Summation formulae | ScienceGate

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.

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Weil type representations and automorphic forms

Nagoya Mathematical Journal ◽

10.1017/s0027763000018730 ◽

1980 ◽

Vol 77 ◽

pp. 145-166 ◽

Cited By ~ 3

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).

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

Journal of the Institute of Actuaries ◽

10.1017/s0020268100016929 ◽

1971 ◽

Vol 97 (2-3) ◽

pp. 325-330 ◽

Cited By ~ 1

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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