In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
Several(generalized)hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors.In the present paper,we investigate four new hypergeometric functio...Several(generalized)hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors.In the present paper,we investigate four new hypergeometric functions in four variables and then establish several recursion formulas for these new functions.Also,some interesting particular cases and consequences of our results are discussed.展开更多
The main object of this paper is to deduce the bibasic Humbert functions Ξ_(1) and Ξ_(2)Some interesting results and elementary summations technique that was successfully employed,q-recursion,q-derivatives relations...The main object of this paper is to deduce the bibasic Humbert functions Ξ_(1) and Ξ_(2)Some interesting results and elementary summations technique that was successfully employed,q-recursion,q-derivatives relations,the q-differential recursion relations,the q-integral representations for Ξ_(1) and Ξ_(2)are given.The summation formula derives a list of p-analogues of transformation formulas for bibasic Humbert functions that have been studied,also some hypergeometric functions properties of some new interesting special cases have been given.展开更多
This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampe...This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampe de Feriets series of double hypergeometric series F;.展开更多
We introduce and study two subclasses ？_（[α_1]）（A, B, λ） and ？_（[α_1]）~＋ （A, B, λ） of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function....We introduce and study two subclasses ？_（[α_1]）（A, B, λ） and ？_（[α_1]）~＋ （A, B, λ） of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class ？_（[α_1]）~＋（α, β, γ, λ）.展开更多
In this study,we aim to explore a novel class of twenty-five double integrals involving generalized hypergeometric functions.These integrals take the form:_(3)F_(2)[∫_(0)^(1)∫_(0)^(1)y^(c)(1-x)^(c-1)(1-y)^(c-1)(1-xy...In this study,we aim to explore a novel class of twenty-five double integrals involving generalized hypergeometric functions.These integrals take the form:_(3)F_(2)[∫_(0)^(1)∫_(0)^(1)y^(c)(1-x)^(c-1)(1-y)^(c-1)(1-xy)^(1-2c)1/2(a+b+i+1),2c+j;4y(1-x)(1-y)/(1-xy)^(2)]dxdy for i,j=0,±1,±2.The results are derived using generalized versions of Watson’s summation theorem,as established in earlier work by Lavoie et al.Additionally,fifty integrals,split into two sets of twenty-five,are presented as special cases of our main findings,offering further insights into the structure of these integrals.展开更多
we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 < c 5d/6 < ∞ andδ∈(-1/2,1/2),and find the la...we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 < c 5d/6 < ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) <F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.展开更多
The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, <img src="Edit_699140d3-f569-463e-b835-7ccdab822717.png" width="290" height="22" ...The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, <img src="Edit_699140d3-f569-463e-b835-7ccdab822717.png" width="290" height="22" alt="" /><img src="Edit_bdd10470-9b63-4b2d-9cec-636969547ca5.png" width="90" height="22" alt="" /><span style="white-space:normal;">and <img src="Edit_e9cd6876-e2b8-45cf-ba17-391f054679b4.png" width="90" height="21" alt="" /></span>where <span style="white-space:nowrap;"><em>α</em>,<span style="white-space:nowrap;"><em>η</em></span><em></em></span> and <span style="white-space:nowrap;"><em>β</em></span> are real or complex constants are evaluated in terms of the confluent hypergeometric function <sub>1</sub><em>F</em><sub>1</sub> and the hypergeometric function <sub>1</sub><em>F</em><sub>2</sub>. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions <sub style="white-space:normal;">1</sub><em style="white-space:normal;">F</em><sub style="white-space:normal;">1</sub> and <sub style="white-space:normal;">1</sub><em style="white-space:normal;">F</em><sub style="white-space:normal;">2</sub>. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (<span style="white-space:nowrap;"><em>x</em><sup>2</sup></span>) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.展开更多
In the paper the new subclasses■and■of the function class∑of bi-univalent functions involving the Hohlov operator are introduced and investigated.Then,the corresponding Fekete-Szeg functional inequalities as well a...In the paper the new subclasses■and■of the function class∑of bi-univalent functions involving the Hohlov operator are introduced and investigated.Then,the corresponding Fekete-Szeg functional inequalities as well as the bound estimates of the coefficients a2 and a3 are obtained.Furthermore,several consequences and connections to some of the earlier known results also are given.展开更多
Abstract. In this paper, we study the quotient of hypergeometric functions μα （r） in the theory of Ramanujan＇s generalized modular equation for α ∈（0, 1/2]. Several new inequalities are given for this and rela...Abstract. In this paper, we study the quotient of hypergeometric functions μα （r） in the theory of Ramanujan＇s generalized modular equation for α ∈（0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.展开更多
The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-fun...The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.展开更多
Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is g...Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.展开更多
The purpose of the present paper is to introduce a new subclass of p-valent meromorphic functions by using certain integral operator and to investigate various properties for this subclass.
