scholarly journals Voronoi summation formulae on GL(n)

2016 ◽  
Vol 162 ◽  
pp. 483-495 ◽  
Author(s):  
Fan Zhou
Keyword(s):  
Summation Formulae ◽  
Voronoi Summation

Voronoi summation formula for Gaussian integers

2021 ◽  
Author(s):  
Debika Banerjee ◽  
Ehud Moshe Baruch ◽  
Daniel Bump
Keyword(s):  
Summation Formula ◽  
Gaussian Integers ◽  
Voronoi Summation

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

scholarly journals Approach of q-Derivative Operators to Terminating q-Series Formulae

2018 ◽  
Vol 26 (2) ◽  
pp. 99-111
Author(s):  
Xiaoyuan Wang ◽  
Wenchang Chu
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Operator Approach ◽  
Derivative Operator ◽  
Summation Formulae

AbstractThe q-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

scholarly journals Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

2020 ◽  
pp. 114-126
Author(s):  
T.G. Ergashev ◽  
A. Hasanov
Keyword(s):  
Elliptic Equation ◽  
Fundamental Solution ◽  
Hypergeometric Function ◽  
Explicit Form ◽  
Green's Function ◽  
Explicit Solution ◽  
Hypergeometric Functions ◽  
Summation Formulae ◽  
Singular Coefficients ◽  
Abc Method

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem. В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

scholarly journals Weil type representations and automorphic forms

1980 ◽  
Vol 77 ◽  
pp. 145-166 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Theta Series ◽  
Type Representation ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Reciprocity Law ◽  
Weil Type ◽  
Generalized Theta Series ◽  
Poisson Summation

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

The Application of the Chi-Square Test of Goodness-of-Fit to Mortality Data Graduated by Summation Formulae.

1971 ◽  
Vol 97 (2-3) ◽  
pp. 325-330 ◽  
Author(s):  
J. H. Pollard
Keyword(s):  
Degrees Of Freedom ◽  
Goodness Of Fit ◽  
Summation Formula ◽  
Mortality Data ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Summation Formulae ◽  
Chi Square Test

In his paper of 1941, Seal included details of some experiments he performed in an attempt to estimate the appropriate number of degrees of freedom for the chi-square goodness-of-fit test of a summation formula graduation. These results are referred to by Tetley and by Benjamin and Haycocks in their textbooks when they mention the difficulty of determining the number of degrees of freedom or mean chi-square value.

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