Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust ...Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.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
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.
We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized...We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized operator on harmonic function and investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.展开更多
Very recently, it was observed that the temperature of nanofluids is finally governed by second-order ordinary differential equations with variable coefficients of exponential orders. Such coefficients were then trans...Very recently, it was observed that the temperature of nanofluids is finally governed by second-order ordinary differential equations with variable coefficients of exponential orders. Such coefficients were then transformed to polynomials type by using new independent variables. In this paper, a class of second-order ordinary differential equations with variable coefficients of polynomials type has been solved analytically. The analytical solution is expressed in terms of a hypergeometric function with generalized parameters. Moreover, applications of the present results have been applied on some selected nanofluids problems in the literature. The exact solutions in the literature were derived as special cases of our generalized analytical solution.展开更多
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
Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a di...Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.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
The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form , for...The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form , for?k > 0, β > 0, y > 0, 0 is the confluent k-hypergeometric functions, by using k-fractional integration.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
In this paper, we use the generalized hypergeometric series method the high-order inverse moments and high-order inverse factorial moments of the generalized geometric distribution, the Katz distribution, the Lagrangi...In this paper, we use the generalized hypergeometric series method the high-order inverse moments and high-order inverse factorial moments of the generalized geometric distribution, the Katz distribution, the Lagrangian Katz distribution, generalized Polya-Eggenberger distribution of the first kind and so on.展开更多
The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other...The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.展开更多
Hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this ar...Hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this article, we try to clarify several points and give a general overview of the topic, going from the univariate case to the matrix case, in one and then in several arguments. We also survey some results in fields close to Statistics, where hypergeometric functions are actively used, studied and developed.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 work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x)...In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function;the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.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
In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(&...In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.展开更多
In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslo...In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslov’s Theorem.We also establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices,and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well.The particular solutions of the adjoint equation are then obtained.As an appliction of these particular solutions,we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.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
文摘Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.
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 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We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized operator on harmonic function and investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.
文摘Very recently, it was observed that the temperature of nanofluids is finally governed by second-order ordinary differential equations with variable coefficients of exponential orders. Such coefficients were then transformed to polynomials type by using new independent variables. In this paper, a class of second-order ordinary differential equations with variable coefficients of polynomials type has been solved analytically. The analytical solution is expressed in terms of a hypergeometric function with generalized parameters. Moreover, applications of the present results have been applied on some selected nanofluids problems in the literature. The exact solutions in the literature were derived as special cases of our generalized analytical solution.
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Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss Hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.
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The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form , for?k > 0, β > 0, y > 0, 0 is the confluent k-hypergeometric functions, by using k-fractional integration.
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In this paper, we use the generalized hypergeometric series method the high-order inverse moments and high-order inverse factorial moments of the generalized geometric distribution, the Katz distribution, the Lagrangian Katz distribution, generalized Polya-Eggenberger distribution of the first kind and so on.
文摘The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.
文摘Hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this article, we try to clarify several points and give a general overview of the topic, going from the univariate case to the matrix case, in one and then in several arguments. We also survey some results in fields close to Statistics, where hypergeometric functions are actively used, studied and developed.
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supported by the projects UAM-A-CBI-2232004 and 009.JGR thanks to the Instituto Politécnico Nacional for the financial support given through the COFAA-IPN project SIP-200150019.
文摘In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function;the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.
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 the National Natural Science Foundation of China(Grant No.12001288)。
文摘In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.
基金Supported by the Fundamental Research Funds for the Central Universities of China(Grant No.20720150006)Natural Science Foundation of Fujian Province of China(Grant No.2016J01032)。
文摘In this article,we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation,its expression is different from Suslov’s Theorem.We also establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices,and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well.The particular solutions of the adjoint equation are then obtained.As an appliction of these particular solutions,we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.
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.
