Mathematics & Statistics
http://hdl.handle.net/10222/22277
2019-06-27T12:09:05ZNonlinear Identities for Bernoulli and Euler Polynomials
http://hdl.handle.net/10222/75064
Nonlinear Identities for Bernoulli and Euler Polynomials
Dilcher, Karl
It is shown that a certain nonlinear expression for Bernoulli polynomials, related to higher-order convolutions, can be evaluated as a product of simple linear polynomials with integer coefficients. The proof involves higher-order Bernoulli polynomials. A similar result for Euler polynomials is also obtained, and identities for Bernoulli and Euler numbers follow as special cases.
2018-01-01T00:00:00ZDerivatives and Special Values of Higher-Order Tornheim Zeta Functions
http://hdl.handle.net/10222/75063
Derivatives and Special Values of Higher-Order Tornheim Zeta Functions
Dilcher, Karl; Tomkins, Hayley
We study analytic properties of the higher-order Tornheim zeta function, defined by a certain $n$-fold series ($n\geq 2$) in $n+1$ complex variables. In particular, we consider the function $\omega_{n+1}(s)$, obtained by setting all variables equal to $s$. Using a free-parameter method due to Crandall, we first give an alternative proof of the trivial zeros of $\omega_{n+1}(s)$ and evaluate $\omega_{n+1}(0)$. Our main result, however, is the evaluation of $\omega_{n+1}'(0)$ for any $n\geq 2$. This is again achieved by using Crandall's method, and it generalizes recent results in the cases $n=2, 3$. Properties of Bernoulli numbers and of higher-order Bernoulli numbers and polynomials play an important role throughout this paper.
2018-01-01T00:00:00ZA role for generalized Fermat numbers
http://hdl.handle.net/10222/71449
A role for generalized Fermat numbers
Cosgrave, John B.; Dilcher, Karl
We define a Gauss factorial $N_n!$ to be the product of all positive integers
up to $N$ that are relatively prime to $n\in\mathbb N$. In this paper we study
particular aspects of the Gauss factorials $\lfloor\frac{n-1}{M}\rfloor_n!$
for $M=3$ and 6, where the case of $n$ having exactly one prime factor
of the form $p\equiv 1\pmod{6}$ is of particular interest. A fundamental
role is played by those primes $p\equiv 1\pmod{3}$ with the property that the
order of $\frac{p-1}{3}!$ modulo $p$ is a power of 2 or 3 times a power of 2;
we call them Jacobi primes. Our main results are characterizations of those
$n\equiv\pm 1\pmod{M}$ of the above form that satisfy
$\lfloor\frac{n-1}{M}\rfloor_n!\equiv 1\pmod{n}$, $M=3$ or 6, in terms
of Jacobi primes and certain prime factors of generalized Fermat numbers. We
also describe the substantial and varied computations used for this paper.
Post-print version of the article, issued prior to publication.
2016-01-01T00:00:00ZA pilot study to quantify parental anxiety associated with enrollment of an infant or toddler in a phase III vaccine trial.
http://hdl.handle.net/10222/62391
A pilot study to quantify parental anxiety associated with enrollment of an infant or toddler in a phase III vaccine trial.
Langley, J. M.; Halperin, S. A.; Smith, B.
2003-01-01T00:00:00Z