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In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums R n (p , t) = ∑ m = 0 n m p H m t for t = 2 , 3 , 4 , where H m are harmonic numbers. When t ≤ 4 these sums were first studied by Spieß around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums R n (p , t) for all t ≥ 0 . [ABSTRACT FROM AUTHOR]

In this paper we investigate a class of integrals that were encountered in the course of a work on statistical plasma physics, in the so-called Sommerfeld temperature-expansion of the electronic entropy. We show that such integrals, involving some parameters, can be fully described in closed form represented by special functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate one in the family of four parametric linear Euler sums as given by Alzer and Choi and later extended by Sofo and Choi. We give explicit expressions for a variety of special cases involving a free parameter. Our analysis incorporates the properties of the polylogarithm function, commonly referred to as the Fermi–Dirac integral. A reciprocity property is utilized to highlight an explicit representation of a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature. [ABSTRACT FROM AUTHOR]

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

In the study, the authors introduce Qi's normalized remainder of the Maclaurin power series expansion of the function ln sec x = − ln cos x ; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function (2 x − 1) ζ (x) on (1 , ∞) , they prove the logarithmic convexity of Qi's normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi's normalized remainders. [ABSTRACT FROM AUTHOR]

We present several formulas involving the classical Bernoulli numbers and polynomials. Among others, we extend an identity for Bernoulli polynomials published by Wu et al. (Fibonacci Quart 42:295-299, 2004). [ABSTRACT FROM AUTHOR]

By examining the connection coefficients, we systematically review and extend (with an extra integer parameter) several double sum expressions for the Bernoulli numbers. New summation formulae are also established explicitly. [ABSTRACT FROM AUTHOR]

By examining the connection coefficients, we systematically review and extend (with an extra integer parameter) several double sum expressions for the Bernoulli numbers. New summation formulae are also established explicitly.

In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.

In this study we investigate a family of skew linear harmonic Euler sums involving some free parameters. Our analysis involves using the properties of the polylogarithm function, commonly referred to as the Bose-Einstein integral. A reciprocity property is utilized to highlight an explicit representation for a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature and introduces some new results.

In the paper, the authors intrinsically observe that Hoffman’s combinatorial identity discovered in 1992 and Genčev’s combinatorial identities extended in 2024 can be reformulated in terms of complete Bell polynomials. They provide alternative proofs and offer some elementary generalizations of these combinatorial identities. [ABSTRACT FROM AUTHOR]

In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function. [ABSTRACT FROM AUTHOR]

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers.

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2) a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ?= 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1,p2 (x1, x2). By using results for S(1,x2,p2) 1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses lacunary recurrence relations with gaps of length 8 for the Bernoulli and Euler polynomials. The authors use standard notations for these polynomials and numbers and provide formulas and expansions for them. They also mention the history of lacunary recurrence relations for these numbers and polynomials and highlight their own contributions in finding such relations. The authors describe the method of proof for their main result, Theorem 1, and provide corollaries that extend the relations to the Bernoulli and Euler numbers. The article concludes by acknowledging funding support and declaring no conflicts of interest. [Extracted from the article]

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polynomials and the Bernoulli numbers, present two identities among the differences between the Bernoulli polynomials and the Bernoulli numbers in terms of a determinant and a partial Bell polynomial, and derive a determinantal formula of the differences between the Bernoulli polynomials and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial E m p (m p) (x) is in Eisenstein form with respect to p if and only if p does not divide m (2 m − 1) B m . As a consequence, we deduce that at least 1 / 3 of the generalized Euler polynomials E n (n) (x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q. [ABSTRACT FROM AUTHOR]

We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing lower bounds on their number which in turn point out a rich array of fruitful connections to seemingly unrelated topics such as the Titchmarsh divisor problem, Mersenne primes, Fermat quotients, partitions into cycles, quadratic reciprocity, Gauss and Jacobi sums, and density results in class field theory resulting from Chebotarev's theorem. Moreover, representations of an element taken from a prime field, in terms of the sums in a covering sumset, furnish us with new formulas for the class numbers of quadratic fields, Bernoulli numbers and Bernoulli polynomials. In this way, curious tendencies among the number of representations are discovered over half intervals. Lastly, our findings show in different circumstances that the summands of a covering sumset can seldom form an arithmetic progression, thereby indicating a tension between additive and multiplicative structures in a prime field. [ABSTRACT FROM AUTHOR]

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. [ABSTRACT FROM AUTHOR]

In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in the expansion $ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $, where $ a \neq 0, b $ are complex numbers, have studied in [16]. In this paper, we show that Bernoulli polynomials $ B_{p}(x) $ can be written in terms of the numbers $ S_{1,x}\left(p,k\right) $, and then use the known results for $ S_{1,x}\left(p,k\right) $ to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials. [ABSTRACT FROM AUTHOR]

In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

Abstract By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper.

In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime p ∑ i , j , k ≥ 1 gcd (i j k , p) = 1 i + j + k = p 1 i j k ≡ − 2 B p − 3 (mod p) , where B n is the nth Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime p in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer n with an odd prime factor p such that p r | | n where r ≥ 1 ∑ i , j , k ≥ 1 gcd (i j k , n) = 1 i + j + k = n 1 i j k ≡ − 2 B p − 3 ⋅ n p ⋅ ∏ prime q | n q ≠ p 1 − 2 q 1 − 1 q 3 (mod p r). In this paper, we establish the following generalization of their conjecture: for any positive integer n with an odd prime factor p such that p r | | n where r ≥ 1 ∑ i , j , k ≥ 1 gcd (i j k , n) = 1 a 1 i + a 2 j + a 3 k = A n 1 i j k ≡ − 2 B p − 3 ⋅ n p ⋅ A g 3 3 1 a 1 2 g 1 2 + 1 a 2 2 g 2 2 + 1 a 3 2 g 3 2 × ∏ prime q | n q ≠ p 1 − 2 q 1 − 1 q 3 (mod p r) , where a 1 , a 2 and a 3 are positive integers coprime to p , and A is a positive common multiple of a 1 , a 2 and a 3 . Also, g 1 = gcd (a 2 , a 3) , g 2 = gcd (a 3 , a 1) , g 3 = gcd (a 1 , a 2) and g = gcd (a 1 , a 2 , a 3). [ABSTRACT FROM AUTHOR]

