In this paper,the expressions of two classes of infinite series in terms of finite series involving Bernoulli numbers are obtained.As applications,we derive some special series including Dirichlet beta functionβ(s)wi...In this paper,the expressions of two classes of infinite series in terms of finite series involving Bernoulli numbers are obtained.As applications,we derive some special series including Dirichlet beta functionβ(s)with argument 2n+1 and Dirichlet lambda functionλ(s)with argument 2n.In addition,we solve the problem proposed recently by Zhou(2021).展开更多
Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying...Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.展开更多
Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursi...Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1<sup>st</sup> order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.展开更多
We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient p...We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.展开更多
In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and dedu...In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.展开更多
Zhao (2003a) first established a congruence for any odd prime p〉3, S(1,1,1 ;p)=-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β, γ,ρ) (modp) is...Zhao (2003a) first established a congruence for any odd prime p〉3, S(1,1,1 ;p)=-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β, γ,ρ) (modp) is considered for all positive integers α,β, γ. We refer to w=α+β+ γ as the weight of the sum, and show that if w is even, S(α,β, γ,ρ)=0 (mod p) for p≥w+3; if w is odd, S(α,β, γ,ρ)=-rBp-w (mod p) for p≥w, here r is an explicit rational number independent ofp. A congruence of Catalan number is obtained as a special case.展开更多
Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizatio...Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.展开更多
This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts throu...This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts through built-in properties of(number-theoretic) special functions.We follow roughly the historical order of events from somewhat more modern point of view.§1 deals with Euler's fundamental ideas as expounded in [6] and ,from a more advanced standpoint.§2 gives some rudiments of Bernoulli numbers and polynomials as consequences of the partial fraction expansion.§3 states sieve-theoretic treatment of the Euler product.Thus,the events in §1-§3 more or less belong to Euler's era.§4 deals with RSA cryptography as motivated by Euler's function,with its several descriptions being given.§5 contains a slight generalization of Dirichlet's test on uniform convergence of series,which is more effectively used in §6 to elucidate Riemann's posthumous Fragment II than in [1].Thus §5-§6 belong to the Dirichlet-Riemann era.§7 gives the most general modular relation which is the culmination of the Riemann-Hecke-Bochner correspondence between modular forms and zeta-functions.Appendix gives a penetrating principle of the least period that appears in various contexts.展开更多
In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix method...In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.展开更多
In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithm...In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithmic derivative of the ratio,and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.展开更多
In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with ...In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with the prime recognitions.展开更多
The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, ...The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, and to give four interesting hybrid mean value formulae.展开更多
The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions between a quadratic residue and its inverse modulo p value formula. to study ...The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions between a quadratic residue and its inverse modulo p value formula. to study the asymptotic property of the difference (a prime), and to give an interesting hybrid mean展开更多
We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform...We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^(3) logN)operations.Numerical examples have confirmed the accuracy and robustness of the new scheme.展开更多
Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then...Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.展开更多
Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-a...Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-adic L-fuctions[12]. Herethe author shows how a more general sums (npl +j)-r,l N, may be studied by elementary methods.展开更多
Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain som...Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.展开更多
In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m(m ∈ N) and λ(λ 0).And the constant factor expressed by the Bernoulli number and π is pro...In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m(m ∈ N) and λ(λ 0).And the constant factor expressed by the Bernoulli number and π is proved to be the best possible.And then some important and especial results are enumerated.As applications,some equivalent forms are given.展开更多
文摘In this paper,the expressions of two classes of infinite series in terms of finite series involving Bernoulli numbers are obtained.As applications,we derive some special series including Dirichlet beta functionβ(s)with argument 2n+1 and Dirichlet lambda functionλ(s)with argument 2n.In addition,we solve the problem proposed recently by Zhou(2021).
基金Supported by the National Natural Science Foundation of China(51276017)
文摘Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.
文摘Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1<sup>st</sup> order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.
文摘We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.
