On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

On partial sums of a spectral analogue of the Möbius function

Sankaranarayanan and Sengupta introduced the function μ ^*(n) corresponding to the Möbius function. This is defined by the coefficients of the Dirichlet series 1/L _f (s), where L _f (s) denotes the L-function attached to an even Maaß cusp form f. We will examine partial sums of μ ^*(n). The main result is $\sum_{n\leq x}\mu^{*}(n)=O(x\exp(-A\sqrt{\log x}))$ , where A is a positive constant. It seems to be the corresponding prime number theorem.     

How small are the increments of partial sums of a discrete random field?

In this paper we establish inferior limit results and path properties for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) discrete random field with multidimensional indices.     

用(sinα±cosα)~2=1±2sinαcosα解题

由平方关系sin2α+cos2α=1不难得到(sinα±cosα)2=1±2sinαcosα.它揭示了sinα+cosα、sinα-cosα、sinαcosα三者之间的密切关系,知其一必能求出另二.在一些解方程、求最值问题中,恰当运用此关系有助于简化运算、发现解题途径.例1已知sinα+cosα=1/5(0<α<π),求tanα的值.分析本题可先求出sinα-cosα的值,再和sinα+cosα=15联立方程组求出sinα,cosα     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.     

Approximation of Fejér partial sums by interpolating functions

The interpolating spline or trigonometric polynomial to a function at equally spaced points approximates the Dirichlet partial sums of its Fourier series with accuracy depending only on the neglected coefficients. We show that the Fejér mean of the Dirichlet sums can be approximated by the arithmetic mean of two Fejér trigonometric interpolants, one at the points with even indexes and one at the points with odd indexes, with an error depending only on the neglected Fourier coefficients and it is positive for positive functions. We also consider the case of Fejér spline interpolants and a constructive relation between Hermite and Fejér interpolants.     

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M _n (nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α _{n, s} denotes the s th zero of M _n (nα;δ, η) , counted from the right, and if α˜ _n,s denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α _n,s and α˜ _n,s as n→∞ . December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999.     

Zeros of Differential Polynomial f（z）f（k）（z） -- a and Its Normality

Suppose that function f(z) is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f(z)f(k)(z) takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f(z)f(k)(z).     

On the Relationship between the Multidimensional Dirichlet Divisor Problem and the Boundary of Zeros of ζ(s)F

Mathematical Notes -     

Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums

It is well-known that the Fourier partial sums of a function exhibit the Gibbs phenomenon at a jump discontinuity. We study the same question for de la Vallée-Poussin sums. Here we find a new Gibbs function and a new Gibbs constant. When the function is continuous, a behavior similar to the Gibbs phenomenon also occurs at a kink. We call it the “generalized Gibbs phenomenon”. Let $F_{n}(x):=\frac{k_{n}(g,x)-g(x)}{k_{n}(g,x_{0})-g(x_{0})}$ , where x ₀ is a kink and where k _n (g,x) represents Fourier partial sums and de la Vallée-Poussin sums. We show that F _n (x) exhibits the “generalized Gibbs phenomenon”. New universal Gibbs functions for both sums are derived.     

Necessary conditions for the weak generalized localization of Fourier series with “lacunary sequence of partial sums”

It has been established that, on the subsets $ \mathbb{T}^N = [ - \pi ,\pi ]^N $ describing a cross W composed of N-dimensional blocks, $ W_{x_s x_t } = \Omega _{x_s x_t } \times [ - \pi ,\pi ]^{N - 2} (\Omega _{x_s x_t } $ is an open subset of [?π, π]²) in the classes $ L_p (\mathbb{T}^N ),p > 1 $ , a weak generalized localization holds, for N ≥ 3, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums $ S_n (x;f)(x \in \mathbb{T}^N ,f \in L_p ) $ of these series corresponds the number n = (n ₁,…, n _N ) ∈ ? ₊ ^N , some components n _j of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating W) of the N-dimensional blocks $ W_{x_s x_t } $ as well as of the structure and geometry of $ W_{x_s x_t } $ . In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block.     

Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series

In this article we discuss the Nörlund means of cubical partial sums of Walsh-Fourier series of a function in L ^p (1 ≤ p ≤ ∞). We investigate the rate of the approximation by this means, in particular, in Lip(α, p), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞ by L ^p we mean C _W , the collection of the uniformly W-continuous functions. Our main theorems state that the approximation behavior of the two-dimensional Walsh- Nörlund means is so good as the approximation behavior of the one-dimensional Walsh- Nörlund means. As special cases, we get the Nörlund logarithmic means of cubical partial sums of Walsh-Fourier series discussed recently by Gát and Goginava [5] in 2004 and the (C, β)-means of Marcinkiewicz type with respect to double Walsh-Fourier series discussed by Goginava [10]. Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series was given by Móricz and Siddiqi [14].     

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.