In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors. [ABSTRACT FROM AUTHOR]

Abstract By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

The main purpose of this article is to derive several convolutions for generalized Bernoulli and Euler–Genocchi polynomials of order (r , m) , B n (r , m) (x) and A n (r , m) (x) , respectively. These polynomials have been introduced recently and contain the generalized Bernoulli, Euler and Genocchi polynomials as special members. Some of our results extend the results of M. Merca and others concerning Bernoulli numbers and polynomials. Probabilistic interpretations of the presented results are also given. [ABSTRACT FROM AUTHOR]

Using the monotone form of the L'Hôspital rule, we discuss the (absolute) monotonicity of the functions $U\left(x\right)=\frac{1}{x^{4}}-% \frac{1}{x^{5}}\frac{3\sin x}{\cos x+2}$, $G(x)=\frac{1}{x^{2}}\left[\frac{% \ln\sin x-\ln x}{\ln\left(2+\cos x\right) -\ln 3}-1\right]$ and $J(x)=\frac{% 1-(\sin x)/x}{1-(2+\cos x)/3}$ to improve the Cusa-Huygens inequality in several directions on wider ranges. Our results are much better than those existing ones.

In the article, we establish three new Wilker type inequalities involving tangent and sine functions by use of a double inequality for the ratio of two consecutive non-zero Bernoulli numbers.

In this work, a new kernel function defined in the whole plane, including both the homogeneous and the non-homogeneous cases, and involving multiple parameters is constructed. By the method of weight coefficient and using some techniques of real analysis, a new Hilberttype inequality with the newly constructed kernel function, as well as its equivalent Hardy-type inequality are established. The constant factors of the obtained inequalities are proved to be the best possible. Furthermore, assuming special values to the parameters, some interesting and special Hilbert-type inequalities with the constant factors involving some special constants, such as the Euler number, Bernoulli number and the Catalan constant are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

By the introduction of a new half-discrete kernel which is composed of several exponent functions, and using the method of weight coefficient, a Hilbert-type inequality and its equivalent forms involving multiple parameters are established. In addition, it is proved that the constant factors of the newly obtained inequalities are the best possible. Furthermore, by the use of the rational fraction expansion of the tangent function and introducing the Bernoulli numbers, some interesting and special half-discrete Hilbert-type inequalities are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

Let p be a prime and let x be a p-adic integer. We prove two supercongruences for truncated series of the form $$\begin{align*}\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}\end{align*}$$ which generalise previous results. We also establish q-analogues of two binomial identities. [ABSTRACT FROM AUTHOR]

In this article, we study about the value of Riemann Zeta Function for even numbers using Bernoulli number. First, we give some basic theory about Bernoulli number and Riemann Zeta function. The method that used in this research was literature study. From our analysis, we have a theorem to evaluate the value of Riemann Zeta function for the even numbers with its proving.

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality. [ABSTRACT FROM AUTHOR]

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius–Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds [ABSTRACT FROM AUTHOR]

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form. [ABSTRACT FROM AUTHOR]

Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s(n, k), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p-adic valuations of s(n, k). In this paper, by using Washington's congruence on the generalized harmonic number and the n-th Bernoulli number Bn and the properties of m-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound on vp(s(ap, k)) witha and k being integers such that 1 ≤ a ≤ p - 1 and 1 ≤ k ≤ a p . This infers that for any regular prime p ≥ 7 and for arbitrary integers a and k with 5 ≤ a ≤ p - 1 and a - 2 ≤ k ≤ a p - 1 , one has v p (H (a p - 1 , k)) < - log (a p - 1) 2 log p with H (a p - 1 , k) being the k-th elementary symmetric function of 1 , 1 2 , ... , 1 a p - 1 . This gives a partial support to a conjecture of Leonetti and Sanna. We also present results on v p (s (a p n , a p n - k)) from which one can derive that under certain condition, for any prime p ≥ 5 , any odd number k ≥ 3 and any sufficiently large integer n, if (a , p) = 1 , then v p (s (a p n + 1 , a p n + 1 - k)) = v p (s (a p n , a p n - k)) + 2 . It confirms partially Lengyel's conjecture. [ABSTRACT FROM AUTHOR]

We say a polytope is Ehrhart positive if all the coefficients in its Ehrhart polynomial are positive. Answering an Ehrhart positivity question posed on MathOverflow, Stanley provided an example of a non-Ehrhart-positive order polytope of dimension 21. Stanley's example comes from a certain family of order polytopes. In this paper, we study the Ehrhart positivity question on this family of polytopes. By giving explicit formulas for the coefficients of the Ehrhart polynomials of these polytopes in terms of Bernolli numbers, we determine the sign of each Ehrhart coefficient of each polytope in the family. As a consequence of our result, we conclude that for any positive integer d ≥ 21 , there exists an order polytope of dimension d that is not Ehrhart positive, and for any positive integer ℓ , there exists an order polytope whose Ehrhart polynomial has precisely ℓ negative coefficients, which answers a question posed by Hibi. We finish this article by discussing the existence of lower-dimensional order polytopes whose Ehrhart polynomials have a negative coefficient. [ABSTRACT FROM AUTHOR]