scholarly journals COUNTING POINTS ON DWORK HYPERSURFACES AND -ADIC HYPERGEOMETRIC FUNCTIONS

2016 ◽  
Vol 94 (2) ◽  
pp. 208-216 ◽  
Author(s):  
RUPAM BARMAN ◽  
HASANUR RAHMAN ◽  
NEELAM SAIKIA
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Counting Points

We express the number of points on the Dwork hypersurface $X_{\unicode[STIX]{x1D706}}^{d}:x_{1}^{d}+x_{2}^{d}+\cdots +x_{d}^{d}=d\unicode[STIX]{x1D706}x_{1}x_{2}\cdots x_{d}$ over a finite field of order $q\not \equiv 1\,(\text{mod}\,d)$ in terms of McCarthy’s $p$-adic hypergeometric function for any odd prime $d$.

scholarly journals A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

2015 ◽  
Vol 11 (08) ◽  
pp. 2431-2450 ◽  
Author(s):  
Dermot McCarthy ◽  
Matthew A. Papanikolas
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Modular Forms ◽  
Hypergeometric Functions ◽  
Siegel Modular Forms ◽  
Hecke Operator ◽  
Special Value ◽  
Forms Of Higher Degree

Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function.

scholarly journals Hypergeometric functions and relations to Dwork hypersurfaces

2017 ◽  
Vol 13 (02) ◽  
pp. 439-485 ◽  
Author(s):  
Heidi Goodson
Keyword(s):  
Finite Field ◽  
Hypergeometric Function ◽  
Finite Fields ◽  
Hypergeometric Functions ◽  
K3 Surfaces ◽  
Point Count ◽  
Higher Dimensional ◽  
The Family ◽  
Trace Of Frobenius ◽  
The Relationship

We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order [Formula: see text] in terms of Greene’s finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy’s [Formula: see text]-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

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.

New subclasses of analytic functions defined by convolution involving the hypergeometric function and the Owa–Srivastava operator

2019 ◽  
Vol 26 (3) ◽  
pp. 449-458
Author(s):  
Khalida Inayat Noor ◽  
Rashid Murtaza ◽  
Janusz Sokół
Keyword(s):  
Hypergeometric Function ◽  
Convolution Operator ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Convolution Properties ◽  
Appl Math ◽  
Inclusion Results

Abstract In the present paper we introduce a new convolution operator on the class of all normalized analytic functions in {|z|<1} , by using the hypergeometric function and the Owa–Srivastava operator {\Omega^{\alpha}} defined in [S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 1987, 5, 1057–1077]. This operator is a generalization of the operators defined in [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365] and [K. I. Noor, Integral operators defined by convolution with hypergeometric functions, Appl. Math. Comput. 182 2006, 2, 1872–1881]. Also we introduce some new subclasses of analytic functions using this operator and we discuss some interesting results, such as inclusion results and convolution properties. Our results generalize the results of [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365].

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650064 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Contiguous Relations

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

scholarly journals Integral representations of generalized Lauricella hypergeometric functions

1992 ◽  
Vol 15 (4) ◽  
pp. 653-657 ◽  
Author(s):  
Vu Kim Tuan ◽  
R. G. Buschman
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Appell Function

The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell function are obtained.

scholarly journals Fractional operators with generalized Mittag-Leffler k-function

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shahid Mubeen ◽  
Rana Safdar Ali
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Euler Integral ◽  
K Function ◽  
Generalized Wright Hypergeometric Function

AbstractIn this paper, our main aim is to deal with two integral transforms involving the Gauss hypergeometric functions as their kernels. We prove some composition formulas for such generalized fractional integrals with Mittag-Leffler k-function. The results are established in terms of the generalized Wright hypergeometric function. The Euler integral k-transformation for Mittag-Leffler k-functions has also been developed.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

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