By using the hypergeometric function defined by the Dziok-Srivastava operator, a new subclass of meromorphic function is introdued. We obtain Fekete-Szeg? inequalities for the meromorphic function f(z) for which α-(1...By using the hypergeometric function defined by the Dziok-Srivastava operator, a new subclass of meromorphic function is introdued. We obtain Fekete-Szeg? inequalities for the meromorphic function f(z) for which α-(1 + α{1 +z[_lI_mf(z)]′′/[_lI_mf(z)]′}/z[_lI_mf(z)]′/_lI_mf(z))■φ(z)(α ∈ C-{1/2, 1}).展开更多
The main aim of this article is to obtain certain Laurent type hypergeometric generating relations. Using a general double series identity, Laurent type generating functions(in terms of Kampede Feriet double hypergeom...The main aim of this article is to obtain certain Laurent type hypergeometric generating relations. Using a general double series identity, Laurent type generating functions(in terms of Kampede Feriet double hypergeometric function) are derived. Some known results obtained by the method of Lie groups and Lie algebras, are also modified here as special cases.展开更多
The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta functio...The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.展开更多
The aim of this research paper is to derive two extension formulas for Lauricella’s function of the second kind of several variables with the help of generalized Dixon’s theorem on the sum of the series obtain...The aim of this research paper is to derive two extension formulas for Lauricella’s function of the second kind of several variables with the help of generalized Dixon’s theorem on the sum of the series obtained by Lavoie et al. [1]. Some special cases of these formulas are also deduced.展开更多
We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential eq...We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.展开更多
The object of the present paper is to investigate some mapping properties of subordinations by certain multivalent functions in the open unit disk associated with fractional integral operator.Furthermore,some applicat...The object of the present paper is to investigate some mapping properties of subordinations by certain multivalent functions in the open unit disk associated with fractional integral operator.Furthermore,some applications to the integral operator are also pointed out.展开更多
Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds t...Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.展开更多
基金supported by the Natural Science Foundation of China(61673169,11401191,11371125)the Tianyuan Special Funds of the Natural Science Foundation of China(11626101)the Natural Science Foundation of the Department of Education of Zhejiang Province(201635325)
文摘In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
文摘Several(generalized)hypergeometric functions and a variety of their extensions have been presented and investigated in the literature by many authors.In the present paper,we investigate four new hypergeometric functions in four variables and then establish several recursion formulas for these new functions.Also,some interesting particular cases and consequences of our results are discussed.
基金Supported by the National Natural Science Foundation of China(11601266)the Natural Science Foundation of Fujian Province of China(2020J01783)。
文摘The main object of this paper is to deduce the bibasic Humbert functions Ξ_(1) and Ξ_(2)Some interesting results and elementary summations technique that was successfully employed,q-recursion,q-derivatives relations,the q-differential recursion relations,the q-integral representations for Ξ_(1) and Ξ_(2)are given.The summation formula derives a list of p-analogues of transformation formulas for bibasic Humbert functions that have been studied,also some hypergeometric functions properties of some new interesting special cases have been given.
文摘This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampe de Feriets series of double hypergeometric series F;.
文摘We introduce and study two subclasses ？_（[α_1]）（A, B, λ） and ？_（[α_1]）~＋ （A, B, λ） of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class ？_（[α_1]）~＋（α, β, γ, λ）.
基金Deanship of Scientific Research at Majmaah University for supporting this work under Project number(ICR-2023-622).
文摘In this study,we aim to explore a novel class of twenty-five double integrals involving generalized hypergeometric functions.These integrals take the form:_(3)F_(2)[∫_(0)^(1)∫_(0)^(1)y^(c)(1-x)^(c-1)(1-y)^(c-1)(1-xy)^(1-2c)1/2(a+b+i+1),2c+j;4y(1-x)(1-y)/(1-xy)^(2)]dxdy for i,j=0,±1,±2.The results are derived using generalized versions of Watson’s summation theorem,as established in earlier work by Lavoie et al.Additionally,fifty integrals,split into two sets of twenty-five,are presented as special cases of our main findings,offering further insights into the structure of these integrals.
