Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

Computation of Hilbert-Poincaré series

We describe a new algorithm for computing standard and multi-graded Hilbert-Poincaré series of a monomial ideal. We compare it with different strategies along with implementation details and timing data.     

On certain Schlömilch-type series

By using a form of the Poisson summation formula together with a generalization due to Srivastava and Exton of the discontinuous integral of Weber and Schafheitlin and the (heretofore not readily available) cosine transform of the hypergeometric function ₁F₂[−b²t²] (b > 0), several new Schlömilch and Fourier-type series are evaluated. By specialization of the latter series numerous results appearing in the literature are obtained in a unified way.     

Complex series for 1/π

Many series for 1/?? were discovered since the appearance of S. Ramanujan??s famous paper ??Modular equations and approximation to ???? published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/?? discovered by Z.-W.?Sun, the authors found that a series for 1/?? involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type.     

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.     

Fourier series and the δ process

We discuss the effect of a particular sequence acceleration method, the δ²${\delta }^2$ process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points.     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Level 17 Ramanujan–Sato series

The Ramanujan Journal - Two level 17 modular functions $$\begin{aligned} r = q^2 \prod _{n=1}^{\infty } (1-q^{n})^{\left( \frac{n}{17} \right) },\qquad s = q^{2} \prod _{n=1}^{\infty } \frac{(1 -...     

Everywhere divergent Φ-means of Fourier series

For a function ? ∈, L ¹( $\mathbb{T}$ ), we investigate the sequence (C, 1) of mean values Φ(¦S _k (x, ?) ? ?(x)¦), where Φ(t): [0, +∞) → [0,+∞), Φ(0) = 0, is a continuous increasing function. We prove that if Φ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier.     

Universality properties of Walsh–Fourier series

It is shown that Walsh–Fourier series of \(W\) -continuous functions can have maximal sets of limit functions on small subsets of the unit interval.     

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.     

What is the alternating series test?

<正>We learned all the convergence tests before,which all terms are positive.Please try to use your brain to imagine how to deal with series whose terms are not always positive.The most simplest series are alternating series,whose terms alternate in sign.     

Hyperbolic Fourier coefficients of Poincaré series

The Ramanujan Journal - Poincaré (Ann Fac Sci Toulouse Sci Math Sci Phys 3:125–149, 1912) and Petersson (Acta Math 58(1):169–215, 1932) gave the now classical expression for the...     

Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Hypergeometric approach to Apéry-like series

By reformulating four hypergeometric series formulae, we derive 36 Apéry-like series expressions for the Riemann zeta function, including a couple of identities conjectured by Sun [New series for some special values of L-functions. Nanjing Univ J Math. 2015;32(2): 189–218].