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57
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Semantics of Exact Real Arithmetic
, 1997
"... In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the exten ..."
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Cited by 29 (8 self)
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In this paper, we incorporate a representation of the nonnegative extended real numbers based on the composition of linear fractional transformations with nonnegative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.
Generalization Of Taylor's Theorem And Newton's Method Via A New Family Of Determinantal Interpolation Formulas
 J. of Comp. and Appl. Math
, 1997
"... The general form of Taylor's theorem gives the formula, f = Pn +Rn , where Pn is the Newton 's interpolating polynomial, computed with respect to a confluent vector of nodes, and Rn is the remainder. When f 0 6= 0, for each m = 2; : : : ; n + 1, we describe a "determinantal interpo ..."
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Cited by 12 (12 self)
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The general form of Taylor's theorem gives the formula, f = Pn +Rn , where Pn is the Newton 's interpolating polynomial, computed with respect to a confluent vector of nodes, and Rn is the remainder. When f 0 6= 0, for each m = 2; : : : ; n + 1, we describe a "determinantal interpolation formula", f = P m;n +R m;n , where P m;n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m = 2, the formula is Taylor's and for m = 3 it gives Halley's iteration function, as well as a Pad'e approximant. By applying the formulas to Pn , for each m 2, Pm;m\Gamma1 ; : : : ; Pm;m+n\Gamma2 , is a set of n rational approximations that includes Pn , and may provide a better approximation to f , than Pn . Thus each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a family of iteration functions for real or complex ...
Extrapolation algorithms and Padé approximations: a historical survey
, 1994
"... This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For ..."
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Cited by 10 (2 self)
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This paper will give a short historical overview of these two subjects. Of course, we do not pretend to be exhaustive nor even to quote every important contribution. We refer the interested reader to the literature and, in particular to the recent books [5, 22, 29, 24, 38, 46, 48, 68, 78, 131]. For an extensive bibliography, see [23]. 1 Extrapolation methods Let (S n ) be the sequence to be accelerated. It is assumed to converge to a limit S. An extrapolation method consists in transforming this sequence into a new one, (T n ), by a sequence transformation T : (S n ) \Gamma! (T n ). The transformation T is said to accelerate the convergence of the sequence (S n ) if and only if lim n!1 T n \Gamma S S n \Gamma S =<F13.
The geometry of continued fractions and the topology of surface singularities, arxiv:math.GT/0506432
 SingularitiesSapporo 2004, Advanced Studies in Pure Mathematics
, 2006
"... Abstract. We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence ..."
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Cited by 9 (1 self)
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Abstract. We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ> 1 and λ λ−1.
The impact of Stieltjes’ work on continued fractions and orthogonal polynomials
, 1993
"... Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials. ..."
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Cited by 9 (0 self)
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Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials.
Scalar Levintype sequence transformations
 81–147 of Brezinski, C. (Editor), Numerical
, 2000
"... Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {sn} a new sequence {s ′ n} = T ( {sn}) where each s ′ n depends on a finite n ..."
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Cited by 7 (1 self)
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Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {sn} a new sequence {s ′ n} = T ( {sn}) where each s ′ n depends on a finite number of elements sn1,..., snm. Often, the sn are the partial sums of an infinite series. The aim is to find transformations such that {s ′ n} converges faster than (or sums) {sn}. Transformations T ( {sn}, {ωn}) that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of socalled remainder estimates ωn are of Levintype if they are linear in the sn, and nonlinear in the ωn. Such remainder estimates provide an easytouse possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the ωn are simple functions of a few sequence elements sn. Then, nonlinear sequence transformations are obtained. Special cases of such Levintype
Prediction properties of Aitken’s iterated ∆ 2 process, of Wynn’s epsilon algorithm, and of Brezinski’s iterated theta algorithm
 Interpolation and Extrapolation, Elsevier
, 2000
"... The prediction properties of Aitken’s iterated ∆ 2 process, Wynn’s epsilon algorithm, and Brezinski’s iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of ..."
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Cited by 7 (6 self)
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The prediction properties of Aitken’s iterated ∆ 2 process, Wynn’s epsilon algorithm, and Brezinski’s iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracythroughorder relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant. 1
The function vMm(s; a, z) and some wellknown sequences
 JOURNAL OF INTEGER SEQUENCES, VOL. 5 (2002), ARTICLE 02.1.6
, 2002
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