基金supported by National Natural Science Foundation of China(Grant Nos.11371125,11171307 and 61374086)Natural Science Foundation of the Hunan Province(Grant No.14JJ2127)Natural Science Foundation of the Zhejiang Province(Grant No.LY13A010004)
文摘we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 < c 5d/6 < ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) <F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.
文摘The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, <img src="Edit_699140d3-f569-463e-b835-7ccdab822717.png" width="290" height="22" alt="" /><img src="Edit_bdd10470-9b63-4b2d-9cec-636969547ca5.png" width="90" height="22" alt="" /><span style="white-space:normal;">and <img src="Edit_e9cd6876-e2b8-45cf-ba17-391f054679b4.png" width="90" height="21" alt="" /></span>where <span style="white-space:nowrap;"><em>α</em>,<span style="white-space:nowrap;"><em>η</em></span><em></em></span> and <span style="white-space:nowrap;"><em>β</em></span> are real or complex constants are evaluated in terms of the confluent hypergeometric function <sub>1</sub><em>F</em><sub>1</sub> and the hypergeometric function <sub>1</sub><em>F</em><sub>2</sub>. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions <sub style="white-space:normal;">1</sub><em style="white-space:normal;">F</em><sub style="white-space:normal;">1</sub> and <sub style="white-space:normal;">1</sub><em style="white-space:normal;">F</em><sub style="white-space:normal;">2</sub>. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (<span style="white-space:nowrap;"><em>x</em><sup>2</sup></span>) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.
基金Supported by Science and Technology Research Project of Colleges and Universities in Ningxia(Grant No.NGY2017011)Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(Grant No.NJYT-18-44)+1 种基金the Natural Science Foundation of Inner Mongolia(Grant No.2018MS01026)the Natural Science Foundation of China(Grant Nos.11561055,11561001,11762016)
文摘In the paper the new subclasses■and■of the function class∑of bi-univalent functions involving the Hohlov operator are introduced and investigated.Then,the corresponding Fekete-Szeg functional inequalities as well as the bound estimates of the coefficients a2 and a3 are obtained.Furthermore,several consequences and connections to some of the earlier known results also are given.
基金Supported by the Zhejiang Provincial Natural Science Foundation of China(LQ17A010010)the National Natural Science Foundation of China(11171307,11671360)the Natural Science Foundation of the Department of Education of Zhejiang Province(Y201328799)
文摘Abstract. In this paper, we study the quotient of hypergeometric functions μα （r） in the theory of Ramanujan＇s generalized modular equation for α ∈（0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.
基金NBHM Department of Atomic Energy,Government of India,Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131
文摘The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.
文摘Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.
文摘The purpose of the present paper is to introduce a new subclass of p-valent meromorphic functions by using certain integral operator and to investigate various properties for this subclass.
基金The NSF(KJ2015A372)of Anhui Provincial Department of Education
文摘By using the hypergeometric function defined by the Dziok-Srivastava operator, a new subclass of meromorphic function is introdued. We obtain Fekete-Szeg? inequalities for the meromorphic function f(z) for which α-(1 + α{1 +z[_lI_mf(z)]′′/[_lI_mf(z)]′}/z[_lI_mf(z)]′/_lI_mf(z))■φ(z)(α ∈ C-{1/2, 1}).
基金Dr.D.S.Kothari Post Doctoral Fellowship(Award letter No.F.4-2/2006(BSR)/MA/17-18/0025)awarded to Dr.Mahvish Ali by the University Grants CommissionGovernment of India,New Delhi。
文摘The main aim of this article is to obtain certain Laurent type hypergeometric generating relations. Using a general double series identity, Laurent type generating functions(in terms of Kampede Feriet double hypergeometric function) are derived. Some known results obtained by the method of Lie groups and Lie algebras, are also modified here as special cases.
文摘The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.
文摘The aim of this research paper is to derive two extension formulas for Lauricella’s function of the second kind of several variables with the help of generalized Dixon’s theorem on the sum of the series obtained by Lavoie et al. [1]. Some special cases of these formulas are also deduced.
文摘We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.
基金supported by Daegu National University of Education Research grant in 2024.
文摘The object of the present paper is to investigate some mapping properties of subordinations by certain multivalent functions in the open unit disk associated with fractional integral operator.Furthermore,some applications to the integral operator are also pointed out.
文摘Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.
