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We prove the following conjecture of Z.-W. Sun ['On congruences related to central binomial coefficients', J. Number Theory 13 (11) (2011), 2219–2238]. Let p be an odd prime. Then $$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$ where $H_n$ is the n th harmonic number and $B_n$ is the n th Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p -adic integers $a, b$. [ABSTRACT FROM AUTHOR]

A Wolstenholme prime is a prime number p ≥ 5 that divides the numerator of the Bernoulli number Bp−3. A number of equivalent definitions for Wolstenholme primes are known, mainly related to congruences of harmonic sums or binomial coefficients. In this paper, we introduce an equivalent definition of Wolstelholme primes related to the number of terms in the group determinant of cyclic groups, and equivalently, the cardinality of certain sets of restricted partitions. [ABSTRACT FROM AUTHOR]

Summary: We provide a simple recursive method for generating the finite sum of whole number powers of integers. Given that we know only the formula for the sum with power p – 1, this method generates the formula for the sum with power p by a simple polynomial integration and a solution to a linear equation in one variable. This method makes no use of Bernoulli's equation or Bernoulli numbers in either its implementation or justification. [ABSTRACT FROM AUTHOR]

This investigation deals with two new subclasses of analytic and bi-univalent functions defined by Bernoulli polynomial. In this paper, coefficient estimation and Fekete-Szego problems are solved for these newly defined function subclasses. In addition, certain remarks are indicated for the subclasses of bi-starlike and bi-convex functions. [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]

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]

Motivated by the work of [K. Girstmair, A "popular" class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of 1 / p in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of ℚ (− p) and digits of 1 / p , Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the ℓ -adic expansion of 1 / n whenever n is a product of two distinct primes or a prime power. More explicitly, if ℓ > 1 is an integer such that gcd (ℓ , n) = 1 , and suppose that 1 / n = ∑ k = 1 ∞ x k ℓ k is the ℓ -adic expansion of 1 / n , then we establish the average of the digits of the ℓ -adic expansion of 1 / n in terms of (ℓ − 1) / 2 and the "trace" of generalized Bernoulli numbers B 1 , χ , where χ 's are odd Dirichlet characters modulo n. As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7). [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

In this paper we define, using the Stirling numbers of the second kind, the Gompertz wavelets and show their basic properties. In particular, we prove that the admissibility condition holds for them. We also compute the normalizing factors in the space of the square-integrable functions L2(ℝ) and present an explicit formula for them in terms of the Bernoulli numbers. Then, after implementing the second-order Gompertz wavelets in the Matlab Wavelet Toolbox, we perform numerical experiments demonstrating the practical applicability and effectiveness of the wavelets. [ABSTRACT FROM AUTHOR]

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev's integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard's numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. [ABSTRACT FROM AUTHOR]

Suppose that M is a closed isotropic Riemannian manifold and that R1,...,Rm generate the isometry group of M. Let f1,...,fm ​ be smooth perturbations of these isometries. We show that the fi are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475-505 (2007)] from Sn to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work [ABSTRACT FROM AUTHOR]

The zeta function is like a milestone in analytic number theory. Considering Euler's definition of the zeta function in this article, we will find some recursion at even integral points. We will also discuss the Dirichlet eta function, even Bernoulli numbers, and the gamma–zeta relation. Toward the end, we will define something known as the odd zeta function and find a recurrence for the same. [ABSTRACT FROM AUTHOR]

The near orthgonality of certain k-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average 1 k s (r + 1) ∑ j = 1 k s j r c k (s) (j) remains positive for all r ≥ 1 . Further, we give a lower bound for max N ∑ j = 1 N s c k (s) (j) . [ABSTRACT FROM AUTHOR]

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol–Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler's identity, we give relations among this characteristic function, the Apostol–Bernoulli polynomials and numbers, and also trigonometric functions including cosine and sine. A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol–Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas. [ABSTRACT FROM AUTHOR]

We compute the full asymptotic expansion of the heat kernel Tr (exp (- t D 2)) where D is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The coefficients of the expansion are explicit expressions involving Bernoulli and Euler numbers. We relate the divergent terms with the heat kernel expansion of the Dirac square root of the prolate wave operator investigated in our joint work with Henri Moscovici. [ABSTRACT FROM AUTHOR]

We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. [ABSTRACT FROM AUTHOR]

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series. [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]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then \begin{align*} &A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4), \end{align*} where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the n th Apéry number, and Bn is the n th Bernoulli number. [ABSTRACT FROM AUTHOR]

In this work, by means of the generating function method and the De Moivre's formula, we derive some interesting combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers. One of them confirms the formula proposed recently by Svinin (2022). [ABSTRACT FROM AUTHOR]

This article describes a novel approach for the calculation of ionization potentials (IPs), or, more generally, electron-detachment energies, based on a unitary coupled-cluster (UCC) parameterization of the ground-state wave function. Explicit working equations for a scheme referred to as IP-UCC3 are given, providing electron-detachment energies and spectroscopic amplitudes of electron-detached states dominated by one-hole excitations correct through third order. In the derivation, an expansion of the UCC transformed Hamiltonian involving Bernoulli numbers as expansion coefficients is employed. Both the secular matrix and the effective transition moments are shown to be essentially equivalent to the strict third-order algebraic-diagrammatic construction scheme for the electron propagator (IP-ADC). Interestingly, due to the Bernoulli expansion, neglecting triple substitutions in the UCC expansion manifold does not affect the third-order consistency of the IP-UCC effective transition moments. Finally, the equivalence between ADC and UCC excited-state schemes is shown to not hold in fourth or higher order due to a different treatment of the correlated excited-state basis. [ABSTRACT FROM AUTHOR]

