closed-form higher order singular value decomposition-based algorithms for the joint estimation of the channels and the unknown impairments. Moreover, we analyze the identifiability and computational complexity of the proposed algorithms and study the effects of various imperfections on the channel estimation quality., Comment: arXiv admin note: text overlap with arXiv:2206.03557

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980)., Comment: LaTeX2e, 48 pages. Version 2 contains a few additional historical remarks, and adds a new Remark 4 at the end of Section 10. To be published in Expositiones Mathematicae

We prove the ergodicity of the Wang--Swendsen--Koteck\'y (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of interest in statistical mechanics., Comment: 27 pages, pdflatex, and 22 pdf figures. Corrected an error in Remark 4 after Theorem 4.4. Final version

Passive intelligent reconfigurable surfaces (IRS) are becoming an attractive component of cellular networks due to their ability of shaping the propagation environment and thereby improving the coverage. While passive IRS nodes incorporate a great number of phase-shifting elements and a controller entity, the phase-shifts are typically determined by the cellular base station (BS) due to its computational capability. Since the fine granularity control of the large number of phase-shifters may become prohibitive in practice, it is important to reduce the control overhead between the BS and the IRS controller. To this end, in this paper we propose a low-rank approximation of the near-optimal phase-shifts, which would incur prohibitively high communication overhead on the BS-IRS controller links. The key idea is to represent the potentially large IRS phase-shift vector using a low-rank tensor model. This is achieved by factorizing a tensorized version of the IRS phase-shift vector, where each component is modeled as the Kronecker product of a predefined number of factors of smaller sizes, which can be obtained via tensor decomposition algorithms. We show that the proposed low-rank models drastically reduce the required feedback requirements associated with the BS-IRS control links. Our simulation results indicate that the proposed method is especially attractive in scenarios with a strong line of sight component, in which case nearly the same spectral efficiency is reached as in the cases with near-optimal phase-shifts, but with a drastically reduced communication overhead., Comment: arXiv admin note: text overlap with arXiv:2205.12024

Reconfigurable intelligent surface (RIS) is a candidate technology for future wireless networks. It enables to shape the wireless environment to reach massive connectivity and enhanced data rate. The promising gains of RIS-assisted networks are, however, strongly depends on the accuracy of the channel state information. Due to the passive nature of the RIS elements, channel estimation may become challenging. This becomes most evident when physical imperfections or electronic impairments affect the RIS due to its exposition to different environmental effects or caused by hardware limitations from the circuitry. In this paper, we propose an efficient and low-complexity tensor-based channel estimation approach in RIS-assisted networks taking different imperfections into account. By assuming a short-term model in which the RIS imperfections behavior, modeled as unknown amplitude and phase shifts deviations, is non-static with respect to the channel coherence time, we formulate a closed-form higher order singular value decomposition based algorithm for the joint estimation of the involved channels and the unknown impairments. Furthermore, the identifiability and computational complexity of the proposed algorithm are analyzed, and we study the effect of different imperfections on the channel estimation quality. Simulation results demonstrate the effectiveness of our proposed tensor-based algorithm in terms of the estimation accuracy and computational complexity compared to competing tensor-based iterative alternating solutions.

In this paper, we propose a rank-one tensor modeling approach that yields a compact representation of the optimum IRS phase-shift vector for reducing the feedback overhead. The main idea consists of factorizing the IRS phase-shift vector as a Kronecker product of smaller vectors, namely factors. The proposed phase-shift model allows the network to trade-off between achievable data rate and feedback reduction by controling the factorization parameters. Our simulations show that the proposed phase-shift factorization drastically reduces the feedback overhead, while improving the data rate in some scenarios, compared to the state-of-the-art schemes.