基金Supported by the PCSIRT of Education of China(IRT0621)Supported by the Innovation Program of Shanghai Municipal Education Committee of China(08ZZ24)Supported by the Henan Innovation Project for University Prominent Research Talents of China(2007KYCX0021)
文摘In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.
基金Project (No. 10371107) supported by the National Natural Science Foundation of China
文摘Zhao (2003a) first established a congruence for any odd prime p〉3, S(1,1,1 ;p)=-2Bp-3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β, γ,ρ) (modp) is considered for all positive integers α,β, γ. We refer to w=α+β+ γ as the weight of the sum, and show that if w is even, S(α,β, γ,ρ)=0 (mod p) for p≥w+3; if w is odd, S(α,β, γ,ρ)=-rBp-w (mod p) for p≥w, here r is an explicit rational number independent ofp. A congruence of Catalan number is obtained as a special case.
文摘Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.
文摘This paper contains material presented by the first authors in CIMPA School at Kathmandu University.,July 26,27,28,2010,to be included in ,and is intended for a rambling introduction to number-theoretic concepts through built-in properties of(number-theoretic) special functions.We follow roughly the historical order of events from somewhat more modern point of view.§1 deals with Euler's fundamental ideas as expounded in [6] and ,from a more advanced standpoint.§2 gives some rudiments of Bernoulli numbers and polynomials as consequences of the partial fraction expansion.§3 states sieve-theoretic treatment of the Euler product.Thus,the events in §1-§3 more or less belong to Euler's era.§4 deals with RSA cryptography as motivated by Euler's function,with its several descriptions being given.§5 contains a slight generalization of Dirichlet's test on uniform convergence of series,which is more effectively used in §6 to elucidate Riemann's posthumous Fragment II than in [1].Thus §5-§6 belong to the Dirichlet-Riemann era.§7 gives the most general modular relation which is the culmination of the Riemann-Hecke-Bochner correspondence between modular forms and zeta-functions.Appendix gives a penetrating principle of the least period that appears in various contexts.
文摘In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.
基金partially supported by the National Nature Science Foundation of China(12061033)。
文摘In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithmic derivative of the ratio,and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.
基金Supported partly by NNSFCA Presidential Faculty Fellowsupported in part by the NSF of the United States
文摘In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with the prime recognitions.
文摘The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, and to give four interesting hybrid mean value formulae.
基金This work is supported by the N.S.F.(10271093)P.N.S.F.of P.R.China
文摘The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions between a quadratic residue and its inverse modulo p value formula. to study the asymptotic property of the difference (a prime), and to give an interesting hybrid mean
基金The research of Liu was supported by the NSF grant DMS 10-11738The research of Wang was supported by National Science Council of Taiwan under grant 97-2115-M-007-005In addition,this work is also supported in part by National Center for Theoretical Sciences of Taiwan.
文摘We present a fast Poisson solver on spherical shells.With a special change of variable,the radial part of the Laplacian transforms to a constant coefficient differential operator.As a result,the Fast Fourier Transform can be applied to solve the Poisson equation with O(N^(3) logN)operations.Numerical examples have confirmed the accuracy and robustness of the new scheme.
文摘Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.
文摘Recently Hong Shaofang[6] has investigated the sums (np + j)-r ( with an odd prime number p 5 and n, r N) by Washington’s p-adic expansion of these sums as a power series in n where the coefficients are values of p-adic L-fuctions[12]. Herethe author shows how a more general sums (npl +j)-r,l N, may be studied by elementary methods.
基金Supported by the Natural Science Foundation of Gansu Province (Grant No. 3ZS041-A25-007)
文摘Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.
基金Supported by the Project of Scientific Research Fund of Hunan Provincial Education Department (GrantNo.09C789)
文摘In this paper it is shown that a new Hilbert-type integral inequality can be established by introducing two parameters m(m ∈ N) and λ(λ 0).And the constant factor expressed by the Bernoulli number and π is proved to be the best possible.And then some important and especial results are enumerated.As applications,some equivalent forms are given.