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
The importance of basic hypergeometric series has been widely recognized. For non specialists, it is necessary to have a quick introduction to this classical but flourishing subject. An effort along this direction wil...The importance of basic hypergeometric series has been widely recognized. For non specialists, it is necessary to have a quick introduction to this classical but flourishing subject. An effort along this direction will be made in the present article.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
The purpose of this paper is to establish several transformation formulae for bivariate basic hypergeometric series by means of series rearrangement technique. From these transformations, some interesting summation fo...The purpose of this paper is to establish several transformation formulae for bivariate basic hypergeometric series by means of series rearrangement technique. From these transformations, some interesting summation formulae are obtained.展开更多
The Abel＇s lemma on summation by parts is employed to evaluate terminating hypergeometric series. Several summation formulae are reviewed and some new identities are established.
The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-ser...The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).展开更多
Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(...Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.展开更多
In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about a...In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about asymptotic formula and error estimation in terms of modules of continuity.展开更多
The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q...The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established.展开更多
A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hy...A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hypergeometric series, the limits of those formulas are also obtained.展开更多
The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studyi...The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential.We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor(1-x^2)^(-p-1) as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction.The calculations are obtained systematically using some properties of the generalized hypergeometric series.展开更多
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.
文摘The importance of basic hypergeometric series has been widely recognized. For non specialists, it is necessary to have a quick introduction to this classical but flourishing subject. An effort along this direction will be made in the present article.
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.
基金the National Natural Science Foundation of China (No.10771093) the Natural Science Foundation of the Education Department of Henan Province (No.2007110025)
文摘The purpose of this paper is to establish several transformation formulae for bivariate basic hypergeometric series by means of series rearrangement technique. From these transformations, some interesting summation formulae are obtained.
基金Supported by Shanghai Leading Academic Discipline Project (Grant No.J50101)
文摘The Abel＇s lemma on summation by parts is employed to evaluate terminating hypergeometric series. Several summation formulae are reviewed and some new identities are established.
文摘The unifiedΩ-series of the Gauss and Bailey2F1(1/2)-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts.Several remarkable transformation theorems for theΩ-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type,including a couple of beautiful expressions forπand the Catalan constant discovered by Guillera(2008).
文摘Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.
文摘In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about asymptotic formula and error estimation in terms of modules of continuity.
基金supported by National Natural Science Foundation for the Youth (Grant No. 10801026)
文摘The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established.
文摘A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hypergeometric series, the limits of those formulas are also obtained.
基金Supported by the National Natural Science Foundation of China under Grant No.11275165Partially by 20160978-SIP-IPN,Mexico
文摘The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential.We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor(1-x^2)^(-p-1) as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction.The calculations are obtained systematically using some properties of the generalized hypergeometric series.
