By recent work of the author, Wilson's theorem as well as the Wilson quotient can be described by supercongruences of power sums of Fermat quotients modulo every higher prime power. We translate these congruences into congruences of power sums and Bernoulli numbers. This together provides relatively short proofs of the congruences compared to former approaches. As an application, we compute, e.g., the Wilson quotient up to p^4 and equivalently the factorial (p-1)! up to p^5, which can be extended to any higher prime power with some effort. As a by-product, we determine some power sums of the Fermat quotients up to p^4.

We show that Wilson's theorem as well as the Wilson quotient can be described by supercongruences modulo any higher prime power involving terms of power sums of Fermat quotients. The new approach uses Bell polynomials and Newton's identities relating elementary symmetric polynomials to power sums. This enables us to compute certain multivariate polynomials recursively that are needed to establish the supercongruences. Subsequently, we give a recurrence formula for these polynomials and show further properties.

It is well known that the Bernoulli polynomials $\mathbf{B}_n(x)$ have nonintegral coefficients for $n \geq 1$. However, ten cases are known so far in which the derivative $\mathbf{B}'_n(x)$ has only integral coefficients. One may assume that the number of those derivatives is finite. We can link this conjecture to a recent conjecture about the properties of a product of primes satisfying certain $p$-adic conditions. Using a related result of Bordell\`es, Luca, Moree, and Shparlinski, we then show that the number of those derivatives is indeed finite. Furthermore, we derive other characterizations of the primary conjecture. Subsequently, we extend the results to higher derivatives of the Bernoulli polynomials. This provides a product formula for these denominators, and we show similar finiteness results.

The main purpose of this paper is to study generalized (self-) reciprocal Appell polynomials, which play a certain role in connection with Faulhaber-type polynomials. More precisely, we show for any Appell sequence when satisfying a reflection relation that the Appell polynomials can be described by Faulhaber-type polynomials, which arise from a quadratic variable substitution. Furthermore, the coefficients of the latter polynomials are given by values of derivatives of generalized reciprocal Appell polynomials. Subsequently, we show some applications to the Bernoulli and Euler polynomials. In the context of power sums the results transfer to the classical Faulhaber polynomials.

We consider certain generalized binomial sums $\mathcal{S}_{(r,n)}(\ell)$ and discuss the nonintegrality of their values for integral parameters $n,r \geq 1$ and $\ell \in \mathbb{Z}$ in several cases using $p$-adic methods. In particular, we show some properties of the denominator of $\mathcal{S}_{(r,n)}(\ell)$. Viewed as polynomials, the sequence $(\mathcal{S}_{(r,n)}(x))_{n \geq 0}$ forms an Appell sequence. The special case $\mathcal{S}_{(r,n)}(2)$ reduces to the sum $\sum_{k=0}^{n} \binom{n}{k} \frac{r}{r+k}$, which has recently received some attention from several authors regarding the conjectured nonintegrality of its values. So far, only a few cases have been proved. The generalized results imply, among other things, for even $|\ell| \geq 2$ that $\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$ when $\binom{r+n}{r}$ is even, e.g., $r$ and $n$ are odd. Although there exist exceptions where $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$, ``almost all'' values of $\mathcal{S}_{(r,n)}(\ell)$ for $n,r \geq 1$ are nonintegral for any fixed $|\ell| \geq 2$. Subsequently, we also derive explicit inequalities between the parameters for which $\mathcal{S}_{(r,n)}(\ell) \notin \mathbb{Z}$. Especially, this is shown for certain small values of $\ell$ for $r \geq n$ and $n > r \geq \frac{1}{5} n$. As a supplement, we finally discuss exceptional cases where $\mathcal{S}_{(r,n)}(\ell) \in \mathbb{Z}$.

In this note, we consider asymptotic products of binomial and multinomial coefficients and determine their asymptotic constants and formulas. Among them, special cases are the central binomial coefficients, the related Catalan numbers, and binomial coefficients in a row of Pascal's triangle. For the latter case, we show that it can also be derived from a limiting case of products of binomial coefficients over the rows. The asymptotic constants are expressed by known constants, for example, the Glaisher–Kinkelin constant. In addition, the constants lie in certain intervals that we determine precisely. Subsequently, we revisit a related result of Hirschhorn and clarify the given numerical constant by showing the exact expression.

We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers $\mathcal{B}_{r,s}$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers do not vanish in general. As a main result among other properties, we show the non-vanishing of these numbers, apart from exceptional cases. We further derive an explicit product formula for their denominators, which follows from a von Staudt--Clausen type relation.

We give a new characterization of the set $\mathcal{C}$ of Carmichael numbers in the context of $p$-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-$p$-digits function. More precisely, we show that such a denominator obeys a triple-product identity, where one factor is connected with a $p$-adically defined subset $\mathcal{S}$ of the squarefree integers that contains $\mathcal{C}$. This leads to the definition of a new subset $\mathcal{C}'$ of $\mathcal{C}$, called the primary Carmichael numbers. Subsequently, we establish that every Carmichael number equals an explicitly determined polygonal number. Finally, the set $\mathcal{S}$ is covered by modular subsets $\mathcal{S}_d$ ($d \geq 1$) that are related to the Knodel numbers, where $\mathcal{C} = \mathcal{S}_1$ is a special case.