I analyze an unexpected connection between multiple orthogonal polynomials, $d$-orthogonal polynomials, production matrices and branched continued fractions. This work can be viewed as a partial extension of Viennot's combinatorial theory of orthogonal polynomials to the case where the production matrix is lower-Hessenberg but is not necessarily tridiagonal., Comment: 41 pages, LaTeX2e. Version 2 corrects a few small typos, uses the term "coefficient matrix" instead of "representing matrix" for a sequence of monic polynomials, and adds two references. To be published in the Transactions of the AMS

I show that a hypergeometric function ${}_{p}F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;\,\cdot\,)$ with $p \le q$ belongs to the Laguerre--P\'olya class $LP^+$ for arbitrarily large $b_{p+1},\ldots,b_q > 0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are nonnegative integers. This result arises as an easy corollary of the case $p=q$ proven two decades ago by Ki and Kim. I also give explicit examples for the case ${}_{1}F_2$., Comment: 10 pages, LaTeX2e

I give a short and completely elementary proof of Takagi's 1921 theorem on the zeros of a composite polynomial $f(d/dz) \, g(z)$., Comment: LaTeX2e, 4 pages. Accepted for publication in the American Mathematical Monthly

We present an on-device real-time hand gesture recognition (HGR) system, which detects a set of predefined static gestures from a single RGB camera. The system consists of two parts: a hand skeleton tracker and a gesture classifier. We use MediaPipe Hands as the basis of the hand skeleton tracker, improve the keypoint accuracy, and add the estimation of 3D keypoints in a world metric space. We create two different gesture classifiers, one based on heuristics and the other using neural networks (NN)., Comment: 5 pages, 6 figures; ICCV Workshop on Computer Vision for Augmented and Virtual Reality, Montreal, Canada, 2021

We present fundamental parameters for 110 canonical K- & M-type (1.3$-$0.13$M_\odot$) Taurus-Auriga young stellar objects (YSOs). The analysis produces a simultaneous determination of effective temperature ($T_{\rm eff}$), surface gravity ($\log$ g), magnetic field strength (B), and projected rotational velocity ($v \sin i$). Our method employed synthetic spectra and high-resolution (R$\sim$45,000) near-infrared spectra taken with the Immersion GRating INfrared Spectrometer (IGRINS) to fit specific K-band spectral regions most sensitive to those parameters. The use of these high-resolution spectra reduces the influence of distance uncertainties, reddening, and non-photospheric continuum emission on the parameter determinations. The median total (fit + systematic) uncertainties were 170 K, 0.28 dex, 0.60 kG, 2.5 km s$^{-1}$ for $T_{\rm eff}$, $\log$ g, B, and $v \sin i$, respectively. We determined B for 41 Taurus YSOs (upper limits for the remainder) and find systematic offsets (lower $T_{\rm eff}$, higher $\log$ g and $v \sin i$) in parameters when B is measurable but not considered in the fit. The average $\log$ g for the Class II and Class III objects differs by 0.23$\pm$0.05dex, which is consistent with Class III objects being the more evolved members of the star-forming region. However, the dispersion in $\log$ g is greater than the uncertainties, which highlights how the YSO classification correlates with age ($\log$ g), yet there are exceptionally young (lower $\log$ g) Class III YSOs and relatively old (higher $\log$ g) Class II YSOs with unexplained evolutionary histories. The spectra from this work are provided in an online repository along with TW Hydrae Association (TWA) comparison objects and the model grid used in our analysis., Comment: 21 pages, 10 figures, 5 tables. Accepted for publication in ApJ

We consider the lower-triangular matrix of generating polynomials that enumerate $k$-component forests of rooted trees on the vertex set $[n]$ according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight $m! \, \phi_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\phi$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays., Comment: LaTex2e, 65 pages. arXiv admin note: text overlap with arXiv:1907.02645. Version 2 makes some minor changes. To be published in Monatshefte f\"ur Mathematik

I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$., Comment: LaTeX2e, 11 pages. Version 2 fixes Remark 2 on page 5. To be published in Proceedings of the AMS

