Legendre polynomials and Ramanujan-type series for 1/π

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.     

Sporadic sequences, modular forms and new series for 1/π

Two new sequences, which are analogues of six sporadic examples of D.?Zagier, are presented. The connection with modular forms is established and some new series for 1/?? are deduced. The experimental procedure that led to the discovery of these results is recounted. Proofs of the main identities will be given, and some congruence properties that appear to be satisfied by the sequences will be stated as conjectures.     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

Ramanujan-type formulae for 1/π: q-analogues

The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of π, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some q-analogues of Ramanujan-type evaluations and of related supercongruences.     

Hypergeometric series summations and π-formulas

By two Slater's hypergeometric series identities, we establish two general summation formulas with six free parameters. Specializing certain parameters in these formulas, we obtain a list of Ramanujan-type series formulas for π  , π²${\pi }^2$ and π³${\pi }^3$.     

Dihedral quartic approximations and series for π

Imaginary quadratic fields with class groups that have C(4) as a subgroup are analyzed in depth, and the units of associated dihedral quartic fields are thereby evaluated using Epstein zeta functions. Emphasis is placed on extreme examples such as Q(√?3502) which is probably the last case having an even discriminant and C(4) × C(4) as its class group. These extreme examples lead to very remarkable approximations and series for π such as

$\text{π=}\text{6}\text{3502}\text{log}\text{(2π)+7.37×10}{}^{\mathrm{?82}}$

where u is the product of four, rather simple, quartic units. The approximations and series relate to Baker's theory of linear forms in logarithms and to certain modular identities. Related topics are discussed briefly.     

New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π

New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the ₅ F ₄ hypergeometric function. We use these results to obtain two explicit ₅ F ₄ evaluations, and several new formulas for 1/π.      

A new Ramanujan-like series for 1/<Emphasis Type="Italic">π</Emphasis><Superscript>2</Superscript>

Our main results are a WZ-proof of a new Ramanujan-like series for 1/π ² and a hypergeometric identity involving three series.     

arctg1/2+arctg1/3=π/4几何证法举隅

用几何法证明恒等式,不仅能使我们了解数形之间的许多巧妙联系,而且还能培养我们的创造性思维。下面以arctg1/2 +arctg2/3=π/4的几何证明为例,说明数形联系的多样性。证法一(利用勾股定理及逆定理): 作宽为1,长分别为2、3的矩形ABCD和AGEF,如图,连AC、AE、CE、则A     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

Fejér type theorems for Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2], then the nth partial sum of its formally differentiated Fourier series divided by n converges to ^-1 [F(x+0) - F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of nonperiodic functions of bounded variation is also known. These theorems can be interpreted in such a way that the terms of the Fourier-Stieltjes (or Fourier) series of F determine the atoms of the finite Borel measure on the torus T:= [0, 2) induced by an appropriate extension of F (or by F itself in the periodic case). The aim of the present paper is to extend all of these results to the Cesàro as well as Abel-Poisson means of Fourier-Stieltjes (or Fourier) series of a nonperiodic (or periodic) function F of bounded variation. At the end, we sketch a possible extension of these results to linear means defined by more general kernels.     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Seeking a proof of Xie’s inequality: analogues for series and Fourier series

Seeking to free the existence and regularity theory for the Navier–Stokes equations from assumptions about the regularity of a fluid’s boundary, we continue efforts of Wenzheng Xie and myself to prove a certain domain independent inequality for solutions of the steady Stokes equations. For the Laplacian, Xie proved an analogue of the desired inequality by using the maximum principle in obtaining an intermediary result. His conjecture that an analogue of this intermediary result is also valid for the Stokes equations remains unproven. My efforts to circumvent the need for it have led, so far, only to further interesting conjectures. Here, we seek to better understand both Xie’s arguments and mine by applying them to simpler problems concerning series and Fourier series. First, a bound is proven for a series of real numbers that can be interpreted as a bound for the sup-norm of a Fourier cosine series, in terms of the \(L^{2}\) -norms of its fractional-order derivatives of orders 1/3 and 2/3. This is generalized to a bound for a weighted sum of a sequence of real numbers. We conjecture that the hypotheses concerning the weights are satisfied by the sequence of numbers \(\{ \sin ny\}\) , for any nonzero \(y\in (-\pi ,\pi )\) . If so, we obtain an inequality for the sup-norm of a Fourier sine series, similar to that for a cosine series. Remarkably, the hypotheses for the weights are analogous to those we have been seeking to verify in trying to prove the original inequality for the Stokes equations. We conclude with a remark showing that Xie’s central argument provides a possibly new, very straightforward, proof of Hölder’s inequality for series.     

Multi–regime models for nonlinear nonstationary time series

Nonlinear nonstationary models for time series are considered, where the series is generated from an autoregressive equation whose coefficients change both according to time and the delayed values of the series itself, switching between several regimes. The transition from one regime to the next one may be discontinuous (self-exciting threshold model), smooth (smooth transition model) or continuous linear (piecewise linear threshold model). A genetic algorithm for identifying and estimating such models is proposed, and its behavior is evaluated through a simulation study and application to temperature data and a financial index.     

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bohr’s strip for vector valued Dirichlet series

Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series , converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr’s strip for a Dirichlet series with coefficients a _n in Y is bounded by 1 - 1/Cot (Y), where Cot (Y) denotes the optimal cotype of Y. This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210.