The primary Carmichael numbers were recently introduced as a special subset of the Carmichael numbers. A primary Carmichael number m has the unique property that s_p(m) = p holds for each prime factor p, where s_p(m) is the sum of the base-p digits of m. The first such number is Ramanujan's famous taxicab number 1729. Due to Chernick, all Carmichael numbers with three factors can be constructed by certain squarefree polynomials U_3(t) ∈ℤ[t], the simplest one being U_3(t) = (6t+1)(12t+1)(18t+1). We show that the values of any U_3(t) obey a special decomposition for all t ≥ 2 and besides certain exceptions also in the case t=1. These cases further imply that if all three factors of U_3(t) are simultaneously odd primes, then U_3(t) is not only a Carmichael number, but also a primary Carmichael number. Together with the exceptional cases, all Carmichael numbers with three factors have at least the property that s_p(m) = p holds for the greatest prime factor p of m. Subsequently, we show some connections to taxicab and polygonal numbers, involving the number 1729 as an example again.

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit criterion for integrality of the coefficients of these polynomials. As applications, we obtain new results on the sequence of denominators of the Bernoulli polynomials. A consequence is that certain quotients of successive denominators are infinitely often integers, which we characterize.

Let {·} denote the fractional part and n ≥ 1 be a fixed integer. In this short note, we show for any prime p the one-to-one correspondence ∑_ν≥ 1{n/p^ν} > 1 p |denom( B_n(x) - B_n ), where B_n(x) - B_n is the nth Bernoulli polynomial without constant term and denom(·) is its denominator, which is squarefree.

The power sum 1(n) + 2(n) + ... + x(n) has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in x of degree n + 1 with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n + 1 is at least p. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

We study the properties of the product, which runs over the primes,P-n = Pi(sp(n)>= p) p (n >= 1), sp 01) > Pwhere s(p)(n) denotes the sum of the base-p digits of n. One important property is the fact that p(n) equals the denominator of the Bernoulli polynomial B-n(x) - B-n, where we provide a short p-adic proof. Moreover, we consider the decomposition P-n = P-n(-) . P-n(+), where p(n)(+) contains only those primes p > root n. Let omega(.) denote the number of prime divisors. We show that omega(p(n)(+)) < root n, while we raise the explicit conjecture thatomega(P-n(+)) similar to kappa (log n)/(root n) as n -> infinitywith a certain constant kappa > 1, supported by several computations. (C) 2017 Elsevier Inc. All rights reserved.

We consider two types of polynomials $F_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) x^\nu$ and $\hat{F}_n (x) = \sum_{\nu=1}^n \nu! S_2(n,\nu) H_\nu x^\nu$, where $S_2(n,\nu)$ are the Stirling numbers of the second kind and $H_\nu$ are the harmonic numbers. We show some properties and relations between these polynomials. Especially, the identity $\hat{F}_n (-\tfrac{1}{2}) = - (n-1)/2 \cdot F_{n-1} (-\tfrac{1}{2})$ is established for even $n$, where the values are connected with Genocchi numbers. For odd $n$ the value of $\hat{F}_n (-\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we discuss some of these convolutions, which are connected with Miki type convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic valuations of them.

We consider iterations of integer-valued functions ϕ, which have no fixed points in the domain of positive integers. We define a local function ϕ_n, which is a sub-function of ϕ being restricted to the subdomain {0, ..., n }. The iterations of ϕ_n can be described by a certain n × n sparse matrix M_n and its powers. The determinant of the related n × n matrix M̂_n = I - M_n, where I is the identity matrix, acts as an indicator, whether the iterations of the local function ϕ_n enter a cycle or not. If ϕ_n has no cycle, then M̂_n = 1 and the structure of the inverse M̂_n^-1 can be characterized. Subsequently, we give applications to compute the inverse M̂_n^-1 for some special functions. At the end, we discuss the results in connection with the 3x+1 and related problems.

We show for even positive integers n that the quotient of the Riemann zeta values zeta(n + 1) and zeta(n) satisfies the equationzeta(n + 1)/zeta(n) = (1 - 1/n) (1 - 1/2(n+1) - 1) L-star(p(n))/p(n)'(0),where p(n) is an element of Z[x] is a certain monic polynomial of degree n and L-star : C[x] -> C is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition p(n)(x) = x(x + 1)q(n)(x). If n = p + 1 where p is all odd prime, then q(n) is an Eisenstein polynomial and therefore irreducible over Z[x] (C) 2013 Elsevier Inc. All rights reserved.

The Erdos-Moser conjecture states that the Diophantine equation S-k(m) = m(k), where S-k(m) = 1(k) + 2(k) + ... +(m-1)(k), has no solution for positive integers k and in with k >= 2. We show that stronger conjectures about consecutive values of the function S-k, that seem to be more naturally, imply the Erdos-Moser conjecture. (C) 2011 Elsevier Inc. All rights reserved.

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., ∏_ν=1^n |B_2ν| and ∏_ν=1^n (k ν)!^ν^r as n →∞ for integers k ≥ 1 and r ≥ 0. Our main interest is to determine exact expressions, in terms of known constants, for the asymptotic constants of these expansions and to show some relations among them.

WediscusstheasymptoticexpansionsofcertainproductsofBernoullinumbersandfac- torials,e.g., n ! !=1 |B 2! |and n ! !=1 (k!)! ! r asn!" forintegersk#1andr#0.Ourmaininterestistodetermineexactexpressions,in termsofknownconstants,fortheasymptoticconstantsoftheseexpansionsandtoshow somerelationsamongthem.