The projected stellar rotational velocity ($v \sin i$) is critical for our understanding of processes related to the evolution of angular momentum in pre-main sequence stars. We present $v \sin i$ measurements of high-resolution infrared and optical spectroscopy for 70 pre-main sequence stars in the Taurus-Auriga star-forming region, in addition to effective temperatures measured from line-depth ratios, and stellar rotation periods determined from optical photometry. From the literature, we identified the stars in our sample that show evidence of residing in circumstellar disks or multiple systems. The comparison of infrared $v \sin i$ measurements calculated using two techniques shows a residual scatter of $\sim$ 1.8 km s$^{-1}$, defining a typical error floor for the $v \sin i$ of pre-main sequence stars from infrared spectra. A comparison of the $v \sin i$ distributions of stars with and without companions shows that binaries/multiples typically have a higher measured $v \sin i$, which may be caused by contamination by companion lines, shorter disk lifetimes in binary systems, or tidal interactions in hierarchical triples. A comparison of optical and infrared $v \sin i$ values shows no significant difference regardless of whether the star has a disk or not, indicating that CO contamination from the disk does not impact $v \sin i$ measurements above the typical $\sim$ 1.8 km s$^{-1}$ error floor of our measurements. Finally, we observe a lack of a correlation between the $v \sin i$, presence of a disk, and H-R diagram position, which indicates a complex interplay between stellar rotation and evolution of pre-main sequence stars., Comment: 20 pages, 11 figures, accepted for publication in ApJ

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle., Comment: LaTeX2e, 12 pages with 2 figures: extended abstract submitted to FPSAC 2021

We study the triangular array defined by the Graham--Knuth--Patashnik recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n + \beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) = \delta_{k0}$ and parameters $\mathbf{\mu} = (\alpha,\beta,\gamma, \alpha',\beta',\gamma')$. We show that the family of arrays $T(\mathbf{\mu})$ is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$. Our main result is to determine all parameter sets $\mathbf{\mu} \in \mathbb{C}^6$ for which the ordinary generating function $f(x,t) = \sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type continued fraction in $t$ with coefficients that are polynomials in $x$., Comment: The document contains the main paper (pdflatex, 72 pages), a Supplementary Material file (pdflatex, 31 pages, 1 pdf figure) with the detailed proof of Theorem 3.1 of the main paper, and the e-jc.sty file. Version published in journal

We present a detailed characterization of 2MASS J04435750+3723031, a low-mass companion orbiting the young M2 star, 2MASS J04435686+3723033, at 7.6 arcseconds (550 AU) with potential membership in the 23 Myr $\beta$ Pictoris moving group ($\beta$PMG). Using near-infrared spectroscopy of the companion from IRTF/SpeX we have found a spectral type of M6 $\pm$ 1 and indications of youth through age-sensitive absorption lines and a low surface gravity index (VL-G). A young age is supported by H$\alpha$ emission and lithium absorption in the host. We re-evaluate the membership of this system and find that it is a marginally consistent kinematic match to the $\beta$PMG using $Gaia$ parallaxes and new radial velocities for the host and companion. If this system does belong to the $\beta$PMG, it would be a kinematic outlier and the companion would be over-luminous compared to other similar ultracool objects like PZ Tel B; this would suggest 2M0443+3723 B could be a close brown dwarf binary ($\approx$52+52 M$_\mathrm{Jup}$ if equal-flux, compared with 99 $\pm$ 5 M$_\mathrm{Jup}$ if single), and would make it the sixth substellar companion in this group. To test this hypothesis, we acquired NIR AO images with Keck II/NIRC2, but they do not resolve the companion to be a binary down to the diffraction limit of $\sim$3 AU. If 2M0443+3723 AB does not belong to any moving group then its age is more uncertain. In this case it is still young ($\lesssim$30 Myr), and the implied mass of the companion would be between $\sim$30--110 M$_\mathrm{Jup}$., Comment: 18 pages, 18 figures, Accepted to ApJ

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them., Comment: LaTeX2e, 122 pages, includes 9 tikz figures. Version 2 includes a new footnote 27 on page 81. To be published in Advances in Applied Mathematics