With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harm...With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.展开更多
We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spira...We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment.展开更多
The higher harmonics of the free electron laser(FEL)with a two-wavenumber plane wiggler magnetic field,which may induce a lot of higher harmonics,are investigated.Using a two-wavenumber wiggler comprised of two separa...The higher harmonics of the free electron laser(FEL)with a two-wavenumber plane wiggler magnetic field,which may induce a lot of higher harmonics,are investigated.Using a two-wavenumber wiggler comprised of two separate periodicities,k_(ω)and k_(ω)+Δk,numerical simulations show that the higher harmonic may be amplified and the first harmonic may be suppressed in this kind of FEL with an optimized parameterΔk.The coupling coefficient and gain coefficient in low-gain regime of the harmonic are also obtained.展开更多
In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r&...In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r</sub> (α,β) with Changhee sequences, Daehee sequences, Degenerate Changhee-Genoocchi sequences, Two kinds of degenerate Stirling numbers. Using Riordan arrays, we explore interesting relations between these polynomials, Apostol Bernoulli sequences, Apostol Euler sequences, Apostol Genoocchi sequences.展开更多
Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coeff...Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.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;p) (mod p) is consider...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;p) (mod p) 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;p)≡0 (mod p) for p≥w+3; if w is odd, S(α,β,γ;p)≡rBp≥w (mod p) for p≥w, here r is an explicit rational number independent of p. 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
设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1...设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.展开更多
The factorization of harmonic maps from a simply-connected domain to the unitary group is studied, showing that the theory of isotropic harmonic maps is equivalent to that of 2-unitons. Furthermore, a positive answer ...The factorization of harmonic maps from a simply-connected domain to the unitary group is studied, showing that the theory of isotropic harmonic maps is equivalent to that of 2-unitons. Furthermore, a positive answer is given to the Uhlenbeck’s conjecture on the upper bound of minimal uniton numbers.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
文摘With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.
基金Project supported by the Fundamental Research Funds for the Central Universities,China (Grant No.2017KFYXJJ029)。
文摘We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment.
文摘The higher harmonics of the free electron laser(FEL)with a two-wavenumber plane wiggler magnetic field,which may induce a lot of higher harmonics,are investigated.Using a two-wavenumber wiggler comprised of two separate periodicities,k_(ω)and k_(ω)+Δk,numerical simulations show that the higher harmonic may be amplified and the first harmonic may be suppressed in this kind of FEL with an optimized parameterΔk.The coupling coefficient and gain coefficient in low-gain regime of the harmonic are also obtained.
文摘In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r</sub> (α,β) with Changhee sequences, Daehee sequences, Degenerate Changhee-Genoocchi sequences, Two kinds of degenerate Stirling numbers. Using Riordan arrays, we explore interesting relations between these polynomials, Apostol Bernoulli sequences, Apostol Euler sequences, Apostol Genoocchi sequences.
文摘Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.
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;p) (mod p) 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;p)≡0 (mod p) for p≥w+3; if w is odd, S(α,β,γ;p)≡rBp≥w (mod p) for p≥w, here r is an explicit rational number independent of p. 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.
基金Supported by the National Natural Science Foundation of China(11971222)the initial version was posted to arXiv in 2014 with the ID arXiv:1407.8465.
文摘设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.
基金supported by the National Natural Science Foundation of China Natural Science Foundation of Zhejiang Province.
文摘The factorization of harmonic maps from a simply-connected domain to the unitary group is studied, showing that the theory of isotropic harmonic maps is equivalent to that of 2-unitons. Furthermore, a positive answer is given to the Uhlenbeck’s conjecture on the upper bound of minimal uniton numbers.
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.
