On Solutions of the q-Hypergeometric Equation with q N = 1

We consider the q-hypergeometric equation with q ^N = 1 and , , . We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0 < |q| < 1 and at |q| = 1.     

Multi-variable Poincaré series of algebraic curve singularities over finite fields

Let be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.      

Algebraic Independence Results Related to 〈q, r〉-Number Systems

We define 〈q, r〉-linear arithmetical functions attached to the 〈q, r〉-number systems and give a necessary and sufficient condition for their generating power series to be algebraically independent over . We also deduce algebraic independence of the functions values at a nonzero algebraic number in the circle of convergence.     

Scalar Products of Symmetric Functions and Matrix Integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions, and integrals defining matrix-model partition functions. Using the fermionic Fock space representation, we prove an expansion of an associated class of KP and 2-Toda tau functions _r,n in a series of Schur functions generalizing the hypergeometric series and relate it to the scalar product formulas. We show how special cases of such tau functions can be identified as formal series for partition functions. A closed form expansion of log _r,n in terms of Schur functions is derived.     

О множествах расходи мости рядов по некото рым ортогональным систе мам функций

The main results concern the structure of divergence sets of power series on the unit circle. A class of setsF ¹,F ¹F , is characterized which are not divergence sets for power series on the unit circle. It is also emphasized that, the divergence sets of power series can be of sufficiently complicated structure. Divergence sets of power series on the unit circle are studied, whose partial sums satisfy lim sup ¦s _n(x)¦< outside a set of first category. Analogous results hold for trigonometric series on [0,2] and also for series with respect to Vilenkin sets on corresponding zero measure compact Abelian groups. Nonsummability sets for Abel's method on the unit circle are also studied for power and trigonometric series.     

Two results on maximum nonlinear functions

Maximum nonlinear functions are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions , gcd(k, m) = 1, and the Kasami power functions , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.      

Transcendence of Binomial and Lucas' Formal Power Series

The formal power series[formula]is transcendental over (X) whentis an integer ≥ 2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn = ∑n_ipⁱis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over _p(X), the field of rational functions over the Galois field _p, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over (X).     

Bounded analytic structure of the banach space of formal power series

Let be a sequence of positive numbers and 1≤p<∞. We consider the spaceH ^p(β) of all power series such that Σ| (n)|^{p β}(n ^p<∞. We investigate regions on which our formal power series represent bounded analytic functions. Research partially supported by the Shiraz University Research Council Grant No. 79-SC-1311-675.     

Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics

Weber integrals and Beltrami integrals are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions with Gaussian or exponentially cut power‐law densities. Finite series representations of the integrals are derived for integer power‐law index μ, which admit high‐precision evaluation at low and moderate Bessel index n. At high n, numerically tractable uniform asymptotic approximations are obtained on the basis of the Debye expansion of modified spherical Bessel functions in the case of Weber integrals. The high‐n approximation of Beltrami integrals can be reduced to Legendre asymptotics. The Airy approximation of Weber and Beltrami integrals is derived as well, and numerical tests are performed over a wide range of Bessel indices by comparing the exact finite series expansions of the integrals with their high‐index asymptotics. Copyright © 2013 John Wiley & Sons, Ltd.     

Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Ax^r+B)/(Cx^r+D), where r is a power of p and the coefficients belong to . These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.     

Continued fractions for square series generating functions

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled “Square Series Generating Function Transformations” (arXiv:1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of \(q^{n^2}\) for some fixed non-zero q with \(|q| < 1\), we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the hth convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists as \(h \rightarrow \infty \). We also prove new infinite q-series representations of special square series expansions involving square-power terms of the series parameter q, the q-Pochhammer symbol, and double sums over the q-binomial coefficients. Applications of the new results we prove within the article include new q-series representations for the ordinary generating functions of the special sequences, \(r_p(n)\), and \(\sigma _1(n)\), as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.     

On Some Integrals Involving the Hurwitz Zeta Function: Part 2

We establish a series of indefinite integral formulae involving the Hurwitz zeta function and other elementary and special functions related to it, such as the Bernoulli polynomials, ln sin(q), ln (q) and the polygamma functions. Many of the results are most conveniently formulated in terms of a family of functions A _k(q) := k(1 – k, q), k , and a family of polygamma functions of negative order, whose properties we study in some detail.     

A generalization of the inversion formulas of systems of power series in systems of implicit functions

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk in a neighborhood of the point (0, 0)C _(w,x) ^m+n , in terms of the coefficients of the power series of the functions F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.     

Convergence of Series of Fourier Coefficients of p-Absolutely Continuous Functions

In the paper, we generalize the well-known criteria of Bernstein and Stechkin on the absolute convergence in terms of best approximations and moduli of smoothness of continuous functions. We give conditions for the convergence of the series of Fourier coefficients raised to the power in terms of best approximations in the space of p-absolutely continuous functions and in terms of fractional moduli of continuity with respect to this space. We also prove the sharpness of our conditions for 0 < 1 with no restriction and for 1 < 2 under some restrictions.     

On an Approximation of Functions Regular in Convex Polygons by Linear Means of Their Exponent Series

We study linear summation methods for exponent series which are expansions of functions f(z) that are regular on an open convex polygon M and continuous on . We find estimates for deviations of the arithmetic mean and the Vallée-Poussin and Rogozinskii means.     

The asymptotic behaviour of certain entire functions of order zero

Summary A certain class of entire functionsF(s) of order zero which are asymptotically equal to the sum of just two neighbouring terms of their power series when |s| with |args| < – for any fixed > 0, is investigated. Which two terms one has to take, depends upons. It is shown that these functions have infinitely many negative zeros, and the asymptotic behaviour of the zeros is also determined.     

Representation of real discrete Fourier transform in terms of a new set of functions based upon Möbius inversion

The trigonometric functions sin(2n/N) and cos(2n/N) are transformed into a new set of basis functions using Möbius inversion of certain types of series. The new basis functions are number theoretic series. They are used to represent the real discrete Fourier transform (RDFT) in terms of 2 matrices of factorization. The first matrix, with elements 1, -1 and 0 is obtained by replacing cos(2k/N) and sin(2k/N) by (k/N + 1/4) and (k/N), where (x) is the bipolar rectangular wave function. The second matrix is block-diagonal where each block is a circular correlation and consists of the new basis functions. Some applications of the new representation are discussed.     

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let _t be the heat kernel on G; that is, _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on . Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on in the limit of small t, where the Hermite functions are viewed as functions on via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of . For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.     

Self-Affine Fractal Functions and Wavelet Series

We consider functions represented by series ∑_g  G c_gψ(g ^− 1(x)) of wavelet-type, where G is a group generated by affine functions L₁,…,L_n and ψ is piecewise affine. By means of those functions we characterize the class of self-affine fractal functions, previously studied by Barnsley et al. We compute their global and local Hölder exponents and investigate points of non-differentiability. Wavelet-representations for various continuous nowhere differentiable and singular functions are presented. Another application is the construction of functions with prescribed local Hölder exponents at each point.