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
In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of t...In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of the generating functions and summation transformations,we also present an explicit expression for the coefficient sequence of the expansion.Moreover,we establish two(general)lacunary harmonic number asymptotic expansions,which contain only even or odd power terms in the logarithmic term.展开更多
In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as...In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.展开更多
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 define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
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
The beam matching status between the two isochronous cyclotrons in the Heavy Ion Research Facility a the Lanzhou-Cooling Storage Ring(HIRFL-CSR)is described.Several methods which can be used to accomplish 100% matchin...The beam matching status between the two isochronous cyclotrons in the Heavy Ion Research Facility a the Lanzhou-Cooling Storage Ring(HIRFL-CSR)is described.Several methods which can be used to accomplish 100% matching are proposed.By comparing of them,the best method is determined.The advantage due to this method i discussed.展开更多
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.
基金the National Natural Science Foundation of China(Grant No.11501081).
文摘In this paper,by exponential complete Bell polynomials,we establish a(general)harmonic number asymptotic expansion,and give the corresponding recurrence of the coefficient sequence in the expansion.By the methods of the generating functions and summation transformations,we also present an explicit expression for the coefficient sequence of the expansion.Moreover,we establish two(general)lacunary harmonic number asymptotic expansions,which contain only even or odd power terms in the logarithmic term.
基金Supported by Zhoukou Normal University High-Level Talents Start-Up Funds Research Project(Grant No.ZKNUC2022007)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX240725).
文摘In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.
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.
文摘We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic 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.
文摘The beam matching status between the two isochronous cyclotrons in the Heavy Ion Research Facility a the Lanzhou-Cooling Storage Ring(HIRFL-CSR)is described.Several methods which can be used to accomplish 100% matching are proposed.By comparing of them,the best method is determined.The advantage due to this method i discussed.
