Mathematics & Statistics http://hdl.handle.net/10222/22277 2022-10-07T15:39:04Z 2022-10-07T15:39:04Z Nonlinear Identities for Bernoulli and Euler Polynomials Dilcher, Karl http://hdl.handle.net/10222/75064 2019-01-03T08:30:16Z 2018-01-01T00:00:00Z 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. 2018-01-01T00:00:00Z Derivatives and Special Values of Higher-Order Tornheim Zeta Functions Dilcher, Karl Tomkins, Hayley http://hdl.handle.net/10222/75063 2019-01-03T08:30:10Z 2018-01-01T00:00:00Z 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. 2018-01-01T00:00:00Z A role for generalized Fermat numbers Cosgrave, John B. Dilcher, Karl http://hdl.handle.net/10222/71449 2017-12-21T16:53:48Z 2016-01-01T00:00:00Z 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. 2016-01-01T00:00:00Z 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. http://hdl.handle.net/10222/62391 2017-12-21T16:53:37Z 2003-01-01T00:00:00Z 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