In this study, we use discrete Appell convolution to define the sequence of Laguerre type twice-iterated Appell polynomials. We obtain explicit representation, recurrence relation, determinantal representation, lowering operator, integro-partial raising operator and integro-partial differential equation. In addition, the special cases of this new family are investigated using Euler and Bernoulli numbers. We also state their corresponding characteristic properties. [ABSTRACT FROM AUTHOR]

The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n \gt 0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int _0^1 b_n(x) dx = 0$. We offer a related characterization: $b_1(x) = x - 1/2$ and $({-}1)^{n-1} b_n(x)$ for $n \gt 0$ is the $n$ -fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta $(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$. This result has a novel combinatorial analog, the Bernoulli clock : mark the hours of a $2 n$ hour clock by a uniformly random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$ , meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$ , then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$ , and so on. Starting from hour $0 = 2n$ , move clockwise to the first hour marked $1$ , continue clockwise to the first hour marked $2$ , and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$. We show that for each positive integer $n$ , the event $(I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$ , where $n! b_n(0) = B_n(0)$ is the $n$ th Bernoulli number. For $ 1 \le k \le 2 n$ , the difference $\delta _n(k)\,:\!=\, 1/(2n) -{\mathbb{P}}(I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $\delta _n(2 n + 1 - k) = ({-}1)^n \delta _n(k)$ , which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials $b_n(1-x) = ({-}1)^n b_n(x)$. [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]

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0). [ABSTRACT FROM AUTHOR]

We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. [ABSTRACT FROM AUTHOR]

This paper presents various numerical simulations to improve the calculation of nuclear reactivity while considering the effect of feedback temperature. The Euler-Maclaurin series approximation with the second Bernoulli number is used to discretise the inverse equation of point kinetics. Additionally, the Doppler effect is taken into account, considering its dependency on the feedback temperature, which is recursively calculated from an initial value. Several numerical experiments are conducted with different time steps, simulation times, and neutron population density distributions to validate the proposed method. [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. [ABSTRACT FROM AUTHOR]

In this note, we prove that x n 1 - e - x (n) > 0 for all x ∈ (log 2 , ∞) and n ∈ N . This result improves a theorem of Al-Musallam and Bustoz (Ramanujan J 11:399–402, 2006). [ABSTRACT FROM AUTHOR]

In this paper, we establish some congruences mod $ p^3 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l} $, where $ p > 3 $ is a prime number and $ B_{p, k} $ are generalized Catalan numbers. We also establish some congruences mod $ p^2 $ involving the sums $ \sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2} $, where $ m, l_1, l_2, d $ are positive integers and $ 1\leq d\leq p-1 $.

Recently, Kim-Kim introduced the truncated degenerate Bell polynomials and numbers. In this paper, weintroduce the truncated Lah-Bell polynomials and numbers. We obtain some identities, recurrence relations and properties. Furthermore, we also introduce the truncated modified Lah-Bell polynomials and numbers, truncated degenerate Lah-Bell polynomials and numbers and we derive some properties and identities involving with those polynomials and numbers. [ABSTRACT FROM AUTHOR]

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable. [ABSTRACT FROM AUTHOR]

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function G (z ; τ). Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as z → ∞. Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms C (τ) and D (τ). [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]

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind. [ABSTRACT FROM AUTHOR]

Let ℓ be any fixed prime number. We define the ℓ -Genocchi numbers by G n : = ℓ (1 − ℓ n) B n , with B n the n -th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ -Genocchi irregular if it divides at least one of the ℓ -Genocchi numbers G 2 , G 4 , ... , G p − 3 , and ℓ -regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ -Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ = 2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1 − ℓ n) B 2 n / 2 n and (1 + ℓ n) B 2 n / 2 n. This allows us to estimate the number of primes p ≤ x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ. [ABSTRACT FROM AUTHOR]

In this paper, based on Serre's p-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series G k in the form ∑ i = 1 n g i (p) G f i (p) ≡ g 0 (p) (m o d p N) , where f 1 (t) , ... , f n (t) ∈ Z [ t ] are non-constant integer polynomials with positive leading coefficients and g 0 (t) , ... , g n (t) ∈ Q (t) are rational functions. This generalizes the classical von Staudt–Clausen and Kummer congruences of Eisenstein series, and also yields some new congruences. [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]

The main result of this paper presents a series expansion of the generalized Mathieu series Sm(r) with computable coefficients. This generalizes Elbert's work from m = 1 to arbitrary m ∈ Z>0, and provides a method for calculating the asymptotic formula of Sm(r) within a given error. Moreover, this paper revises an integral representation for Sm(r) given by Cerone and Lenard. [ABSTRACT FROM AUTHOR]

The restricted partition function p N (n) counts the number of partitions of n into at most N parts. In the nineteenth century, Sylvester showed that these partition functions can be expressed as a sum of k-periodic quasi-polynomials ( 1 ≤ k ≤ N ) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called q-partial fractions) of the generating function of p N (n) . In this paper, we show that the coefficients of these q-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the Degenerate Bernoulli numbers, the Degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian–Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of p N (n) but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the I-adic completion of the ring of polynomials, where I is an ideal generated by the cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli's numbers of the second kind. Let φ B S (ξ) = ξ e ξ − 1 2 = ∑ n = 0 ∞ ξ n B n 2 n ! , where the coefficients of B n 2 are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions such that ξ ′ (ξ) (ξ) ≺ φ B S (ξ). We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class. [ABSTRACT FROM AUTHOR]

We present computational algorithms and complexity estimates for power series in which all exponents are positive integers raised to one and the same integer power . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums. [ABSTRACT FROM AUTHOR]